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MULTICULTURAL FUN for AGES 9 & up

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Claudia Zaslavsky

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Page 1: Math Games & Activities from around the world

Math, history, art, and world cultures come together in this delightful book for kids.

More than seventy math games, puzzles, and projects from all over the world encourage kids to hone their math skills as they use geometry to design game boards, probability to analyze the outcomes of games of chance, and logical thinking to devise strategies for the games.

Many of the games have been played for centuries, like Tic-tac-toe, played in ancient Egypt; Nine Men’s Morris, once played in England with living game pieces; and Mankala, the oldest and most popular game in the world. Kids will learn that math is everywhere, from the geometry reflected in buildings to the border patterns of Eskimo parkas. Activities include building a model pyramid, testing the golden ratio of the Parthenon, and working mazelike African network puzzles.

Claudia Zaslavsky is the author of many books for children and adults, including Africa Counts, Multicultural Math, and Fear of Math.

“Connects youngsters with ‘friends’ around the globe and through time in compelling math play.” ——Dr. Lorraine Whitman, Executive Director, Salvadori Center

Math G

ames & A

ctivities from Around the W

orld Zaslavsky

• MULTICULTURAL FUN for AGES 9 & up •

IPG

ISBN 978-1-55652-287-1

9 7 8 1 5 5 6 5 2 2 8 7 1

5 1 6 9 5

$16.95 (CAN $18.95)

Math Games Cover [Converted].pdf 1 12/7/10 10:12 AM

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Math Games & Activitiesfrom around the

World

Claudia Zaslavsky

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Zaslavsky, Claudia.Math games and activities from around the world / Claudia Zaslavsky

p. cm.Includes bibliographical references (p. 144 – 145).Summary: Presents games and other activities from different countries

and cultures that teach a variety of basic mathematical concepts.ISBN 1-55652-287-81. Mathematical recreations—Juvenile literature.[1. Mathematical recreations 2. Games.] I. Title.QA95.z37 1998793.7'4—dc21

Cover, interior design, and illustrations by Mel Kupfer

Photo credits: p. 20—D. W. Crowe; p. 102 courtesy of the Kenya Mission to the U. N.; pp. 21, 52, 133—Sam Zaslavsky.

Figure credits: Figs. 61a and b reprinted by permission of Claudia Zaslavsky: The Multicultural Math Classroom: Bringing in the World (Heinemann, A Division of Reed Elsevier, Inc., Portsmouth, NH, 1996). Figs. 43b and 57a, b, and c courtesy of J. Weston Walch, reprinted from Multicultural Mathematicsby Claudia Zaslavsky. Copyright 1987, 1993.

© 1998 by Claudia ZaslavskyAll rights reservedFirst EditionPublished by Chicago Review Press, Incorporated814 North Franklin StreetChicago, Illinois 60610ISBN 1-55652-287-8Printed in the United States of America5 4 3 2

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This book is dedicated to all the children of the world. May they have a bright future and enjoy peaceful games.

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ACKNOWLEDGMENTS

I want to thank the many educators who shared their expertise to make this bookpossible. In particular, Judith Hankes contributed her material about the dreamcatcher; Esther Ilutsik shared her knowledge of Yup’ik border patterns; MarciaAscher set me straight on the solution to the river-crossing puzzle involving the

jealous husbands; and Beverly Ferrucci shared Japanese paper-cutting and severalgames. I am grateful to the authors of the many books listed in the Bibliography

that were a source of information and inspiration for this collection. Cynthia Sherrywas a most concerned and involved editor; she played all the games and carried outthe activities, pointing out my mistakes and occasional lack of clarity. I take full

responsibility for any remaining errors.

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INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . viii

1: THREE-IN-A-ROW GAMES. . . . . . . . . . . . . . . . . . 2Shisima from Kenya . . . . . . . . . . . . . . . . . . . . 4Tapatan from the Philippines . . . . . . . . . . . . . 6Tsoro Yematatu from Zimbabwe . . . . . . . . . . . 8Picaría, Native American . . . . . . . . . . . . . . . 109 Men’s Morris from England . . . . . . . . . . . . 12Trique from Colombia . . . . . . . . . . . . . . . . . 15Nerenchi from Sri Lanka . . . . . . . . . . . . . . . 16Dara from Nigeria. . . . . . . . . . . . . . . . . . . . . 18

2: MANKALA: BOARD GAMES OF TRANSFER . . . . . . . 20Easy Oware from Ghana . . . . . . . . . . . . . . . 22The Real Oware Game from Ghana . . . . . . . 24Sungka from the Philippines . . . . . . . . . . . . 26Giuthi from Kenya . . . . . . . . . . . . . . . . . . . . 28

3: MORE BOARD GAMES. . . . . . . . . . . . . . . . . . . . . 30Pong Hau K’i from Korea & China . . . . . . . . 32Mu Torere from New Zealand . . . . . . . . . . . . 34Pentalpha from Crete . . . . . . . . . . . . . . . . . . 36Kaooa from India . . . . . . . . . . . . . . . . . . . . . 38Awithlaknannai, Native American. . . . . . . . . 39Butterfly from Mozambique . . . . . . . . . . . . . 41Yoté from West Africa . . . . . . . . . . . . . . . 42

4: GAMES OF CHANCE . . . . . . . . . . . . . . . . . . . . . . 44Lu-lu from the Hawaiian Islands . . . . . . . . . 46Native American Games . . . . . . . . . . . . . . . . 48The Game of Dish, Native American . . . . . . . 49Stick Game, Native American . . . . . . . . . . . . 51Igba-Ita from Nigeria . . . . . . . . . . . . . . . . . . 52Spin the Dreidel, Jewish . . . . . . . . . . . . . . . 54Toma-Todo from Mexico . . . . . . . . . . . . . . . . 56Trigrams & Good Luck from East Asia . . . . . 58The Hexagrams of I Ching from China . . . . . 59

5: PUZZLES WITH NUMBERS. . . . . . . . . . . . . . . . . . 60Secret Code, Part I, Ancient Hebrew & Greek . . . . . . . . . . . . . . 61

Secret Code, Part II, Ancient Hebrew & Greek . . . . . . . . . . . . . . 63

Magic Squares, Part I, from West Africa . . . . 64Magic Squares, Part II, from China . . . . . . . 66Magic Squares, Part III . . . . . . . . . . . . . . . . . 68Counting Your Ancestors . . . . . . . . . . . . . . . 70Rice Multiplies from Asia . . . . . . . . . . . . . . . 72Dividing the Camels from North Africa . . . . . 73The Ishango Bone from Congo . . . . . . . . . . . 75Postal Codes from the U.S.A. . . . . . . . . . . . . 76

Table of Contents

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6: PUZZLES WITHOUT NUMBERS . . . . . . . . . . . . . . . 78Crossing the River in the Sea Islands . . . . . . 79Crossing the River in Liberia . . . . . . . . . . . . 81Crossing the River with Jealous Husbandsfrom Kenya . . . . . . . . . . . . . . . . . . . . . . . . 82

Crossing the River in Colonial America. . . . . 83The Snake & the Swallow’s Nest from Angola . . . . . . . . . . . . . . . . . . . . . . . 84

The Chokwe Storytellers from Angola . . . . . . 85Decorations on the Walls from Angola . . . . . 87How the World Began from Angola . . . . . . . . 88Children’s Networks from Congo . . . . . . . . . 90

7: GEOMETRY ALL AROUND US . . . . . . . . . . . . . . . . 92The Olympic Games Symbol. . . . . . . . . . . . . 94The Yin-Yang Symbol from China . . . . . . . . . 95The Dream Catcher, Native American . . . . . . 96The Tipi, Native American . . . . . . . . . . . . . . 98Round Houses in Kenya . . . . . . . . . . . . . . . 100Cone-Cylinder Houses in Kenya . . . . . . . . . 101Tangram Polygons from China . . . . . . . . . . 103The Pyramids of Ancient Egypt. . . . . . . . . . 105The Parthenon in Greece . . . . . . . . . . . . . . 107Pueblo Buildings in the U.S.A. . . . . . . . . . . 108

8: DESIGNS & SYMMETRY . . . . . . . . . . . . . . . . . . 110Masks and Faces from the U.S.A.. . . . . . . . 112Native American Masks . . . . . . . . . . . . . . . 114Hopi Flat Baskets, Native American . . . . . . 116Pennsylvania Dutch Love Pattern from the U.S.A. . . . . . . . . . . . . . . . . . . . . 118

Mon-Kiri Cutouts from Japan. . . . . . . . . . . 120

9: REPEATING PATTERNS. . . . . . . . . . . . . . . . . . . 122Yup’ik Eskimo Border Patterns from Alaska. . . . . . . . . . . . . . . . . . . . . . . 123

The Covenant Belt, Native American. . . . . . 125African Patterns from Congo . . . . . . . . . . . 127Patchwork Quilts from the U.S.A.. . . . . . . . 130Adinkra Cloth from Ghana . . . . . . . . . . . . . 133Tessellations in Islamic Culture . . . . . . . . . 135Polygon Patterns, Islamic . . . . . . . . . . . . . . 136Be a Tessellation Artist, Islamic . . . . . . . . . 138

10: SELECTED ANSWERS . . . . . . . . . . . . . . . . . . . 140

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . 144

A WORD ABOUT UNICEF . . . . . . . . . . . . . . . . . . . 146

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viii

INTROD

UCTION

id you know that some ofthe games that kids playwere invented hundreds,

even thousands, of years ago?Today you can play computerversions of Tic-tac-toe and Oware,games that go back at least 3,300years to ancient Egypt.The games, puzzles, and

projects in this book come from allparts of the world—Africa, Asia,Europe, North America, and theisland nations of Hawaii, thePhilippines, and New Zealand(called Aotearoa by the Maoripeople who first lived there). Theseactivities will introduce you to thepeople who played the games, whosolved the puzzles, and whodesigned the art.

You will exercise your brain asyou solve puzzles like the Africanchildren’s network that aEuropean scientist said was“impossible.” You will follow thelead of Islamic artists who madebeautiful patterns using only acompass and a straightedge(unmarked ruler). You will designand decorate game boards andmake the game pieces you willneed to play some of the games.You will make models of thehomes that different peoplearound the world live in.In all of these activities you will

be using math. Many of thesemath ideas are probably differentfrom the math you learn inschool. If a puzzle or activitydoesn’t work out at first, just keeptrying. Read the hints andsuggestions carefully. You mightwant to discuss the problem witha friend, teacher, or familymember. Two heads are betterthan one!

Most of the activities are self-checking—you will know whetherthey are correct. See Chapter 10for the answers to some of theseactivities, but you probably won’tneed them. It’s much moresatisfying to work out the solutionyourself, even if it takes a while.In this book you will read about

two types of games for two ormore people—games of strategyand games of chance. Somegames require a game board andcertain types of playing pieces orcounters. Players must decide howthey are going to move their pieceson the board. These are calledgames of strategy. Another kind ofgame depends upon the way thepieces fall. The players have nocontrol over the outcome of thegame, and such games are calledgames of chance.

D

Introduction

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ix

INTRODUCTION

It is interesting to see how agame changes as it travels fromone place to another and ispassed along from ancient timesto the present. Three-in-a-rowgames and mankala games aregood examples. You will learnseveral versions of each of thesegames.The games of strategy require

several types of game boards.Although you can draw them onpaper, you will probably wantboards that last for a while. Usesections of cardboard or matboard, and draw the lines neatlywith the help of a ruler. It’s a goodidea to make a pattern on a sheetof scratch paper first. Be sure youmeasure carefully.

Many of these games call for twokinds of counters or markers.Kings and princes used to playwith beautiful pieces made of goldand ivory. Ordinary people usedpebbles, seeds, or bits of twig,peeled and unpeeled. You can usered and black checkers or twokinds of coins, beans, or buttons.Or you might like to make yourown special counters.Most of the games of strategy

are for two players or two teams.You can also play them by your-self. Pretend that you are twopeople, and play on both sides ofthe board. This is a good way tolearn a new game or to work outthe fine points of strategy, asthough you were solving a puzzle.Some people play games just to

win and get upset when they lose.Playing a game should be fun.When one player always wins, theother player must always lose andmay give up after a while. Helpingan opponent to improve his or herskills makes the game moreinteresting for both players.

In traditional games of strategyfor two players, one side wins andthe other side loses. Each playershould have an equal chance ofwinning. In some games the firstplayer to move is more likely towin. Players should take turnsgoing first in this type of game.Perhaps you can figure outchanges in the rules so that bothplayers are winners. Cooperationmay be more rewarding thancompetition.You may want to vary the

games. A slight change in therules, or in the shape of the gameboard, or in the number ofcounters may call for an entirelydifferent strategy. Just be surethat both players agree on thenew rules before the game starts.Most important—have fun!

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Three-in-a-Row Gamesll over the world children playsome form of a three-in-a-rowgame for two players. Tic-tac-

toe is one example of such a game.The object of the game is to be thefirst player to get three markers in arow on the game board. It seemsthat people were playing suchgames long before the time of yourgreat grandparents.More than one hundred years

ago, scientists examining therooftop of an ancient Egyptiantemple found several strangediagrams carved in the sandstoneslabs. They looked like this: Figure 0It turned out that every one of

these diagrams is used as a gameboard for a three-in-a-row gamesomewhere in the world! Did theancient Egyptians really play suchgames? How could the scientistsfind out?The temple was built 3,300 years

ago to memorialize the king,Pharaoh Seti I. It stands in thetown of Qurna. Royal tombs were

built on the west side of the NileRiver. This was where the settingsun entered the spirit world for thenight, according to Egyptian beliefs.Ancient Egyptians believed thatpeople would have a life after deathand would need all the things thatthey enjoyed while they were alive.So the tombs contain many itemsthat were important to them in life,like clothing, jewelry, tools, andeven their pets!The Egyptians painted the walls

of their tombs and temples with thescenes from the lives of their kingsand queens and other wealthypeople. Game boards and carvedgame pieces for Senet and othergames were buried with themummies of important Egyptians.That’s how we know about thegames that these people playedwhen they were alive. But no gameboards for three-in-a-row gameshave been found inside the tombs,and no pictures of people playingsuch games appear on temple walls.

A

Figure 0

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How did these diagrams come tobe on the roof of the Pharaoh’stemple? Probably the workmen whobuilt the temple played three-in-a-row games on the stone slabsduring their lunch break. Insteadof drawing a fresh game board inthe sand for each game, theycarved permanent diagrams instone. You may wonder whether

Egyptian children played thesegames. Perhaps fathers playedsuch games with their children athome. But they probably drewgame boards in the dirt outsidethe house and wiped them awaywhen the game was over, leavingno trace.From Egypt the games could

easily have spread all over theworld. Greek scholars traveled toEgypt for higher education, just aspeople nowadays go to college. TheRomans, who probably learned thegames from the Greeks, spreadthem when they conquered parts ofEurope, the Middle East, and NorthAfrica. By that time the Chinesehad already been playing three-in-a-row games for centuries. Gamediagrams carved on the tops ofstone walls and the steps ofimportant buildings can still be

found in many parts of the world.The first European picture of

children playing a three-in-a-rowgame appeared in Spain more thanseven hundred years ago in theBook of Games. In the picture twochildren sit on either side of a largeboard for a game called Alquerquede Tres. The Spanish name means“mill with three.” The game boardthey used is just like the board forTapatan (see page 6). Arabic-speaking Moors came to

Spain from North Africa in theeighth century. They taught theSpanish people how to play gameslike Chess and Alquerque. Laterthe Spanish king Alfonso the Wisehad this information written downin the Book of Games. Soon thesegames spread to other parts ofEurope and, later, to America.Now you will have a chance to

learn several three-in-a-row gamesthat children play in other parts ofthe world. As you will see, some ofthese games are more complicatedthan Tic-tac-toe. But the object ofthe game is always the same—to bethe first player to get three markersin a row.

MAKE A MILL

Three-in-a-row games arecalled “mill” in manyEuropean countries. InEngland they are oftenreferred to as “Morris,”with a number telling howmany counters each playeruses. The word Morris mayhave come from the wordMoor, the name of thepeople who brought thegame to Europe by way ofSpain.

Children in the United Statesplay a three-in-a-row gamecalled Tic-tac-toe. If youdon’t already know thegame, you might ask afriend or an older person toteach it to you. In Englandthey call the same gameNoughts and Crosses. Anought is a zero, or 0, anda cross is an X. The playerstake turns marking X or 0 inthe nine spaces of the gameboard.

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Kenya is a country in East Africa.Children in western Kenya play athree-in-a-row game calledShisima (Shi-SEE-Mah). The wordshisima means “body of water” inthe Tiriki language. They call thecounters imbalavali or water bugs.Water bugs move so rapidlythrough the water that it is hardto keep them in sight. That’s howquickly players of Shisima movetheir counters on the game board.Children in Kenya draw the

game board in the sand and playwith bottle caps, pebbles, orbuttons. You might also use coins.Just be sure that you can tell thedifference between your countersand the other player’s counters.

MATERIALS• Sheet of unlined paper, at least 8 inches (20cm) square

• Pencil with eraser• Compass, or about 10 inches (25cm)of string

• Ruler

• Scissors• Glue• Piece of cardboard, at least 9 inches(22.5cm) square

• Colored markers or crayons• 3 counters for each player, of 2different kinds (buttons, bottle caps,or coins)

DRAWING THE GAME BOARDThe game board has the shape ofan octagon (eight-sided polygon).1.Mark the center of the paper.Use a compass to draw a largecircle. If you don’t have acompass, attach a piece ofstring to a pencil. Hold thepencil upright near the edge ofthe paper. Extend the string tothe center and hold it downthere. Then draw the circle.Figure 1a

2.Draw a line, called the diameter,through the center of the circle.

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Figure 1b

Figure 1a

Shisima from KenyaTWO P L A Y E R S

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Figure 1d

Figure 1c

SHISIMA3.Draw another diameter, so thatthe two lines form a cross. Thesetwo lines are perpendicular toeach other.

4.Draw two more diameters, eachhalfway between the first two.

5.Connect the endpoints of thediameters with straight lines toform an octagon. Erase thecircle. Figure 1b

6.Draw the shisima, or body ofwater, in the center. Erase thelines in the center.

7.Glue your game board to thecardboard and decorate withcolored markers. You mightwant to draw a border aroundyour game board.

PLAYING THE GAMEPlace the counters on the board,as shown in the diagram. Figure 1cPlayers take turns moving their

counters one space along a line to the next empty point. They con-tinue to take turns moving onecounter at a time. A player maymove into the center, the shisima,at any time. Jumping over acounter is not allowed.Each player tries to make a row

with his or her three counters. Arow must go through the shisima.There are four different ways to

make a row. This diagram showsthree black counters in a row.Figure 1dThe first player to get all three

counters in a row is the winner. Ifthe same set of moves is repeatedthree times, the game ends in adraw—no winner or loser. It’s timeto start a new game. Take turnsbeing Player One.After a few games you may be

able to move your counters as fastas the imbalavali swim in thewater.

THINGS TO THINK ABOUTIs it a good idea to move into theshisima on your first move? Whyor why not?If each player has four counters,

can they still play the game? Tryit and see what happens.

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Tapatan (TAP-uh-tan) is a gamethat people play in the Philippines,a country of many islands off thesoutheast coast of the Asiancontinent. Some families keepbeautiful wooden game boards forTapatan. Other families have thediagrams marked on floors or ondoorsteps of their homes. They usespecial round counters for thisgame, three of light wood for oneplayer and three of dark wood forthe other.

MATERIALS• Sheet of unlined paper, at least 8 inches (20cm) square

• Pencil• Ruler• Colored markers or crayons• Scissors• Glue• Piece of cardboard, at least 9 inches(22.5cm) square

• 3 counters for each player, 3 light and3 dark (beans, buttons, or coins)

DRAWING THE GAME BOARD1.Draw a square that measuressix inches (15cm) on each side.

2.With your pencil, draw thediagonals.

3.Draw lines that connect themidpoints of the opposite sides.

4.Use a marker or crayon to markthe nine points where the linesmeet as shown in the diagram.

Figure 2a5.Glue the paper to the cardboardand decorate your game board.

PLAYING THE GAMEThis game is played on the ninepoints where the lines intersect.Players take turns going first.Player One places a light counteron any point. Then Player Twoplaces a dark counter on anyempty point. They take turns untilall the counters have been placedon the game board.Then Player One moves one of

her counters along a line to thenext empty point. Jumping over a6

Figure 2a

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Tapatan fromthe Philippines TWO

P L A Y E R S

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counter is not allowed. Player Twodoes the same with one of hiscounters. They continue this waytaking turns.Each player tries to make a row

of three counters of one color andblock the other player from doingthe same. A row can be made ineight different ways: three across,three down, and two along thediagonal. Figure 2bThe winner is the first player to

make a row. If neither player canget three in a row and the sameset of moves is repeated threetimes, the game ends in a draw—no winner or loser.

THINGS TO THINK ABOUTWhere should Player One place

the first counter in order to win?Can you play the game with

four counters for each player?How is Tapatan like Tic-tac-toe?

How is it different?Player One can place the first

counter on any one of the ninepoints on the board. Show thatthere are really only threedifferent ways to place the firstcounter: center, corner, and side.Figure 2c

CHANGING THE RULESChildren and grown-ups playgames similar to Tapatan in manyparts of the world, but the rulesmay be somewhat different. Hereare some other versions of thegame you might want to try:Marelle (France). Neither player

may make the first move in thecenter.Achi (Ghana and Nigeria). Each

player may have four countersinstead of three.Tant Fant (India). The game

opens with each player’s threecounters already in position, as inthis diagram. In Tant Fant, a rowmay not be made on the startinglines. There are just six differentways to make a row in thisversion. Figure 2d

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F igure 2c

Figure 2d

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Zimbabwe is a country insouthern Africa. It is named afterthe complex of buildings calledGreat Zimbabwe, or the “GreatStone House.” In these buildingsonce lived the rulers of a vastancient kingdom known for itsrich gold mines.Children in Zimbabwe play

Tsoro Yematatu (TSOH-roh Yeh-mah-TAH-too), the “stone gameplayed with three.” Today they aremost likely to use bottle caps ascounters, as soft drinks are justas popular in Africa as they are inthe United States.

MATERIALS• Sheet of unlined paper, at least 8 inches (20cm) square

• Pencil• Ruler• Colored markers or crayons• Scissors• Glue• Sheet of cardboard, slightly largerthan the paper

• 3 counters for each player, of 2 different kinds (coins, buttons, or bottle caps)

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Tsoro Yematatu TWO P L A Y E R S

Figure 3

fromZimbabwe

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DRAWING THE GAME BOARD1.The game board is in the shapeof an isosceles triangle (it hastwo equal sides). With a penciland ruler, draw a triangle onyour sheet of paper as shown inthe diagram. Figure 3

2.Draw an altitude that dividesthe triangle in half. Thenconnect the midpoints of theequal sides.

3.Go over your lines with amarker and mark the sevenpoints where the lines intersect.

4.Glue the paper to the piece ofcardboard. You may want todecorate the game board andkeep it to use again.

PLAYING THE GAMEPlayers take turns placing theircounters on the empty points ofthe board. After all the countershave been placed on the board,one empty point remains. Theneach player in turn moves one ofhis or her counters to the emptypoint on the board. Jumping overa counter is allowed.Each player tries to make a row

of three. There are five differentways to do this. The winner is thefirst to make a row of three. Thisgame can go on for a long timewithout a winner. In that case, theplayers should decide to call it adraw.

THINGS TO THINK ABOUTWhy can’t you play the game withfour counters for each player?

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The Pueblo Indians of New Mexicoplay three-in-a-row games similarto those found halfway around theworld. Did they make up thesegames themselves, or did theylearn them from other people?One clue is the name of the

game. Some of the Pueblo peoplecalled their games Pitarilla orPicaría (Pick-ah-REE-ah). Thesewords sound like the Spanishname for the game Pedreria,which means “little stone.” Mostlikely the Native Americans of theSouthwest learned the gamesfrom the Spanish.In the sixteenth century, the

Spanish conquistadores sailedfrom Spain to America searchingfor riches. They had heard thatsome towns in the Southwest werefilled with gold. They attacked thetowns but found no gold.The Spanish conquistadores

gave the name Pueblo to thepeople of this region. In Spanish,pueblo means both “people” and

“town.” The Spanish forced thePueblo Indians to work like slaves.In 1680 the Pueblos revolted butwere free from slavery for onlytwelve years. Imagine how muchthe Native Americans must havedisliked their Spanish conquerors,and yet they continued to play thegames they learned from them.Pueblo children scratch their

game boards on flat stones. Forcounters they use pebbles, driedcorn kernels, or bits of pottery.

MATERIALS• Sheet of unlined paper, at least 8 inches (20cm) square

• Pencil• Ruler• Colored markers or crayons• Scissors• Glue• Piece of cardboard, at least 9 inches(22.5cm) square

• 3 counters for each player, of 2 different kinds (pebbles, coins, or bottle caps)10

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Picaría TWO P L A Y E R S

Figure 4a

Figure 4b

Native American

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DRAWING THE GAME BOARD1.Draw a square that measuressix inches (15cm) on each side.

2.Using your pencil, connect themidpoints of the opposite sidesto form four small squares.

3.Then draw the diagonals of eachof the four smaller squares.Figure 4a

4.Go over the lines with a marker.Mark the nine points on whichthe game is played—one in thecenter and eight along the sides,as shown in the diagram.

5.Glue the game board to thecardboard and decorate withmarkers or crayons. You maywant to try a border design likethese from Pueblo Indianartwork. Figure 4b

PLAYING THE GAMEThe two players take turns placingone counter at a time on an emptypoint on the board. When all sixcounters have been placed, theplayers take turns moving onecounter at a time along any line tothe next empty point. Jumpingover a counter is not allowed.Each player tries to make a row

with his or her three counters. Arow can be made across, up anddown, or along a diagonal—eight

ways altogether. The winner is the first player to

make a row. If neither player canget three in a row, call it a drawand start again.

CHANGING THE RULESSome people play Picaría on thethirteen points where the linesintersect, as shown in thediagram. Figure 4c Try playing onthirteen points by the game rulesjust given, with these differences:1.Neither player may place acounter in the center of theboard until all six counters areon the board.

2.Players may make three in a row,with no empty points between,anywhere along a diagonal.There are sixteen different waysto make three in a row.

THINGS TO THINK ABOUTWhich form of Picaría is a bettergame? Why?Can you play the first version of

Picaría with four counters for eachplayer instead of three? How aboutthe second version of the game?

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Figure 4c

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For hundreds of years people inEngland have played three-in-a-row games. Some had names likeThree Men’s Morris, Five Men’sMorris, Nine Men’s Morris, andTwelve Men’s Morris. The numbertold you how many counters eachplayer used in the game.English pageants during the

Renaissance used girls and boysas counters in Nine Men’s Morris.Imagine the scene as men in redvelvet and women in lace-trimmedgowns gathered around the largeMorris diagram marked in theearth. They watched as gamemasters ordered their living gamepieces to move along the lines ofthe squares and would call out“Good Move!” or “Watch Out!”Although Morris games have

been popular in England forcenturies, the game itself goesback thousands of years. Athousand-year-old burial ship of aViking prince was dug up inGokstad, Norway, and among its

possessions was a wooden gameboard of three connected squares.The ship’s sailors had cut thesame diagram in the woodenplanks of the deck for their owngames. It is likely that the Vikingsfrom Scandinavia learned thegame and spread it around theworld when they sailed to otherparts of Europe, northern Africa,Asia, and even America.Norwegians still use this same

board today for the game they callMølle. Germans call the samegame Mühle, Russians call itMelnitsa, in Hungary it is Malom,and in Italy it is called Mulinello.All these names mean “mill.” InNigeria a similar game is known asAkidada, and in Arabic-speakingcountries the game is called Dris.

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Figure 5a

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MATERIALS• Pencil• Ruler• Sheet of unlined paper, at least 8 inches (20cm) square

• Colored markers or crayons• Glue• Piece of cardboard, at least 9 inches(22.5cm) square

• 9 counters for each player, of 2 different kinds (beans, buttons, or coins)

DRAWING THE GAME BOARD1.Using your pencil and ruler,draw three squares one insidethe other on your paper. Planand measure carefully, so thatthe board fits on the paper andthere is enough room to movethe counters.

2.Draw four lines connecting themidpoints of the sides, asshown in the diagram. Figure 5a

3.Go over the pencil lines withmarkers or crayons. Mark thetwenty-four points where thelines intersect.

4.Then glue the paper to thecardboard. Decorate your gameboard with markers or crayons.

PLAYING THE GAMEThe two players take turns placingone counter at a time on an emptypoint on the game board. When alleighteen counters have beenplaced, the players take turnsmoving one counter at a timealong a line to the next emptypoint. Jumping over a counter isnot allowed.Each player tries to make a row

of three counters of the same kindalong any straight line. A row ofthree is called a “mill.” There aresixteen different ways to make amill. Can you find them?Here are two ways to make a

mill: Figure 5b A player who makes a mill may

remove one of the other player’scounters from the board, with oneexception. You may not remove acounter from the other player’s millunless no other counter of thatkind is on the board. Countersthat have been removed from theboard are out of the game.The loser is the player who has

only two counters left on the boardor who is blocked from moving.

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Figure 5b

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HOW TO BE A GOOD PLAYERIt is a good strategy to spread yourcounters out on the board. Thiswill make it harder for the otherplayer to block you. Don’t try tomake a mill while you are placingyour counters on the board. When you put your counters on

the board, place them so that youwill be able to move each counterin more than one direction. Move your counters so that you

will have a choice of more thanone way to form a mill on a futuremove.Position three counters so that

you can move one back and forthto close and then open a mill.Every time you close a mill, youcapture one of the other player’scounters. Figure 5c

THINGS TO THINK ABOUTCan you think of a plan that willalways work so that you can closeand open a mill many times? Canyou work out a strategy to blockthe other player from using an“open and close” plan to makemore than one mill? What otherstrategies can you figure out?

CHANGING THE RULESHave a rule that the same countermay not be moved twice in twosuccessive moves.In a version called Going Wild, a

player who has only threecounters left on the board maymove them, one at a time, to anyempty point on the board.

Figure 5c

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Children in Colombia, a country inSouth America, play a three-in-a-row game that they call Trique(TREE-keh). A player who makes arow of three calls out “Trique,” theSpanish word for a clever trick.

MATERIALS• Pencil• Ruler• Sheet of unlined paper, at least

8 inches (20cm) square• Colored markers or crayons• Glue• Piece of cardboard, at least 9 inches

(22.5cm) square• 9 counters for each player, of

2 different kinds (beans, coins, orbuttons)

DRAWING THE GAME BOARD1.Using your pencil and ruler,draw three squares on the sheetof paper, one inside the other, asshown in the diagram. Figure 6

2.Then draw four lines connectingthe midpoints of the sides of the

squares, and four more linesconnecting the corners of thesquares. Plan your layoutcarefully so that you haveenough space to move thecounters.

3.Go over the lines with markers orcrayons and mark the twenty-fourpoints where the lines intersect.

4.Glue the sheet of paper to thecardboard. If you already have aboard for Nine Men’s Morris(page 12), all you need to do isadd four lines connecting thecorners of the squares. Decorateyour game board with markersor crayons.

PLAYING THE GAMEFollow the rules for Nine Men’sMorris (page 12), with oneexception. In Trique, you maymove your counters and makerows of three along the diagonallines connecting the corners. Findthe twenty different ways to makea row of three.

Trique from Colombia TWO P L A Y E R S

Figure 6

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Nerenchi is an ancient three-in-a-row game played in the Asiancountry of Sri Lanka. Diagrams forsuch games were carved in templesteps about two thousand yearsago. Nerenchi has long been aspecial favorite of women and girlsin Sri Lanka. They often play inteams, with the members of eachteam taking turns.

MATERIALS• Sheet of unlined paper, at least 8 inches (20cm) square

• Ruler• Pencil• Colored markers or crayons• Glue• Piece of cardboard or constructionpaper, at least 9 inches (22.5cm)square

• 12 counters for each player or team,of 2 different kinds (beans, buttons,or coins)

DRAWING THE GAME BOARDThe game board for Nerenchi isexactly like the board for Trique(page 15). Figure 7a

PLAYING THE GAMEThe object of the game is to get arow of three counters, called a“nerenchi.” The row can be madealong the side of a square, along aline joining the midpoints of thesides of the squares, or along adiagonal line joining the corners.There are twenty ways to make anerenchi.Here are three different ways to

make a nerenchi: Figure 7bTo Begin. The players, or teams,

take turns placing one counter ata time on an empty point on theboard. This part of the game endswhen twenty-two counters are onthe board, leaving two emptypoints. The remaining twocounters may or may not be usedin the game.

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Figure 7a

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Nerenchi from Sri LankaTWO P L A Y E R SO R TWOT E AMS

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A player who makes a nerenchiduring the “placing” stage of thegame takes an extra turn, and heor she may do so for eachnerenchi made. One player mayhave twelve counters on theboard, while the other player mayhave only ten counters.

To Move. The last player to placea counter on the board makes thefirst move. The players, or teams,take turns moving one counter ata time along a line to the nextempty point. You may not movealong the diagonal lines, and youmay not jump over a counter. Notethat although you may not moveyour counters along a diagonalline, you are allowed to make anerenchi along a diagonal line.Each player or team tries to

make as many nerenchis aspossible. A player who makes anerenchi during the “moving”stage of the game may remove oneof the opponent’s counters fromany position on the board.

To Finish. The loser is the playeror team that has lost all but twocounters or is blocked frommoving.

Children in southern Africa playMurabaraba, a three-in-a-rowgame with the same number ofcounters. The board is very muchlike the board for Nerenchi, andthe rules are almost the same. Asimilar game called Twelve Men’sMorris was popular in the NewEngland colonies more than twohundred years ago. Other three-in-a-row games played with twelvecounters for each player are: SamK’i in China, Kon-tjil in Korea, DigDig in Malaysia, and Shah inSomalia.

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Figure 7b

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This game for two players or twoteams is popular among the menand boys in northern Nigeria,Niger, Mali, and other parts ofnorthwestern Africa. They playthe game with stones or sticksplaced in holes dug in the earthor in the desert sand. You mayplay on a 5-row, 6-column boardsimilar to part of a checkerboard.Good Dara players are held in

great honor. After the day’s workis done, champions travel fromvillage to village challenging localplayers. The contests maycontinue into the night for as longas the moon shines brightly. Achampion will teach the game tohis son as soon as the child is oldenough to learn the rules. Laterthe father tells the boy the secretsof the game, secrets that helearned from his father orgrandfather.

MATERIALS• Sheet of scratch paper• Pencil• Ruler• Piece of construction paper orcardboard, at least 8 inches (20cm)square

• Marker• 12 counters for each player, of 2 different kinds (beans, buttons, or bottle caps)

DRAWING THE GAME BOARD1.Practice drawing your gameboard on scratch paper in pencilfirst. You need a rectangledivided into five rows of sixsquares each.

2.Then draw the game board onconstruction paper orcardboard. First use a ruler tomeasure carefully, and markthe main points in pencil. Thendraw the lines. Figure 8a

3.Go over the lines with a marker.

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Figure 8a

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Dara from Nigeria TWO P L A Y E R SO RTWO T E AMS

Figure 8b

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PLAYING THE GAMETwo players, or teams, take turnsplacing one counter at a timeinside any empty square, until alltwenty-four counters have beenplaced. Then the players taketurns moving one counter at atime to the next empty space.Moves may be made up, down, orsideways, but not diagonally.Jumping over a counter is notallowed.Each player tries to get three

counters in a row with no spacesbetween them. A row can be eitheracross or up and down. A playerwho makes a row may remove oneof the opponent’s counters fromthe board. This is called “eating”the enemy, just as a lion eats itsprey.A player may not have more

than three counters in acontinuous line at any time.A row made during the “placing”

stage does not count. A playerwho makes two rows in one movemay capture only one of theopponent’s counters. See thediagram for an example. Figure 8bThe game ends when one player

can no longer make a row. Thenthe opponent is the winner.

THINGS TO THINK ABOUTCan you plan how to arrange fivecounters so that you can make arow on each move? This is called a“horse” and is a sure way to win.Here are two ways to do it. Canyou find other ways? Figure 8c

CHANGING THE RULESSome African players follow one ormore of these rules for Dara:Play on a checkerboard having

six rows and six columns.A counter may be captured from

the opponent’s row only if theopponent has no other counterson the board.Neither player may remove a

counter from a row. Therefore, the“horse” strategy cannot be used inthis game. The player who makesthree rows before the opponentmakes one row wins the game.

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Figure 8c

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Mankala: Board Games of Transfer

ankala is considered bygame experts to be amongthe best games in the

world. Mankala games arewidespread. You will find them inmost African countries, as well asin India, Indonesia, thePhilippines, Sri Lanka, CentralAsia, and Arabic countries. Ascaptives in the horrendous slavetrade, Africans brought the gamesto parts of the Americas—easternBrazil, Suriname, and theCaribbean islands—where they arepopular even today.Games of this type are

thousands of years old. Gameboards were cut into the stones ofseveral temples in ancient Egypt.Other very old rock-cut boardswere discovered in Ghana,Uganda, and Zimbabwe.

The word mankala is Arabic for“transferring.” Stones or seeds aretransferred from one cup toanother on a board having two,three, or four rows of cups. Ineach region the game has its ownname and its own set of rules. Thetwo-row board is popular in NorthAfrica, West Africa, and parts ofEast Africa, under such names asWari, Oware, Ayo, and Giuthi.People in Asia play Sungka,Dakon, and Congklak on two-rowboards. In eastern and southernAfrica the four-row board game ismost common, with names likeBao (meaning “board” in Swahili),Nchuba, and Mweso. Ethiopia hasthree-row versions. Figure 9a

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Figure 9a

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Oware game board

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F igure 9b

The game has been played bykings on beautiful carved woodenboards or boards of gold, and bychildren who scoop out holes inthe ground. About four hundredyears ago a king in central Africa,King Shyaam aMbul aNgoong,brought the game to his people,the Kuba, living in Congo. Heinduced them to give up warlikeactivities in favor of the peacefularts. A statue of the king, now inthe British Museum, shows himseated in front of a mankala gameboard.

When I was in Nigeria I watchedas two teenaged carvers of gameboards played each other. Theseeds moved so fast around theboard that I had no idea what theplayers were doing. In a fewminutes, the game was over. Theywere experts because they hadplayed the game since they weresmall children. Then one of thecarvers offered to play a gamewith me. He showed me how tomake each move so that I wouldwin. Of course, I bought one ofhis game boards. The board hadhinges in the middle so that youcould close it up and keep theseeds inside. It made a fine gamefor travel. Figure 9b

Ayo game board

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Asante people in Ghana, a countryin West Africa, play the gamecalled Oware (oh-WAHR-ee).Children and grown-ups alikeenjoy the game. Two people faceeach other with the game boardbetween them. Children oftenscoop out holes in the ground fortheir “board” and gather pebblesor large seeds to use as counters.The Yoruba people of southwestNigeria play the same game, butthey call it Ayo (EYE-oh). They usecertain gray seeds as counters.If you and your opponent have

never played the game, you mightwant to start with this simpleform. Playing on a flat sheet ofpaper helps you to see exactlywhat happens with each move. Inthe next section you can learnabout the real Oware game.

MATERIALS• Sheet of unlined paper, at least

10 inches (25cm) long• Ruler• Pencil• Marker• 2 small bowls or cups• 16 counters of 1 kind (beans, buttons,

or shells)

DRAWING THE GAME BOARD1.Draw a game board in the shapeof a large rectangle having tworows of four squares each. Eachsquare should be large enoughto hold several beans. Use aruler and pencil and measurecarefully. Figure 10a

2.Go over the lines with a marker.3.Place one bowl or cup, called“endpot,” at each end of thegame board.

Figure 10b

Easy Oware TWO P L A Y E R S

Player Two

Player One

Figure 10a

from Ghana

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PLAYING THE GAMEThe players sit facing each otherwith the game board betweenthem. Place two beans or othercounters in each space. The fourspaces, called “cups,” on each sideof the board belong to the playernearest them. The endpot to theright of each player belongs to thatplayer. Figure 10b

To Move. Player One picks up allthe beans in any one of her cupsand drops one bean into eachcup, going to her right (counter-clockwise). This is called “sowingthe seeds.” Some beans may fallinto the cups on Player Two’s sideof the board. Figure 10cThen Player Two picks up all the

beans in any one of his cups. Hedrops one bean in each cup goingaround the board to his right. Theplayers take turns in this way. Donot “sow” into the endpots.

To Capture. Captures are madefrom the opponent’s side of theboard. If the last bean in any movemakes a group of two in a cup onthe opponent’s side, the last playercaptures these two beans andplaces them into her endpot. Then,going backward, if the cup justbefore the previous one on theopponent’s side also has two

beans, the player may capturethem and place them in herendpot. Continue to capture aslong as each cup has just twobeans and is on the far side of theboard. Figure 10d

To Finish. The game ends whenone person has no beans left onhis or her side of the board. Thenthe beans in the endpots arecounted, and the player who hascaptured more beans is thewinner.

THINGS TO THINK ABOUT AND DOPractice playing this game byyourself to learn how to makegood moves. Practice differentkinds of moves and see how manybeans you can capture with eachmove.Try playing with twenty-four

beans. Start with three beans ineach cup. How would you changethe rule about captures?

MANKALA: BOARD GAM

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Player One is ready to move

Player One has moved

Player One has captured 4 beans

Player One’s first move

Figure 10c

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Once you have practiced playingEasy Oware (page 22) try thisversion of the game, which is a bitmore challenging, but lots of fun.

MATERIALS• Empty (one dozen) egg carton with the

lid removed• Colored markers• 2 small bowls or cups• 48 counters, of 1 kind (beans,

buttons, or shells)

MAKING THE GAME BOARDThe board has six cups on eachside, twelve cups altogether. Anegg carton makes a perfect gameboard. You might want to decorateit with African patterns and colors.Place one bowl or cup at each endof the board as an endpot to holdthe captured beans. Figure 11a

PLAYING THE GAMEThe players sit facing each otherwith the game board betweenthem. Place four beans or othercounters in each space. The sixspaces, called “cups,” on each sideof the board belong to the playernearest them. The endpot to theright of each player belongs to thatplayer.

To Move. As in Easy Owareplayers take turns picking up allthe beans in any one of their cupsand dropping them, one bean intoeach cup, going to their right(counterclockwise). This is called“sowing the seeds.” Some beansmay fall into the cups on theother player’s side of the board.Do not “sow” into the endpots.When players sow from a cup

that has twelve or more beans,they must skip over the cup thebeans came from and leave thatcup empty, as they sow aroundthe board.

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Figure 11a

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The Real from Ghana

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To Capture. If the last beandropped into a cup on theopponent’s side of the boardmakes a group of two or three,those beans may be captured.Then, going backward, groups oftwo or three beans may becaptured from cups that are nextto each other on the far side of theboard. Figure 11b

To Finish. If all the opponent’scups are empty, a player mustmove beans into them on his orher turn. If the player cannot doso, the game ends. The playeradds the beans on his or her sideto those in the endpot. Beans thatgo around and around with nocaptures may be divided equallybetween the players. The playerwith more beans in his or herendpot is the winner.

THINGS TO THINK ABOUT AND DOPlay both sides by yourself. Planthe best moves that will lead tocapturing the opponent’s beans.When you are learning the

game, you may want to placethree beans in each cup, or thirty-six beans altogether. What otherrules might you change? Perhapsyou will invent a version of Owarethat people already play in otherparts of the world. The next fewpages will describe some of theseversions.

MANKALA: BOARD GAM

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PLAYER ONE

P layer One can capture 5

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Children in the Philippine Islandslike to play a mankala-type gameknown as Sungka (Soon-KAH).Similar games are called Chonkain Malaysia and Congklak inIndonesia. A book editor who grewup in the Philippines told me thatshe had learned more mathplaying Sungka than she learnedin school! She probably meantthat she enjoyed the game morethan she enjoyed schoolwork.The rules for this game differ

from those for Oware in severalimportant respects: 1.All moves are clockwise, going tothe player’s left.

2.Players may make several lapsin one move.

3.Players drop a counter into theirown endpot as they go aroundthe board.

4.Each player’s endpot is to his orher left; and capturing is donedifferently.

MATERIALS• Empty (one dozen) egg carton with the

lid removed• Scissors• 1 or 2 small bowls or cups• 50 counters, of 1 kind (beans, shells,

pebbles, or buttons)

MAKING THE GAME BOARDThis game board has two rows offive cups and two endpots, one ateach end. Using your scissors, cutthe partition between two cups ofthe egg carton at one end, anduse the spare as an endpot forcaptured beans. Place a bowl, orendpot, at the other end of theboard to store captured beans.Figure 12a

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Sungka fromthe Philippines TWO

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Figure 12a

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PLAYING THE GAMEThe fifty beans are distributed, fiveto a cup. Two players face eachother with the game boardbetween them. The five spaces, orcups, on each side belong to theplayer nearest them. The endpot tothe left belongs to that player.

To Move. Players take turnspicking up the beans from any ofthe cups on their side, and sowingthem one-by-one into the cupsgoing to their left (clockwise)around the board. Players dropone bean into their endpot as theygo around the board, but not intothe opponent’s endpot.If the last bean drops into a cup

that already has beans, the playerpicks up all the beans in that cupand continues to sow in aclockwise direction.If the last bean falls into an

empty cup on the opponent’s sideor into their own storage cup, it isthe opponent’s turn to move.

To Capture. If the last bean fallsinto an empty cup on the player’sside of the board and the cupopposite it has beans, the playermay capture those beans andplace them in his or her endpot.Then it is the opponent’s turn.Figure 12b

To Finish. When there are nobeans left on a player’s side, theopponent adds the beans on hisor her side to those in the endpot.The winner is the player withmore beans in his or her endpot.

THINGS TO THINK ABOUTHow is Sungka similar to Oware?Can you play Sungka on an Owareboard that has six cups in eachrow? How would you change therules?

CHANGING THE RULESPeople often play on a board thathas two rows of seven cups andstart by placing seven shells ineach cup. How many shells areneeded?

MANKALA: BOARD GAM

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Figure 12b

5 5 5 5 5

5 5 5 5 50 0

Player One moves 5 beans

6

6 6 5 5 5

6 6 0 5 51 0

Player One continues, and drops 6 beans

6

6 0 6 6 6

6 6 1 6 61 0

Player One captures 6 beans

P

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The Kikuyu people of centralKenya play a version of mankalathat they call Giuthi (Ghee-YOU-thee). The board has two rows ofsix cups. At each end is a storagecup for captured seeds. TheKikuyu have traditionally beenbreeders of cattle, and the wordsthey use in the game reflect theiroccupation. The counters arereferred to as “cattle in the fields,”and the captured pieces are placedin “sheds.” Most people play thegame outdoors. Boys and girlsoften play the game as they lookafter their cattle and goats.

MATERIALS• Empty (one dozen) egg carton with the

lid removed• 2 small bowls or cups for storage,

called “sheds”• 48 counters, of 1 kind (beans,

buttons, or shells)

PLAYING THE GAMEUsing the egg carton as a game

board, the two players, or teams,face each other across the gameboard. Each player owns the sixcups on his or her side of theboard and the bowl or cup forstorage at his or her right. Thebeans are distributed four to eachcup. Figure 13a

To Move. Player One picks up allthe beans in any cup on his orher side and drops them, one byone, in each cup going around theboard. The player chooses thedirection—either to the right(counterclockwise) or to the left(clockwise). The player picks upall the beans in the last cup he orshe dropped a bean into anddistributes them, one by one, inthe opposite direction. Player Onecontinues in this way, changingdirection each time, until the lastbean falls into an empty cup.Players do not drop beans into thestorage cups.

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Giuthi from Kenya TWO P L A Y E R SO RTWO T E AMS

Figure 13a

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If the last bean falls into anempty cup on the player’s side,but he or she has not droppedany beans in the cups on theopponent’s side, the player movesagain. The Kikuyu say: “You can’tsteal the other person’s cattleunless you cross into his land.”That means you must drop beansin your opponent’s cups beforeyou may capture your opponent’sbeans.If the last bean falls in an empty

cup on the opponent’s side of theboard, it is the opponent’s turn.The game continues in this way,unless a player captures theopponent’s beans.To begin any move, a player

must pick up the beans in a cupthat contains at least two beans.

To Capture. If, in the course of amove, Player One has droppedbeans in the opponent’s cups, andthe last bean in Player One’s handfalls into an empty cup on his orher own side, then Player Onemay capture all the beans in theopponent’s cup opposite that cup(which now has one bean in it). Ifthe cup next to this cup is alsoempty, Player One may capturebeans in the opponent’s cupopposite it. Player One continues

to capture as long as there is anunbroken sequence of empty cupson his or her side in the directionhe or she is moving. Capturingends with an occupied cup on theplayer’s side of the board or withan empty cup on the opponent’sside of the board. Figure 13b

To FInish. The game is overwhen neither player can make amove according to the rules. Eachplayer takes the “cattle” on his orher side of the board and addsthem to those in the “shed.” Theplayer with more cattle is thewinner.

THINGS TO THINK ABOUTHow is Giuthi similar to Oware?How is it different? How is Giuthisimilar to Sungka? How is itdifferent?

CHANGING THE RULESThe Kikuyu play with as many asnine beans in each cup. The boardmay have anywhere from five toten cups in each of the two rows.The rules given above are for

the first stage of the game only.The second part is more comp-licated and is not included here.

MANKALA: BOARD GAM

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Figure 13bOne move for Player One

3 3 0 6 7 2

1 2 4 0 0 5

4 3 0 6 7 2

2 0 4 0 0 5

0 3 0 6 7 2

3 1 5 1 0 5

Player One moves 2 beans to the left

Player One moves 4 beans

Player One captures 14 beans

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More Board Gamesn this section you will learntwo blocking games andseveral games that are more

or less like checkers. All but one ofthe games are for two players andinvolve moving the counters on agame board.Pong Hau K’i, from China, and

Mu Torere, a Maori game fromNew Zealand (the Maori call theirland Aotearoa), are rather similar.We can think of Mu Torere as amore complex version of Pong HauK’i, because it requires morecounters and is played on a largerboard. In both games the winneris the player who has blocked theopponent from making any moves.

A five-pointed star was one ofthe diagrams scratched into theroof stones at the ancient Egyptiantemple to the king, Pharaoh Seti I,at Qurna about 3,300 years ago.Two games that use this diagram,called a pentagram, are Pentalpha,for one player, and Kaooa, for twoplayers. Pentalpha is a Greekname for the game—really morelike a puzzle—that is played on theMediterranean island of Crete.Kaooa, also called Vultures andCrows, is popular in India.Awithlaknannai is based on the

Arabic game called El-quirkat, orAlquerque, as it is known today.The game is similar to checkersand may actually be the origin ofcheckers. The Moors of NorthAfrica, who ruled much of Spainfor more than seven centuries,until 1492, introduced the gameinto Europe. Several versions,played on square boards, aredescribed in the Book of Games,written in 1283 under the

I

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direction of the Spanish kingAlfonso the Wise. A diagram forthis game was also found amongthose scratched in the roofingslabs of the temple for theEgyptian Pharaoh Seti I.Awithlaknannai is played by the

Zuni, a Pueblo (Native American)people of New Mexico. You maywonder how the game traveled sofar from its original home.Probably the Spanish conquist-adores brought it with them whenthey invaded the lands of thePueblo starting in the sixteenthcentury, just as they brought thethree-in-a-row game of Picaría (seepage 10).A game that is similar to

Awithlaknannai, except for theshape of the board, is calledButterfly in Mozambique and LauKata Kati in India andBangladesh. Here is anotherexample of a game traveling far.We don’t know whether it wentfrom Africa to Asia or in theopposite direction.

Yoté, a game like checkers butmore complicated, is popular inWest Africa, especially in Senegal.Generally children scoop out holesin the ground and play with sticksor pebbles. This is a game thatdemands quick thinking, and aplayer who seems to be winningmay suddenly find that he or shehas actually lost the game.

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Pong Hau K’i (Pong-haw-kee) is agame from China. In Korea theycall the game Ou-moul-ko-no, orKono. As you can see, the board isvery simple. Two people play onthe five points where the linesintersect. Each player tries toblock the other from moving.

MATERIALS• Sheet of construction paper• Pen or marker• Ruler• 2 counters for each player, of

2 different kinds (beans, buttons, or coins)

DRAWING THE GAME BOARDDraw two intersecting righttriangles on the sheet of paper, asshown in the diagram. Make theboard large enough so that thereis lots of room to move thecounters. Figure 14a

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Figure 14a

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Pong Hau K’i from Korea& China

TWO P L A Y E R S

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PLAYING THE GAMEThe game begins with the countersplaced on the board, as shown inthe diagram. One player’s countersare on the two lower points andthe other player’s counters are onthe two upper points. Figure 14bPlayer One moves one of his or

her counters onto the centerpoint. Then Player Two moves oneof his or her counters onto theempty space. The playerscontinue, with each player takinga turn to move a counter.The game ends when one player

wins by blocking the other playerfrom moving. If the same set ofmoves is repeated three times, thegame ends in a draw—no winnerand no loser.

THINGS TO THINK ABOUTIs it better to go first or second?How must you place your countersin order to block the other player?Is it possible to play the game

on this board with three countersfor each player? How about onecounter for each player? Explain.Try to design a board that will

allow each player to use threecounters and at the same timefollow the rules for Pong Hau K’i.

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Figure 14b

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Maori children in New Zealandhave played Mu Torere (Moo Toh-RERE-uh) for as long as anyonecan remember. The old people saythat the Maori crossed the seas toNew Zealand in seven canoesmany centuries ago. They calledthe land Aotearoa in theirlanguage, which is related to thelanguages of Hawaii and Tahiti.

MATERIALS• Game board (see directions for

Shisima on page 4)• 4 counters for each player, of two

different kinds (call one set “white”and the other “black”.)

DRAWING THE GAME BOARDThe game board is an eight-pointed star with a space in thecenter called the putahi, or meetingplace. The eight rays are calledkawai, or branches. Figure 15aUse the game board you created

for Shisima (page 4), or follow thedirections for making theoctagonal Shisima board. Maorichildren draw the board on theground with a pointed stick, or ona flat rock with a piece ofcharcoal.

PLAYING THE GAMEThis is a blocking game; the goal isto block the other player frommoving. To start, place thecounters on the board. The fourwhite counters occupy adjacentpoints of the star, and the fourblack counters occupy the fourother points, as in the diagram.Figure 15b

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Figure 15a

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Mu Torere from NewZealand

TWO P L A Y E R S

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Black starts, and players taketurns moving their counters oneat a time. A move can be made inone of three ways:1. A counter can move from one

point on the star (or octagon) tothe next point, but only if thepoint is empty.2. A counter can move from the

putahi to an empty point.3. A counter can move from the

point to the putahi, but only if theopponent occupies the point(s) onone or both sides of that point. Forexample, Black can move into theputahi from point E or point H,but not from point F or point G.The game ends when one player

wins by blocking the other playerfrom making any moves.

THINGS TO THINK ABOUTWhat is the reason for rulenumber three above? See whathappens when that rule isdisobeyed.Is it necessary for the winner to

have one counter in the putahi?Why or why not?What formation on the board

leads to winning the game? (Hint:Some people call Mu Torere a“three-in-a-corner” game.)

CHANGING THE RULESAfter each player has made twomoves, any counter can be movedfrom a point to the putahi. Howdoes this new rule change themoves leading to winning thegame?Try to play the game with three

counters for each player. Howmust you move in order to win?Try to play the game on a six-

pointed star, with three countersfor each player.

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Figure 15b

A

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The name pentalpha comes fromtwo Greek words. Pent means five,and alpha is the first letter of theGreek alphabet. The second letteris beta. Can you guess where wegot the word alphabet? This gamefor one person is played on a five-pointed star called a pentagram. Itis popular among people in Crete,an island in the MediterraneanSea. Figure 16a

MATERIALS• Sheet of unlined paper• Pencil, pen, or marker• Ruler• Protractor• 9 counters of any kind

DRAWING THE GAME BOARDHere are directions for drawing areally neat diagram. (Of course,you can draw one freehand thatisn’t so neat!)1.Draw a horizontal line six inches(15cm) long above the center ofthe paper. Use the protractor tomeasure an angle of thirty-sixdegrees at each end of the line.

2.Draw six-inch lines to completethe two angles.

3.Measure an angle of thirty-sixdegrees at the end of each ofthese two lines.

4.Draw six-inch lines to completethe diagram. Mark the tenpoints where the lines intersect.Figure 16b

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Figure 16a

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Pentalpha from Crete ONE P L A Y E R

Figure 16b

36°

36° 36°

36°

6"

6"

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PLAYING THE GAMEThis is a “placing” game. Thecounters are not moved after theyhave been placed on the board.Each counter is placed in threemoves. It helps to count them out.1.Place a counter on an emptypoint. Say “one.”

2.Move it along a straight line sothat it jumps over the next point(empty or not). Say “two.”Remember the line must bestraight.

3.Place it on the third point,which must be empty. Say“three.”Follow these rules for placing

each counter until all ninecounters are on the pentagram.Here is one move: Figure 16cThe goal is to place the nine

counters on nine of the tenmarked points of the board, oneat a time, according to the rulesabove. It sounds easy but it issurprisingly tricky. M

ORE BOARD GAMES

Start at 1, jump over 2, land on 3

1

2

3

Figure 16c

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Children in India play Kaooa on adiagram in the form of a penta-gram, or five-pointed star. Anothername for the game is Vulturesand Crows. The goal is for thevulture to capture the crows byjumping over them, or for thecrows to corner the vulture sothat the vulture can’t move. Figure 17

MATERIALS• Game board (see the directions for

Pentalpha on page 36)• 1 counter to represent the vulture

(bean, button, or coin)• 7 counters to represent the crows

(beans, buttons, or coins)

PLAYING THE GAMEPlayer One places one crow on anypoint. Player Two places thevulture on an empty point.Player One places the second

crow on any empty point. PlayerTwo moves the vulture one spacealong a line to an empty point.

They continue taking turnsuntil all seven crows are on theboard.Then the players take turns

moving one counter at a time toan adjacent empty point. PlayerOne moves the crows and PlayerTwo moves the vulture. Thevulture may capture a crow byjumping over it along a line to anempty point. The vulture maymake a series of captures on asingle move.The game ends when the

vulture is trapped and can’t moveor when the vulture has capturedat least four crows.

THINGS TO THINK ABOUTWould the game work with fewerthan seven crows? More thanseven?

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Figure 17

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Kaooa from India TWO P L A Y E R S

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Almost one hundred years ago ananthropologist named StewartCulin visited many parts of theworld and wrote about the gamesthat children and grown-upsplayed. He observed Zuni (NativeAmerican) children in New Mexicoplaying a game for two people inwhich they moved pebbles on avery long board scratched on arock. Some children played asimilar game on a shorterdiagram. The game is calledAwithlaknannai, referring tostones that were used to killserpents.

MATERIALS• Sheet of construction paper or

cardboard, at least 10 inches (25cm)long

• Ruler• Pencil• Pen or marker• 9 counters for each player, of 2

different kinds (stones, checkers, or coins)

DRAWING THE GAME BOARDHere is a diagram for a simpleversion of the game. You may wantto make a longer board and usemore than nine stones for eachplayer. Stewart Culin claimed thathe saw two people playing withtwenty-three stones each on aboard that was thirty-three incheslong! The total number of countersis always one less than the totalnumber of points on the board.Figure 18Draw this diagram in pencil,

then go over the pencil lines witha pen or marker. The diagramconsists of two sets of triangles,one set of six triangles on eachside of a long line. You mightmake a pattern of an isoscelestriangle (two sides have the samelength) and trace it twelve times tomake a really neat diagram.

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Figure 18

Awithlaknannai TWO P L A Y E R S

NativeAmerican

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PLAYING THE GAMEPlace the counters on the gameboard as shown in the diagram.Note that when the counters are inplace, the center point, and onlythe center point, is empty, nomatter how long a board you use.Each player in turn moves one

of his or her counters one spacealong a line to an adjacent emptypoint.Or a player may jump over and

capture an opponent’s counter ifthe next space along a straightline is empty. A player maycontinue jumping with the samecounter and capturing as long aspossible.A player who fails to jump loses

the counter to the opponent. If aplayer has a choice of more thanone jump, he or she may choosewhich jump to make.The winner is the player who

has captured all the opponent’scounters, or as many as possible.

THINGS TO THINK ABOUTCan you play with the same ruleson a board that has an oddnumber like five, seven, or ninetriangles on each side? Why orwhy not? Try it. Do you thinkStewart Culin was correct when hewrote that each player in the gamehe watched had twenty-threestones? Where would the centerpoint be on their game board?

CHANGING THE RULESPlay on a shorter board with fewercounters. Remember that only thecenter point is empty when youstart. Then play on a longer boardwith more counters.

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The game may be called Butterflyin Mozambique because of theshape of the board. Children inIndia and Bangladesh call thesame game Lau Kata Kati.

MATERIALS• Sheet of unlined paper, at least 10

inches (25cm) long• Ruler• Pencil• Pen or marker• 9 counters for each player, of 2

different kinds (beans, buttons, orcoins)

DRAWING THE GAME BOARDUsing your pencil and ruler, drawthe board as shown in thediagram. Go over the lines withpen or marker. Figure 19a

PLAYING THE GAMETo start place the eighteencounters on the game board asshown in the diagram, leaving justthe center point empty.

Each player in turn moves oneof his or her counters one spacealong a line to an adjacent emptypoint.Or a player may jump over and

capture an opponent’s counter ifthe next space along a straightline is empty. A player maycontinue jumping with the samecounter and capturing as long aspossible.A player who fails to jump loses

the counter to the opponent. If aplayer has a choice of more thanone jump, he or she may choosewhich jump to make.The winner is the player who

has captured all the opponent’scounters.

CHANGING THE RULESYoung children play on a smallergame board. Each player has sixcounters. Figure 19b

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Figure 19a

Figure 19b

Butterfly TWO P L A Y E R S

fromMozambique

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Yoté (YOH-tay) is similar tocheckers. Children in West Africascoop out holes in the sand,collect pebbles or bits of wood, andare ready to play the game. Youmay use checkers as counters andeither make your own board oruse part of a checkerboard.

MATERIALS• Scratch paper• Pencil• Ruler• Piece of cardboard• Marker• 12 checkers or other counters for each

player, of 2 different kinds

DRAWING THE GAME BOARDThis is the same game board thatis used for playing Dara (page 18).It will have five rows of six spaceseach.1.Practice drawing your gameboard on scratch paper in pencilfirst. You need a rectangledivided into five rows of sixsquares each.

2.Then draw the game board oncardboard. First use a ruler tomeasure carefully, and markthe main points in pencil. Thendraw the lines.

3.Go over the lines with a marker.Figure 20

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Figure 20

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Yoté from West Africa TWO P L A Y E R S

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PLAYING THE GAMETo Begin. Players take turnsplacing one counter at a time inany space on the board. They neednot place all the counters beforegoing on to the next stage. Aplayer may keep some counters tobe placed later.

To Move. Players take turnsmoving one counter at a timealong a straight line to the nextspace, if it is empty. Moves are upor down or sideways, but notalong a diagonal.

To Capture. A player may jumpover an opponent’s counter intothe next space, if it is empty, andremove that counter from theboard. In addition, this playermay remove another of theopponent’s counters from theboard as a bonus.

To Finish. The winner is theplayer who has captured all of theopponent’s counters. If eachplayer has only three or fewercounters on the board, the gameends in a tie or a draw.

CHANGING THE RULESPlayers must place all theircounters on the board in the tworows closest to them before theymove.Play on a board with more or

fewer squares. How manycounters are needed for each typeof board?

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Table 1

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Games of Chanceeople have always hopedthat they could find outwhat will happen in the

future. A person might throw apenny and say: “If it comes upheads, I will do well on the mathtest tomorrow.” Of course, doingwell on the math test has nothingto do with the way a coin falls!Games of chance arose from

these attempts to foretell thefuture. People would throw dice orcoins and look at the outcomes,the way the objects fell. Otherobjects used in such games wereseashells, halves of nutshells, pitsof certain fruits, tops, spinners,and marked sticks. Peopleinvented rules for winning andlosing points.

How did people figure out theserules? They may have looked athow often a certain outcome willhappen. For example, if you tossfour coins many times, you are aslikely to get four heads as fourtails. That is because a coin isbalanced—if it’s a fair coin. Headsand tails are equally likely tooccur. However, you are muchmore likely to get an outcome oftwo heads and two tails thaneither four heads or four tails.Toss four different coins, like a

penny, a nickel, a dime, and aquarter, about one hundred timesand make a record of theoutcomes. Or you can make atable with headings for thedifferent coins. Figure out all thedifferent ways that these fourcoins can fall. You should findsixteen ways.

PPenny Nickel Dime Quarter Total

4

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Copy this table and finish it.How many of these ways are twoheads and two tails? How manyare four heads? How many arefour tails? Table 1What about a shell that is not

well balanced? We cannot predictwhether it is more likely to landwith the opening up or theopening down. You might toss ashell many times and find that itlands with the opening up threetimes out of five. A different shellmay have other outcomes.

When you play these games ofchance, think about whether thegames are fair. Do all the playershave an equal chance of winning?These games are often called

“probability games.” The study ofprobability is important in our lifetoday. For example, probability isused in the insurance businessand for predicting the weather.

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Lu-lu is a game played by thepeople who first came to theHawaiian Islands. Stewart Culin, afamous anthropologist andcollector of games, wrote aboutthis game in an article publishedin 1899.Polynesians sailed great

distances across the Pacific Oceanfrom Asia to land on the HawaiianIslands. The islands are ofvolcanic origin, and even todaysome volcanoes spew forth lava.Hawaiian children play Lu-lu withdisks made of volcanic stone.

MATERIALS• 4 disks, or circles cut from cardboard,

about 1 inch (2.5cm) in diameter• Marker or pen• 60 or more toothpicks or beans (or

pencil and paper) to keep score.

PREPARING THE DISKSMark one side of each disk withone, two, three, or four dots, asshown in the diagram. The markedsides are called the faces. Figure 21

46

Figure 21

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Lu-lu fromthe Hawaiian Islands TWO

O R MOREP L A Y E R S

Lu-lu fromthe Hawaiian Islands TWO

O R MOREP L A Y E R S

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PLAYING THE GAMEThe players decide in advance howmany rounds they will play. Theytake turns tossing the disks. Eachplayer has two tosses beforepassing the disks to the nextperson. To toss the disks, a playerholds the four disks in both handsand drops them onto the table orthe ground.If all the disks fall face up, the

player scores ten points andtosses all four disks again. Thenumber of dots that show on thesecond toss is added to the ten onthe first toss to get the total score.If one or more disks fall face

down on the first toss, the playerpicks up only those face-downdisks and tosses them again. Thescore is the total of all the dots onthe four disks after the secondtoss. Figure 22The winner is the player with

the highest score at the end of theagreed-upon number of rounds. Ifthey wish, the players can agreeto play until one person reachesfifty points, or another agreed-upon number of points. To be fair,each person should have the samenumber of turns.

THINGS TO THINK ABOUTWhy do you suppose that a playerscores ten points when all fourdisks fall face up? What is thehighest score a player can have onone turn? There are two different ways a

player can score five points on thefirst toss. What are they? In howmany ways can you score each ofthe other numbers, from zero tonine, on one toss of the disks?

CHANGING THE RULESLet each player toss the disks justonce on each turn. Which rulesmake for a better game, one tossor two tosses on each turn?

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First toss = 5 points

Second toss = 4 points Score = 5 + 4 = 9 points

Figure 22

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Native Americans have beenplaying games of chance for agesand ages. About one hundredyears ago, Stewart Culin, ananthropologist, traveled aroundthe continent to learn about thesegames. He collected them in abook, Games of the North AmericanIndians, first published in 1907.He found that Native Americanchildren and grown-ups had manygames that were similar to oneanother. The players would tosssix or eight objects—sticks, peachpits, or walnut shells, for example.

The sticks or fruit pits usuallywere plain on one side and coloredor decorated on the other side.After each toss, the playerscounted how many of theseobjects fell with the decorated sideup and how many had the plainside up. Players would earn pointsaccording to the way the objectsfell. Their systems of scoring wereoften so complicated that theobserver couldn’t figure them out!The Game of Dish and the Stick

Game are simple forms of thegames of the Native Americans.Children learned the simple formsto prepare for the complicatedgames played by grown-ups.

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Native American Games

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Some of the Native Americans whoplayed this game, often called theBowl Game, are the Seneca of NewYork, the Passamaquoddy ofMaine, the Cherokee of Oklahoma,and the Yokut of California. Theobjects they used depended uponthe things they found in theirenvironment. The Seneca madebuttons from the horns of elk. TheCherokee might have used beans.The Yokut used half shells of nutsfilled with clay or pitch. Playerstossed the playing pieces on flatbaskets that the women hadwoven.

MATERIALS• 4 playing pieces (bottle caps, peach

pits, or walnut halves)• Markers• 50 toothpicks or beans to keep score• Wooden bowl, pie pan, or flat basket

PREPARING THE PLAYING PIECESDecorate each playing piece onone side with markers. Use adifferent pattern or color for eachone. You might want to decoratethe pieces with Native Americanpatterns. Call the decorated sidethe “face” of the piece. Figure 23

PLAYING THE GAMEPlayers place the pile of toothpicksor beans in the center. Decide inadvance how many rounds to play.Players take turns tossing the

playing pieces in the bowl. Holdthe bowl with both hands and flipthe pieces lightly in the air. Notewhether they fall with thedecorated sides (faces) up ordown.Players should be careful not to

let pieces fall out of the bowl. Theymay decide to impose a penalty ifthat happens. Discuss this.

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F igure 23

TWO O R MOREP L A Y E R S

NativeAmericanThe Game of Dish

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SCORING• All four up = five points• Three up and one down = twopoints

• Two up and two down = onepoint

• One up and three down = twopoints

• All four down = five points

Count the number of points.Take that many toothpicks orbeans from the pile and placethem next to you. The player withthe greatest number of toothpicksor beans at the end of the agreed-upon number of rounds is thewinner.

THINGS TO THINK ABOUTIs the scoring system fair? Notethat an outcome of four up isscored the same as four down. Isthis a fair way to score the game?One way to find out is to toss oneplaying piece twenty times. Keeptrack of the number of times itlands up and the number of timesit lands down. Are they about thesame, or is one outcome morelikely than the other? Repeat theexperiment. Is the scoring fair orunfair?

Can you find six different waysthat the pieces can fall with twofaces up and two faces down?Label the pieces A, B, C, and D,and make a list. Copy this tableand finish it. Table 2

CHANGING THE RULESNative Americans usually playedthese games with five, six, or moreplaying pieces. How would youscore for five pieces? For sixpieces?

50

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Table 2

A B C D

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This Native American game isalmost like the Game of Dish (page49).

MATERIALS• 4 popsicle sticks or tongue depressors• Markers• 50 toothpicks or beans to keep score

PREPARING THE PLAYING PIECESDecorate each popsicle stick onone side with markers. Use adifferent pattern or color for eachone. You might want to decoratethe pieces with Native Americanpatterns. Call the decorated sidethe “face” of the piece. Figure 23

PLAYING THE GAMEPlayers take turns. Each playerholds the sticks in one hand andlets them fall to the ground or thetable.

SCORING• All four up = five points• Three up and one down = twopoints

• Two up and two down = onepoint

• One up and three down = twopoints

• All four down = five points

Count the number of points.Take that many toothpicks orbeans from the pile and placethem next to you. The player withthe greatest number of toothpicksor beans at the end of the agreed-upon number of rounds is thewinner.

THINGS TO THINK ABOUTIs this a fair way to score thegame? Try to think of a better wayto score. There are sixteendifferent ways that four sticks canfall. Here are three ways. Howmany more can you find? Copythis table and finish it. Figure 24

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F igure 23

Figure 24

#1 #2 #3 #4

TWO O R MOREP L A Y E R S

NativeAmericanStick Game

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Igba-Ita (EE-bah-EE-tah) is agame of chance played by the Igbo(EE-boh) people of Nigeria. Thename means “pitch and toss.” Inthe old days, groups of men, fromtwo to twelve people, would gatherin the marketplace for a game,while the women were busybuying and selling. The playingpieces were cowrie shells, whichwere used as coins in formertimes. Later they played the gamewith coins and called it Igba-Ego,which means “pitch the coins.”Early in the twentieth century

an observer, G. T. Basden, wroteabout the speed of the players: “I have watched players at thisgame, and it has always beenquite beyond me to note thepositions of the fall; the cowrieshave been counted and snatchedup again long before I could beginto count.”Here is a simple version of the

game.

MATERIALS• 12 cowrie shells for each player (pasta

shaped like shells are a goodsubstitute)

PLAYING THE GAMETo Begin. Decide in advance howmany rounds to play. One person,called the challenger, picks up fourshells. The other players agreethat each will place one, two, orthree shells into the center, calledthe “pot.” The challenger tossesthe four shells. The players counthow many landed with theopenings up and how many hadthe openings down. Figure 25To Win the Pot. The challenger

wins the pot when the shells landin any one of the following ways:• All four with the openings up• All four with the openings down• Two up and two down

52

Figure 25

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Igba-Ita from Nigeria TWO O R MORE P L A Y E R S

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The challenger takes all theshells in the pot and continues totoss four shells.To Continue. If the challenger

loses, he or she puts the fourshells into the pot. The nextperson becomes the challenger.To Finish. The winner is the

person who has the most shells. Ifat any point a player has too fewshells to play, he or she drops outof the game.

THINGS TO THINK ABOUTIs a shell just as likely to fall withthe opening up as with theopening down? To find out, tossjust one shell about thirty times.Keep a record of the outcomes.What is your conclusion? Repeatthe experiment. Is the result thesame or different?Do the same with a coin. Is the

result the same as with the shell?Compare Igba-Ita with the

Game of Dish (page 49). How arethey the same? How are theydifferent?

CHANGING THE RULESPlay the game by tossing threeshells. What combinations will winso that it is a fair game?

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During Hannukah, the JewishFestival of Lights which usuallyoccurs in December, children loveto spin the dreidel (DRAY-del), afour-sided top. They celebrate themiracle that happened more thantwo thousand years ago in theyear 165 B.C.E. (before theCommon Era), when the braveMaccabees recaptured the templeof Jerusalem from the Syrians.Although there was hardly enoughoil to keep the lamps burning justone night, somehow the oil lastedfor eight days.On each side of the dreidel is a

Hebrew letter, spelling the initialsof the message: nes gadol hayahsham. Figure 26aIn English the message is: “A

great miracle happened there,”and the letters are G, M, H, and T.

MATERIALS• Piece of cardboard, 3 inches (7.5cm)

square• Pencil, pen, or marker• Thin dowel rod, lollipop stick, or short

pencil• Plasticine or molding clay• Glue• 20 counters or coins for each player

MAKING THE DREIDEL1.Print one of the letters G, M, H,and T or the Hebrew lettersnun, gimel, hay, and shin alongeach side of the square.

2.To locate the center of thecardboard square, draw the twodiagonals of the square. Wherethe two lines intersect in thecenter, mark a dot.

3.Make a hole in the center andpush the rod or stick through it.

54

Figure 26a

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Spin the Dreidel TWO O R MOREP L A Y E R S

Jewish

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4.Press plasticine or clay evenlyaround the stick where it meetsthe square underneath so thestick stays in place. You maywant to use a little glue to holdthe stick in place as well. Figure 26b

PLAYING THE GAMEDecide in advance on the numberof rounds to play. Players gatheraround the table. Each player putstwo counters into the pot in thecenter.The first player spins the

dreidel and notes which letter isuppermost when the dreidelcomes to rest. The player scoresaccording to these rules (followeither the English or the Hebrewletters):• G or nun—player wins nothing• M or gimel—player takes theentire pot

• H or hay—player takes half thepot

• T or shin—player places acounter in the pot

Now, it is the next player’s turn.Again, each player places twocounters in the pot, and the gamecontinues.

At the end, the player with thelargest number of counters is thewinner. If a player has no morecounters before the game hasended, he or she is out of thegame, unless the other playersagree to let him or her continue.

THINGS TO THINK ABOUTShould the dreidel fall more oftenon one side rather than another?Spin the dreidel twenty times andmake a record of the outcomes.Suppose that one side of thedreidel is longer than the others.Or suppose that more clay hasbeen put on one side than theothers. How might that change theway a dreidel falls? Do you thinkthat the rules for scoring are fair?Why or why not?

CHANGING THE RULESInvent a different system ofscoring. Play the game with yourfamily or friends. Which system ismore fair?

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F igure 26b

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Children and grown-ups in Mexicolike to play Toma-Todo. They spina top in the shape of a hexagon, asix-sided figure. The name inSpanish is pirinola, or sometimestopa, which probably comes fromthe English word top. Try yourluck with this game.

MATERIALS• Compass• Pencil, pen, or marker• Piece of cardboard, at least 4 inches

(10cm) square• Ruler• Scissors• Dowel rod, lollipop stick, or pencil stub• Glue• Plasticine or molding clay• 10 or more chips for each player, the

same number of chips for each player

MAKING THE TOP1.Open your compass to a radiusof two inches (5cm). Draw acircle on the cardboard.

2.Using a ruler, draw a diameter(a line through the center of thecircle).

3.With the compass set at thesame radius, place the compasspoint at one end of the diameterand make two arcs on thecircle. Do the same at the otherend of the diameter.

4.Connect the points to make aregular hexagon; all the sideshave equal length and theangles have equal measure.

5.Draw the two other diameters.Figure 27a

6.Write the Spanish words (listedunder Playing the Game) ineach part of the hexagon.

7.Cut out the hexagon.8.Make a hole in the center of thehexagon and push the rod orstick through it to serve as thespinning axis. Glue the stick to56

Figure 27a

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Toma-Todo from Mexico TWO O R MOREP L A Y E R S

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the cardboard. Press a bit ofplasticine or clay around thestick where it meets thehexagon underneath so that itstays in place. It also adds somemass to the spinner so that itspins better. Figure 27b

PLAYING THE GAMEHere is the translation of theSpanish instructions:

SIDE SPANISH ENGLISH1 Toma uno Take one2 Toma dos Take two3 Toma todo Take all4 Pon uno Put one5 Pon dos Put two6 Todos ponen All put

Players sit in a circle around thetable or on the floor, each with apile of counters or chips. Theydecide in advance on the numberof rounds to play. Each playerplaces two chips in the pot in thecenter. The first player spins thepirinola once and notes the sidethat is uppermost when the topcomes to rest. The Spanish wordson that side tell what to do. Theplayer may be told to take one ortwo chips or all the chips from thepot. Or the player may be told to

place one or two chips into thepot. “Todos ponen” means thatevery player must place two chipsinto the pot.The next player in the circle

spins the top. Play continues inthis way. If at the end of a turnthe pot has fewer than threechips, each player adds two ormore chips to the center.If a player doesn’t have enough

chips to play, he or she drops outof the game.The player who has the most

chips at the end is the winner.

THINGS TO THINK ABOUTCompare the shape of the pirinolawith that of the dreidel (page 54).Compare the rules for the twogames. How does the shape of thespinner affect the rules of thegame?

CHANGING THE RULESInvent a game with a spinner inthe shape of a regular octagon. Toconstruct an octagon, see theinstructions for Shisima (page 4).What rules would you invent forthe game?

GAM

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Figure 27b

TOMA TODOPON

DOS

TOMA

UNO PON

UN

O

TOMA

DOSTODOS PONEN

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Here is a picture of the flag ofSouth Korea. Figure 28aIn the center of the flag is the

yin-yang symbol. Around it arefour signs. Each sign consists ofthree lines, either solid lines orbroken lines. These signs arecalled trigrams. What does trimean? (Think of words liketriangle and tricycle.) Each signhas a meaning. Figure 28bThese signs were invented in

China several thousand years ago.The Korean and Japanese peoplelater adopted them. Some peoplethought that they brought goodluck. The Koreans liked them sowell that they put them on theirflag.

THINGS TO DOThere are eight different trigrams.Draw the four missing signs. Whatthings in nature do you want themto stand for?Make a game with the eight

trigrams. Here is one way. Make

an eight-sided top or spinner inthe shape of a regular octagon.See Shisima (page 4) for instruc-tions on drawing the octagon.Figure 28cRead the rules for Toma-Todo

(page 56) to help you make up therules for your game.

THINGS TO THINK ABOUTTrigrams are like writing. Peoplecould read three solid lines as“sky” or “heaven.” What sign isopposite “heaven” on the flag?Compare the two sets of lines. Nowlook at the sign for “fire.” What isthe opposite of “fire” on the flag?Compare the arrangement of linesin these two trigrams.

58

Figure 28a

Figure 28b

Figure 28c

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Heaven Water Earth Fire

from East AsiaTrigrams & GoodLuck

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For more than two thousand yearsthe Chinese have used the bookcalled I Ching, or Book of Changes,to try to foretell the future. I Chinglists sixty-four different hexagrams,or sets of six solid or broken lines.A hexagram consists of twotrigrams (see page 58). Here aresome examples: Figure 29A person throws six sticks to

find out his or her specialhexagram. Each hexagram hassome wise advice to go with it. Aperson has to figure out howthese words of wisdom apply tohis or her problem. People use thehexagrams to help them solvetheir own problems.

THINGS TO THINK ABOUT AND DOHow many more hexagrams canyou draw? Try to draw all sixty-four hexagrams. Work out asystem so that you don’t miss any.Why do you think these figures arecalled hexagrams? Do you knowother words that start with “hexa”?

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F igure 29

TheHexagrams of I Ching from China

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S

Puzzles with Numbersome of the puzzles in thissection are very old. Thestories in them may have

been told by merchants travelingwith their goods through Asiancountries—India, China, the Arablands. In all these countries therewas great interest in mathematics.Muslim traders brought thesepuzzles to Africa.People who posed these puzzles

dressed them up to suit their owncustoms. A story about camels inone country might become aproblem about horses or sheep inanother place. A puzzle withletters of the alphabet used theletters that people knew (you willuse the Latin letters of the Englishalphabet). Magic squares haddifferent meanings in variouscultures and even used differentnumerals.

In this book several puzzleswere brought up to date. In twopuzzles we see that doubling thenumber of grains of rice over andover again is like counting yourancestors. The last two puzzlescompare the postal codes we usetoday with the markings thatAfricans made on bone abouttwenty-five thousand years ago!

S

5

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PUZZLES WITH

NUM

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Roy and Barbara use a secret codeto figure out the value of theirnames. Roy’s name has a value offifty-eight. Barbara’s name has avalue of forty-three. Roy’s namehas only three letters, whileBarbara’s has seven letters.Although Barbara’s name is muchlonger than Roy’s, the value issmaller. Does that surprise you?Roy and Barbara are using a

system that is very old. It goesback more than two thousandyears to the ancient Hebrews andGreeks. Instead of inventingsymbols for numbers, as theancient Egyptians had done, theHebrews and Greeks used theletters of their alphabets asnumerals. The symbols that weuse today, like 0, 1, 2, and 3,were invented in India andbrought to Europe much later byArabic-speaking North Africans.We call them Indo-Arabic (orHindu-Arabic) numerals.

The first two letters of theHebrew alphabet are aleph andbet. In the Greek alphabet the firsttwo letters are alpha and beta.You can guess where the Englishword alphabet comes from! TheHebrews and Greeks used thoseletters to stand for the numbers 1and 2.

MATERIALS• Sheet of paper• Pen or pencil

DOING THE PUZZLEWe will use the English alphabet.Copy the alphabet on a sheet ofpaper and write the value of eachletter under it. Use all thenumbers from one to twenty-six.Figure 30aRoy figured out the value of his

name this way:R=18, O=15, Y=25. 18+15+25=58.Barbara’s name adds up like

this: 2+1+18+2+1+18+1=43.

Secret Code Part I

A B C D E F G H I J K L M 1 2 3 4 5 6 _ _ _ 10 _ _ _

N O P Q R S T U V W X Y Z_ 15 _ _ _ _ 20 _ _ _ _ _ 26

Figure 30a

Ancient Hebrew & Greek

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THINGS TO DOHere are some ideas for having funwith numbers:What is the value of your

name? Figure out the values ofthe names of your friends andfamily members. Do longer namesalways have higher values?Here are words with one or two

letters missing. The value of thewhole word is given. Copy themand fill in the missing letters. Figure 30b

Here is a game you can playwith your friends. Each personwrites five words and figures outthe value of each word. Thenrewrite the words and their valueson another sheet of paper, butleave out one or two letters.Exchange papers and fill in themissing letters. Then check theanswers.How many words can you find

that add up to thirty? To forty-five? Select a number and find asmany words as possible havingthat value.

62

Figure 30b

PUZZLE

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BER

S

B I _ = 25 C _ T = 44

R E _ T = 62 _ A R M = 40

C _ M _ = 33

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Using letters as numbers can beclumsy. How would you write anumber like 278?The Hebrews and Greeks solved

that problem for numbers thatwent into the hundreds. This ishow they would have done itusing the English alphabet. We’llcall this the Letter-Numeralsystem. Figure 31aIn this system, ME=40+5=45BOY=2+60+700=762For larger numbers, they placed

bars or accents on the letters.

DOING THE PUZZLEThese words are actually Letter-Numerals. What is each number?Figure 31b

THINGS TO THINK ABOUTCan you use the Letter-Numeralsystem to spell more Englishwords? Why are there so fewwords?Invent a way to show numbers

like 1,000, 2,000, 30,000, andother large numbers.

PUZZLES WITH

NUM

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Figure 31a

Figure 31b

UP AM IT WE

COW SOB BOX

PIT TIP

Secret Code Part II Ancient Hebrew & Greek

A1

J10

S100

B2

K20

T200

C3

L_

U_

D4

M_

V_

E5

N_

W_

F6

O_

X_

G_

P_

Y700

H_

Q80

Z800

I_

R90

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Ahmed popped a peanut into hismouth just as his friend Daoudcame into view.“Let’s play Antelope and

Leopard,” called Daoud. “Here, Ihave three times fifteen stones.”Ahmed scooped out the earth

to make the nine holes in theground while Daoud emptied hispockets. Although Ahmed hadnever played the game himself, hehad watched many times as theolder boys challenged one anotherto construct the magic square.The goal was to place the forty-

five stones in the square of nineholes so that the sum of thenumbers of stones was fifteen in

each row, each column, and thetwo main diagonals. Each holemust contain a different numberof stones, and each of the num-bers from one to nine must beused just once. Figure 32a“You be the antelope; I’ll be the

leopard,” said Daoud. “If you can’tmake the magic square, I will eatyou! What will you put up—some-thing good to eat?”Ahmed held out a handful of

his precious peanuts. Daoud’seyes lit up, and he smacked hislips. Ahmed laid the peanutscarefully on the ground, next tothe three-by-three array of holes.

64

Figure 32a

PUZZLE

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BER

S

= 15 = 15

= 15

= 15

= 15

= 15 = 15 = 15

Magic Squares Part I WestAfricafrom

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Ahmed picked up fifteen of thestones. He remembered that themiddle number was always five.That left ten stones to be dividedbetween two holes on either sideof the middle square. But whichtwo holes? How should he split upthe remaining ten stones? Hadn’the seen the older boys place onestone below the five, and nineabove the five? It was worthtrying. He glanced at his friend’sface as he placed his stones in theholes. Daoud did seem ratherdisappointed. Was he worried thathe might not win the peanuts?Figure 32bAhmed decided to concentrate

on the bottom row as his nextmove. He already had one stone inthe middle square, so he would

need fourteen more stones for theother two spaces. Seven and sevenwould not be right, because eachnumber must be used only once.He picked up the fourteen stonesand divided them into two piles ofnine and five. No, that can’t beright. He had already used five inthe center and nine above it.Just then the skies opened up

and sheets of rain began to fall.Ahmed scooped up his peanuts,and both boys ran for shelter. In amoment their magic square waswashed away without a trace.Well, they could always makeanother one.

THINGS TO DOCan you finish the magic square?

PUZZLES WITH

NUM

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Figure 32b

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According to Chinese legend, thishappened more than four thou-sand years ago. The emperor wassailing down the River Lo with hiscourt. Suddenly a turtle appeared.On its back was a pattern of dots.Each part of the pattern had adifferent number of dots. Figure 33a

MATERIALS• Sheet of paper• Pen or pencil

DOING THE PUZZLEDraw a three-by-three square asshown here: Figure 33b. Count thenumber of dots in each part of thepattern in figure 33a. Place thosenumbers in the cells of the square.What number belongs in themiddle? What numbers are in thecorners?The Chinese thought this

square was magical. To find outwhy, add the three numbers ineach row, each column, and eachdiagonal—eight different sums.The sum is always the same

number: fifteen. This is called the“magic sum” of the “magicsquare.” The ancient Chinese andother peoples thought they couldtell fortunes by using magicsquares.

66

Figure 33a

PUZZLE

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BER

S

Figure 33b

= 15 = 15

= 15

= 15

= 15

= 15 = 15 = 15

Magic Squares Part II from China

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THINGS TO FIGURE OUTCopy the squares shown here:Figure 33c. Complete each square sothat the magic sum is fifteen. Useeach of the numbers from one tonine exactly once. Check all eightsums in each square.What number is in the center?

What numbers are in the corners?What numbers are in the twodiagonals?Copy the squares shown here:

Figure 33d. Complete each squareso that the magic sum is eighteen.Use each of the numbers from twoto ten exactly once. Check alleight sums in each square. What number is in the center?

What kind of numbers are in thecorners? What numbers are in thetwo diagonals?

Make two copies of each square.Use each of the numbers fromzero to eight exactly once.Complete each square in twodifferent ways. Figure 33eWhat is the magic sum? What

number is in the middle?Make a three-by-three magic

square using the even numbersfrom two to eighteen exactly once.What number should go in themiddle? What do you think will bethe magic sum?

PUZZLES WITH

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Figure 33c

Figure 33d

Figure 33e

5 16

1 93

8

4 2

6

627

3

5 92 10

4

3

542

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There are eight different three-by-three magic squares using thenumbers from one to nine exactlyonce each (see Magic Squares,Part I, page 64). You can make alleight magic squares by drawingjust one square in a special way.

MATERIALS• 2 sheets of plain paper• Pen or pencil• 2 colored markers ( black and red, ortwo other colors)

• Sheet of scrap paper• Ruler

DRAWING THE MAGIC SQUARE1.Draw a large square. Divide itinto nine small squares. Go overthe lines with a black or dark-colored marker. Figure 34a

2.With the markers copy thepattern of dots below. Make theeven numbers black and theodd numbers red (or two otherdifferent colors). Place scrappaper under the sheet of paper,because the marks will probablygo through the paper. Figure 34bHow many dots are in the wholemagic square?This pattern is similar to the

markings the Chinese emperor sawon the turtle in the River Lo morethan four thousand years ago,according to a Chinese myth (seeMagic Squares, Part II, page 66).Turn the paper over. If the lines

and dots don’t show throughclearly, go over them withmarkers. Now you have anothermagic square on the back of thesheet.68

Figure 34a

Figure 34b

PUZZLE

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BER

S

Magic Squares Part III

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THINGS TO DOTurn or flip your magic square sothat the dots match the numbersin the magic squares shown here:Figure 34cUse your dot-numeral magic

square to find the other five magicsquares. Draw them on a separatesheet of paper, using numeralsinstead of dots.

THINGS TO FIGURE OUTFor each magic square, state thenumber in the center. What is themagic sum of each square? Whatis the sum of all the numbers inthe magic square? Explain how allthese numbers are connected.

PUZZLES WITH

NUM

BERS

Figure 34c

8 6

4 2 2

6 8

41 9

3

7

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In most cultures, people have agreat deal of respect for theirancestors and other older people.Ancestors are your father andmother, your grandparents, andtheir parents and grandparentsgoing all the way back in time.Lucia’s parents and grand-

parents came to the United Statesfrom Mexico about twenty yearsago. She made a “family tree” thatlooked like this: Figure 35Dolores and Cesar are Lucia’s

parents. Dolores’s parents are Anaand Luis. Cesar’s parents areMaria and Julio. Lucia thinks ofher parents as being one gene-ration back in time from hergeneration. Her grandparents aretwo generations back fromLucia’s. In three generations,including herself, Lucia countsseven people.

How many great-grandparentsdoes Lucia have? How manypeople are in Lucia’s family tree infour generations, counting herself?Lucia wondered how many

ancestors she had going back tengenerations. That would take herfamily back to about the year1730, when some of her ancestorstraveled to Mexico from Spain.She made a table to organize hercalculations. Table 3

MATERIALS• Sheet of paper• Pen or pencil• Ruler

70

Figure 35

PUZZLE

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BER

S

Counting Your Ancestors

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DOING THE PUZZLECopy the table and help Lucia tocomplete it for ten generations.She decided that she had 1,022ancestors (not counting herself). Isthis correct?

THINGS TO THINK ABOUTHow far back in time can youcontinue the table? Study thenumbers in each column. Whatpatterns can you find? Is there away to figure out how manyancestors you have going backtwenty generations without havingto make a table?

PUZZLES WITH

NUM

BERS

Number of

generations

Number of ancestors in

this generation

Total

Table 3

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There is an old tale told in China,India, Persia (now called Iran), andother Asian countries. The story isabout a wise old man who showedthe king the solution to a problemthat no one in the king’s courtcould solve.“Ask for anything you desire,”

the king said to the wise man ingratitude.The wise man thought for a

while. Then he said to the king,“Your majesty, I do not needmuch. Just give me one grain ofrice today, two grains of ricetomorrow, and four grains of ricethe day after tomorrow. Each daygive me twice as many grains asthe day before until the end of themonth.”The king was surprised that the

wise man asked for so little, buthe ordered the treasurer of thekingdom to carry out the request.

THINGS TO FIGURE OUTFigure out how many grains of ricethe wise man received on theeleventh day. How many grains ofrice should the wise man receiveon the twenty-first day? Can youmake a good estimate withoutdoing a lot of arithmetic? On thethirty-first day the wise manshould receive well over a billiongrains of rice. Figure out the exactnumber. You can check youranswer in Chapter 10.

THINGS TO THINK ABOUTCould the treasurer count out somany grains? Were farmers able togrow that much rice?

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Rice Multiplies from Asia

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A merchant was riding home onhis camel after selling all his goodsat the town market. He stopped atan inn for the night. Just as hewas ready to go to sleep after histiring day, he heard loud voicesbelow his window. He went out tosee what was going on and foundthree men with a large herd ofcamels.“What can I do with half a

camel?” shouted the oldest man.“How can we split a camel into

little pieces?” complained another.As the argument went on and

on, the merchant wonderedwhether he would get any sleep.He decided to try to settle thematter.“Pardon me, good fellows,” he

said as sweetly as he could.“Perhaps I can help you solveyour problem.”

The three men started shoutingall at once. The merchant did hisbest to calm them down, and atlast he learned what the argumentwas about.The three men were brothers.

Their father had died, leavingthem his thirty-five camels.According to his will, the oldestbrother was to have half the herd.The middle brother would receiveone-third of the camels, while theyoungest would get one-ninth. Nomatter how they figured it out,using the instructions in the will,they always had to cut camelsinto pieces.“Ah, now I see how I can help

you,” said the merchant, a manwith a great deal of experiencewith numbers. “I will add my finecamel to your thirty-five camels.Then you will see how neatly youwill be able to divide the herd,without having to cut up youranimals.”

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Dividing the Camels from NorthAfrica

cutcut

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The brothers were happy tohave such wise advice. Theysmiled as the merchant dividedthe thirty-six camels among them,because each would receive morethan he expected. To the oldestwent half the herd—eighteencamels. The middle brother gotone-third, or twelve camels. Theyoungest received four camels,one-ninth of the herd. Altogether,the brothers received thirty-fouranimals, and two camels were leftover.“You must grant that I made a

fair division,” said the merchant.“Now I will take back the animal Ilent you. And I will take the othercamel as my reward for dividingthe herd among you to everyone’ssatisfaction.”

THINGS TO FIGURE OUTHow many camels would eachbrother have received if they haddivided thirty-five camelsaccording to their father’s will?Explain how the clever merchantcould divide the herd so that allthe camels remained whole andtwo were left over. What waswrong with the instructions in thefather’s will?

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An archaeologist digging in thevicinity of a village called Ishango,in the eastern part of Congo,found an unusual bone. A bit ofquartz was attached at one end,and the bone was covered withnotches. Who were the people thatmade these marks? When did theylive? What do the marks mean?Here are sketches of two sides

of the Ishango bone. Next to eachsketch is a set of numbers tellinghow many notches appear. Studythe numbers and decide howthey are related to one another.Figure 36Dr. Jean de Heinzelin, the

Belgian scientist who discoveredthe bone in the 1950s, thoughtthat it was about nine thousandyears old. Counting the groups ofnotches, he decided that theyshowed doubling—3 and 6, 4 and8—and prime numbers—11, 13,17, 19. A prime number isdivisible only by itself and one.Another scientist, Alexander

Marshack, examined the boneunder a microscope and decidedthat the notches marked the daysof a six-month lunar calendar, acalendar based on cycles of themoon.In the 1980s Dr. de Heinzelin

and a team of scientists wentback to Ishango to dig further.With new methods of dating suchmaterials, they came to asurprising conclusion. The bonewas twenty thousand to twenty-five thousand years old!

THINGS TO THINK ABOUTWhat evidence do you see fordoubling numbers? Do you findprime numbers? What evidence doyou see for a calendar based onthe cycles of the moon? The periodof one cycle is about 291⁄2 days.Can you combine groups ofnotches to make sums of twenty-nine or thirty?

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Figure 36

3

6

4

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19

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The Ishango Bone from Congo

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Usually printed on the bottom edgeof letters and magazines, we seerows of long and short lines. Wesaw similar-looking lines—notchesmade more than twenty thousandyears ago (see page 75)—on theIshango bone. What do they mean?Let’s look at the codes the

United States Post Office uses.Machines invented by smartscientists can read these codesand sort the mail. Imagine howmuch time and energy that saves!I have a return envelope sent by

the Children’s Defense Fund inBaltimore, Maryland 21298-9642.This combination of long and shortmarks is the code for the ninedigits in the address. Figure 37a

Years ago, when the post officestarted to use postal zip codes,five digits were enough. Later theyadded four more digits to specifyindividual buildings and groups ofbuildings and shorten the time ittook to process the mail. Here aresome five-digit zip codes. Figure 37b

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Figure 37a

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Postal Codes from the U.S.A.

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THINGS TO THINK ABOUT AND DOExamine the five-digit codes. Howmany marks are needed for eachnumber from zero to nine? Howmany are short and how many arelong?Make a list of all the numbers

from zero to nine. Write thecorrect code next to each number.Are all the codes different fromone another?Write all the different ways you

can arrange three short marksand two long marks. How manyways have you found?Look at the nine-digit zip code

of the Children’s Defense Fund.Here are some instructions forreading the code. Ignore the firstbar and the last bar. You nowhave fifty marks for ten digits. Thetenth digit is a checking number.You may ignore it. Can you readthe code for the nine remainingdigits? Does your reading agreewith the numerals in the address?

Can you figure out the systemused by the inventors of the postalcode? Find an envelope with apostal code printed on it. Can youread it? You may want to rewriteit using larger marks so that youcan read it more easily.

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Figure 37b

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Puzzles Without Numbers

any puzzles have nonumbers at all. Theyrequire logical thinking.

You have to use your brain tofigure out patterns.This section has two kinds of

puzzles. The first type is aboutcrossing the river with differentkinds of cargo—people, animals, or food. Some of these people,animals, or food cannot be leftwith the others, but the boat istoo small for everything to fit intoit at once.

The second type of puzzle iscalled a network puzzle and itcomes from two different groupsof people in Africa. Both groups ofpeople told stories and madedrawings in the sand but fordifferent reasons. You will prob-ably make your drawings, ornetworks, with paper and pencil.Some of the puzzles are a realchallenge!

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Jonah stood on the bank of theriver and wondered what to do.Next to him stood a fox. Nearbywas a duck and a bag of corn. Hehad to row them all across theriver before nightfall. But his boatwas too small to hold everything. Itcould carry only two thingsbesides Jonah.Jonah would have to leave one

thing behind. But if he left the foxwith the duck on the other side ofthe river, the fox would soon makea meal of the bird. He couldn’tleave the duck with the cornbecause the duck would eat thecorn.Jonah knew that he must make

more than one trip. How could heget the fox, the duck, and the cornacross the river safely?

This puzzle was told by AfricanAmericans living in the SeaIslands off the coast of SouthCarolina. For more than onethousand years people have told“crossing the river” puzzles.Probably they really had to figureout a way to cross the river withtheir load. They made a puzzle outof a real task. People changed thenames of the objects to suit theirown cultures and to make thepuzzle more interesting.

Crossing the River in theSea Islands

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SOLVING THE PUZZLEOne way to solve this puzzle is toact it out with three other people.Each of you wears a sign showingwhich character you are playing.First discuss a plan of action, thensee whether it works. If not, tryanother way to row all the objectssafely across the river.Another way to work out a

solution is to write the name ordraw a picture of each characteron a bottle cap or piece ofcardboard. Use a jar lid as theboat. Copy this diagram andfinish it, to show how many tripsJonah had to make. Figure 38Is there more than one way to

solve the puzzle? You may checkyour answers in Chapter 10.

CHANGING THE RULESSuppose that the boat can carrytwo items besides Jonah, but notwo objects can be left alonetogether. Plan how he manages toget all the objects to the other sideof the river. How many trips musthe make?

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Figure 38

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Side One Side Two

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Here is one way they tell thecrossing-the-river puzzle inLiberia, a country in West Africa.A man has a leopard, a goat,

and a bundle of cassava leaves.He must get them all across theriver, but his boat can carry onlyone object besides the manhimself. If he leaves the goat withthe leopard, the goat will soon beeaten. He cannot leave the goatwith the cassava leaves becausethe goat will make a meal of them.

What is the fewest number oftrips that the man must make toget all three objects across theriver? Can it be done in seventrips? How many different planscan you think of? You may checkyour answers in Chapter 10.

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Crossing the River in Liberia

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This version of the puzzle comesfrom Kenya, a country in EastAfrica. It is far more complicatedthan the first two crossing-the-river puzzles.Three men have recently

married. The three couples aregoing to the market across theriver. There is only one boat, and itcan hold only two people. But noman will leave his wife withanother man, either in the boat oron shore. Fortunately, all threemen and all three women can row.

Show how they all get across.They will need to make at leasteleven trips if a wife cannot bewithout her husband but withanother man as they transferbetween boat and shore. You maywant to ask a grown-up to workon the puzzle with you. You maycheck your answers in Chapter 10.

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Crossing the Riverwith Jealous Husbands from Kenya

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I was looking through Wingate’s A Treatise of Common Arithmetic, a book used in schools in theAmerican colonies more than twohundred years ago. There I foundtwo river-crossing puzzles. Onewas like the puzzle from Liberia,but with different characters. Afarmer had to carry a fox, a goose,and a peck of corn across theriver. His boat could hold only oneobject besides the man himself. Hecouldn’t leave the fox with thegoose, or the goose with the corn.How many trips did he make?The second riddle was a real

surprise. Here is the exactwording that I read in the book:

Three jealous husbands withtheir wives, being ready to passby night over a river, do find atthe riverside a boat which cancarry but two persons at once,and for want of a watermanthey are necessitated to rowthemselves over the river atseveral times. The question is,how those six persons shall passtwo by two, so that none of thethree wives may be found in thecompany of one or two men,unless her husband be present?

Here was the same puzzle aboutthe jealous husbands that wefound in Kenya!

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Crossing the River in Colonial America

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Word was out. Grandfather wasabout to tell a story and draw alusona, a sand drawing, toillustrate it. The children hurriedto the big shade tree and formed acircle around the old man. Whenthey were all seated, grandfathercleared a space on the ground infront of him. With two fingers hemade a pattern of dots in thesand. Figure 39aThen he began to tell the story.

As he was talking, he traced thispattern in the sand with his indexfinger. Figure 39bA swallow built her nest

between the four branches of atree. A snake heard the little birdscheeping and decided to get a lookat them. He glided close to thenest and went around the firstbranch. He glided on, wentaround the second branch, thenthe third and the fourth. Now hewas back at his starting point. Hekept going round and round thefour branches over the next few

days. One day he didn’t hear thelittle birds and he decided to seewhat happened. When he finallywent into the nest, he found itempty.

THINGS TO THINK ABOUT AND DOWhy did the snake want to findthe nest? Why couldn’t he go intoit? Where were the little birdswhen the snake finally went intothe nest?On a sheet of paper, draw the

pattern of dots. Then draw thepath of the snake as it wentaround the four branches. Don’tlift your pencil from the paper orretrace a line. After you havepracticed drawing the path, tellthe story to your friend or yourlittle sister or brother as you drawthe lusona.

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Figure 39a

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Figure 39b

The Snake & The Swallow’s Nest fromAngola

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The Chokwe people live in thenorthwestern part of Angola, acountry in southwest Africa.Before the time of radio, television,and schools, children would learnabout the ways of their peoplefrom the older folks. When workwas done, they would meet aroundthe fire or under the shade of alarge tree. Children learned abouttheir history, how to get along wellwith other people, and howanimals behaved. As the elder toldthe story, he would draw sona,patterns in the sand. A singledrawing was a lusona. We callsuch patterns “networks.” Anetwork is a system of lines thatconnect points. Think about TVnetworks, telephone networks,networks of streets and highways,and the Internet. Networkssurround us!First the storyteller would make

a pattern of dots in the sand toguide him. Then he drew thelusona in a continuous line as he

spoke. He never lifted his fingerfrom the sand, and he neverstopped talking as he drew thelusona.

A RIDDLEThis lusona is a riddle. First thestoryteller made this pattern ofdots. Figure 40aThen he drew this figure around

the dots. Figure 40bCan you guess what it is? Here

are some hints. It can sendmessages; it can keep the rhythmfor dancing; and the top andbottom are covered with skins.

AN ANIMAL STORYAnother storyteller was tellingchildren about an antelope thatwas running away from a leopard.The antelope left marks on theground as it ran. Here is one pawmark. Figure 40c

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Figure 40a

The Chokwe Storytellers from Angola

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MATERIALS• Sheet of paper• Pencil

THINGS TO THINK ABOUT AND DOCompare the two sona. How arethey the same? How are theydifferent?The answer to the riddle is a

two-headed drum. Skins arestretched across the top and thebottom. Does the drawing looklike a drum? Does the seconddrawing look like a paw mark?

Place a thin sheet of paper overthe lusona. Using a pencil, tracelightly over the pattern that showsthrough the paper. Try to tracewithout lifting your pencil from thepaper or going over a line morethan once. You may cross a line.On a separate sheet of paper,

draw the pattern of dots. Thencopy the network without liftingyour pencil from the paper orretracing a line. You may cross aline. Practice until you can do itwell.

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Figure 40c

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If you were to visit a Chokwehome, you might see sonadrawings on the walls. Men,women, and even children enjoymaking these patterns. Here aresome sona that you might find onthe walls. Figure 41

MATERIALS• Sheet of paper• Pencil

THINGS TO DOLay a thin sheet of paper over alusona so that you can see thepattern. Trace it lightly in pencil.Draw some or all of the sona

on a separate sheet of paperwithout lifting your pencil fromthe paper. First make the patternof dots. Work in pencil, so thatyou can erase any lines that don’tcome out right. Then go over thelines with a marker or pen.Make up a story to go with a

lusona and tell it to a friend asyou draw.

Draw four or five kumbi birds inflight.Make up your own lusona to go

with a story. Write the story anddraw the lusona. You and yourfriends can make a book of storiesand riddles, with sona to illustratethem.

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Figure 41

Decorations

Three kumbi birds in flight

A kumbi bird in flight

Two birds and their young

A bat in flight

on the Walls from Angola

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Most sona are larger and morecomplicated than those in thisbook. Here is the story about thebeginning of the world, and thebeautiful lusona that illustrates it.Once upon a time Sun went to

pay his respects to the godKalunga. He walked and walkeduntil he found the path that led toKalunga. When he arrived,Kalunga gave him a rooster andsaid, “See me in the morning.”In the morning the rooster

crowed and woke Sun. Then Sunwent back to see the god Kalunga,who said, “I heard the roostercrow, the one I gave you forsupper. You may keep him, butyou must return every morning.”That is why Sun goes around theearth and appears every morning.

Moon also went to visit Kalunga.He too received a rooster, and itwoke him the next morning. Whenhe returned with the roosterunder his arm, Kalunga said, “Isee that you did not eat therooster I gave you yesterday. Thatis good. You must come back tosee me every twenty-eight days.”That is why we see the full moonevery twenty-eight days.Man went to see Kalunga and

was given a rooster. But Man wasvery hungry after his long trip. Heate part of the rooster for supper.The next morning the sun wasalready high in the sky when Manawoke. He quickly ate the rest ofthe rooster and hurried to see thegod Kalunga. Kalunga said to himwith a smile, “Where is the roosterI gave you yesterday? I did nothear him crow this morning.”Man was afraid. “I was very

hungry and I ate him,” he said.

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How theWorld Began fromAngola

Figure 42

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Then Kalunga said, “That is allright, but listen. You know thatSun and Moon have been to seeme. Each of them received arooster, just as you did, but theydid not kill theirs. That is whythey will never die. But you killedyours, and so you must die as hedid. And, at your death, you willcome back to see me.”And so it is. Haven’t the sun

and the moon always appeared,just as in the days of our great-grandparents? But men andwomen will not live forever. Figure 42

MATERIALS• Sheet of graph paper• Pencil• Markers or crayons

THINGS TO DOWith your finger, trace the paths ofSun, Moon, and Man. The arrowshows you where to start. Followeach line as far as it goes beforechanging direction. First you willtrace one-half of the pattern, thenthe other half. Can you trace thewhole design without taking yourfinger off the paper? Follow the instructions on

page 86 for tracing the patternand then drawing it. It won’t beeasy!Many sona are even more

complicated than this example.The storytellers learned to drawthem from their fathers andgrandfathers. Often they kept thisinformation a secret from thoseoutside the family. The peoplewho invented these beautifuldesigns must have been veryclever!

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Many years ago, Emil Torday, aHungarian anthropologist, wasvisiting the Kuba people in Congo.He admired their beautiful wovencloth and wood carvings. Theylived near the Kasai River, andmany families made their livingfrom fishing. Figure 43a One day Torday came upon a

group of children drawingpatterns in the sand near theriver. They invited him to sitdown. They asked him to drawtwo designs without lifting hisfinger from the sand or going overa line more than once. Thepatterns in these networks arelike the fishing nets that theirparents made. This is one of thosedesigns. Figure 43bTorday wrote in his book: “I was

at once asked to perform certainimpossible tasks. Great was theirjoy when the white man failed toaccomplish them.” At last theyshowed him how to do it.

Let’s look at the network. It hasten small squares in the longestrow and the longest column. We’llstart with small networks of thesame type. Then we will build upto the larger ones. Let’s look at some simple

networks. Figure 43c The firstnetwork has two small squares inthe longest row. The second hasthree small squares, and the nexthas four small squares in thelongest row.

MATERIALS• Sheet of graph paper• Pencil• Markers or crayons

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Figure 43b

Figure 43a

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Children’s Networks from Congo

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THINGS TO THINK ABOUT AND DOCopy the smallest network infigure 43c without lifting the pencilor going over a line more thanonce. You may cross the lines. Arethere special starting points andfinishing points, or can you startanywhere? Try different startingpoints. Don’t erase your work,even if it doesn’t come out right.You want to find out what worksand what doesn’t work.Following the instructions for

drawing the smallest network,draw the next largest network,with three small squares in thelongest row. Then draw the nextone. The neatest method is todraw each line as far as possiblebefore it changes direction. Markthe starting point and thefinishing point of each network.What can you say about thestarting point and the finishingpoint of each network? Can youspot them without having to trydifferent points?

Draw a network having fivesmall squares in the longest row.Continue with the larger networks,each one having one more smallsquare in its longest row than thelongest row in the network beforeit. Mark the starting and finishingpoints.Draw the network that the Kuba

children drew in the sand—onewith ten small squares in thelongest row. Be sure not to liftyour pencil or go over a line morethan once. You may cross thelines.Color the squares to make

beautiful patterns.Here is a challenge. Draw a large

network on colored constructionpaper. Then glue yarn of a con-trasting color over the lines. Youshould be able to do the wholenetwork with one long strand ofyarn.

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Figure 43c

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Geometry All Around Us

ook around you. How manyround things do you see?Many things in nature are

round. Look at the moon when itis full. In pictures, the sun isusually depicted as round. Earthand other planets are round, oralmost round.Many people have thought that

the circle is the most perfectshape. No matter how you turn it,it always looks the same. Untilscientists learned more aboutnature, people believed that theorbit of a planet around the sunwas circular. Now we know thatthe orbit is an ellipse, a flattenedcircle.Black Elk was a wise man of the

Lakota people, Native Americanswho live on the Great Plains. Hereare his words, spoken many yearsago:

Everything the Power of the Worlddoes is done in a circle. The skyis round, and I have heard thatthe earth is round like a ball, andso are all the stars. . . . The suncomes forth and goes down againin a circle. The moon does thesame, and both are round. . . .The life of a man is a circle fromchildhood to childhood, and so itis in everything where powermoves.

L

Figure 44

7

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Why are so many things innature round? For example,flowers are round in their overallshape. You have never seen asquare flower, have you? Thinkabout the way flowers grow—outfrom a center. Anything thatgrows evenly out from a centermust be round. What parts ofyour body are round, or nearlyso?Many objects that people make

are also round. Look at dishes,cans, jars of food, and otherthings in the kitchen. Try toimagine wheels that are notround! If you look around you,you will find many more round oralmost-round objects. Figure 44

It is no surprise that peoplemake round decorations. Andwhen they invent designs thatstand for events or ideas, theyoften use circles.Some people build houses that

are round. In this section you willlearn why people build roundhouses, and you will read aboutsome examples.

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This is the symbol of the Olympicgames. Symbols are designs thatstand for an idea or an event. Thefive circles stand for the fivecontinents. Most of the countriesof the world take part in thegames. They are held every fouryears in different countries. Youcan make an Olympic flagshowing this symbol. Figure 45

MATERIALS• Sheet of plain white paper• Pencil• Compass• Markers or crayons in 5 colors (black,blue, green, red, and yellow)

DRAWING THE OLYMPIC SYMBOL1.Copy the symbol in pencil. Usea compass to draw the circles. Ajar or bottle cap or a coin maywork as well.

2.Go over the pencil mark of eachcircle with a colored marker orcrayon. Make each circle adifferent color.

THINGS TO DONow make another copy of thesymbol. Use all five markers orcrayons to color it differently fromthe first. How many differentarrangements of the five colors arethere? If you have a lot ofpatience, you may be able to find120 different arrangements!

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TheOlympic Games Symbol

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The yin-yang symbol is Chinese.The symbol stands for the unity ofopposites in nature. For example,light and darkness combine toform a union of opposites. Somethink that the yin half is feminineand the yang half is masculine.Wetness and dryness is anotherexample of opposites. What otheropposites in nature can you thinkof? Here’s how to draw a yin-yangsymbol. Figure 46a

MATERIALS• Sheet of plain white paper• Pencil with eraser• Compass• Ruler• Marker or crayon

DRAWING THE YIN-YANG SYMBOL1.Mark a point in the center ofthe paper. Place the compasspoint on that mark.

2.Draw a circle.3.Draw a diameter lightly inpencil.

4.Use the ruler to find themidpoint of each radius andmark it. Figure 46b

5.Change the opening of thecompass so that it is equal tohalf the radius of the largecircle. Draw a semicircle fromthe midpoint of each radius.

6.Erase the diameter you drew inpencil.

7.Color one-half of the yin-yangsymbol.

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Figure 46a

Figure 46b

The Yin-Yang Symbol fromChina

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The Ojibwa people, Native Amer-icans of the Great Plains, have atradition they call bawa ji guunahbee, which means “dream catch-er.” I learned about this traditionfrom a mathematics teacher at auniversity, the daughter of anOjibwa elder. She also gave me theinstructions for making the dreamcatcher. Figure 47aThe dream catcher was hung on

a baby’s cradleboard. Bad dreamswere caught in the web, just as aspider’s web catches and holdseverything that touches it. Gooddreams would slip through theweb and make their way down thefeather to the sleeping child. Youcan make your own dreamcatcher.

MATERIALS• 12-inch (30cm) willow twig• 4 feet (1.2m) of waxed string ordental floss

• 1 feather• 4 beads (1 white, 1 red, 1 black, and1 yellow)

If you cannot find all thesematerials, you may substituteothers. Think about what wouldbe suitable. You might use plaincord instead of waxed string.Instead of willow, use a thin twigfrom a different tree or bush.Soak it in water for a while tosoften it before you bend it into acircle.

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The Dream Catcher NativeAmerican

Figure 47a

Figure 47b

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MAKING A DREAM CATCHER1.Bend the twig into a circle andwrap the ends with a piece ofthe string.

2.Tie four strands of string acrossthe frame, evenly spaced. Eachstrand marks a diameter of thecircle.

3.Tie all the strands together inthe center. Figure 47b

4.Tie one end of a two-foot stringto one of the cross strands. Thisis the weaving string. Start toweave. Figure 47c

5.As you weave, wrap the stringtightly around the strand in afigure-eight knot. Follow thearrows and the numbers in thisdiagram. Figure 47d

6.Tie a feather onto a short pieceof thread. String the four beadsonto the thread, and tie thethread to the center of the web.

THE DREAM CATCHER AS A SYMBOLThe circle is central in the lives ofmany Native American people.Here it represents the harmony ofMother Earth and nature. Thestring stands for the path of life.The beads are for the fourdirections. White is for the north,red is for the south, yellow is forthe west, and black is for the east.The feather is for the effort neededto overcome the hardships of life.

THINGS TO THINK ABOUT AND DOWhy do you think the circle is soimportant in the culture and lifeof many Native Americans?Write a story to go with the

dream catcher. Draw a dreamcatcher to illustrate your story.Read the story to a young child orto a friend. You may also want toread the lovely book DreamCatcher by Audrey Osofsky.

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2

1 5

3

Figure 47c

Figure 47d

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At the beginning of this sectionabout circles, you read the wordsof Black Elk, the Oglala Lakotaelder. The government had forcedhis people, the Lakota nation (alsocalled Sioux), to resettle on areservation. This is what BlackElk said about the way they hadto live:

We made these little gray housesof logs that you see, and theyare square. It is a bad way tolive, for there can be no power ina square. . . . Our tipis wereround like the nests of birds, andthese were always set in acircle, the nation’s hoop, a nestof many nests. . . . The Native Americans of the

Great Plains were nomadic. Theyfollowed the buffalo herds as theywandered across the grassy plainsin search of food. The buffalosupplied most of the people’sneeds. Every spring the women ofthe family made a new tipi ofbuffalo hides. When the people

moved to a new location, theytook apart the tipi and bundled itonto a travois, a kind of sled,along with their other posses-sions. Strong horses pulled theload.The tipi is considered the most

perfect tent. It is often thought ofas the “good mother who sheltersher children.” The four main polesare called the grandfathers. Theypoint to the four directions. To theeast are the moon and stars. Tothe south are the sky and themountains. To the west are theanimals. To the north are thepeople. The pole to the north, likethe people, needs the support ofthe other three poles. Figure 48aTipis faced east to catch the

rays of the rising sun and to keepout the wind that blew from thewest. The flaps were opened to letout the smoke, or closed to keepout the rain.

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TheTipi Native American

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Today many Native Americanslive in cities and towns, much likeother Americans. They use the tipimainly for ceremonies. You canmake a paper model of a tipi. Youmay decide how large you want tomake it, or you may follow thedirections below.

MATERIALS• White, yellow, or tan constructionpaper

• Compass or bowl with 8-inch diameter• Pencil• Scissors• Colored markers or crayons• 6-inch length of string• 41⁄8-inch dowel rods, about 6 inches(15cm) long

• 6-inch (15cm) square of Styrofoam• Tape• Ruler

MAKING A TIPI1.Use this pattern as a guide.Figure 48b Lay the bowl on thepaper or use a compass todraw a semicircle (half circle)with an eight-inch diameter.With a pencil draw semicirclesfor the opening, and “wings” tomake the flaps. Cut out thetipi.

2.Draw a design on the tipi andcolor it. Red, yellow, black, andgreen are good colors. Fold backthe flaps.

3.With the string, tie the ends ofthe dowel rods together. Setthem up on the Styrofoamsquare. Figure 48c

4.Fit the tipi around the rods.These are the poles that supportthe tent.

5.Push one end of each rod intothe square of Styrofoam. Thenwrap the tipi around the polesand tape the edges together.

THINGS TO THINK ABOUT AND DOBlack Elk said there is no powerin a square. What did he mean?Where is the power, in Black Elk’sview? Do you agree with him?Now that you have made a

small paper tipi, you might wantto make a larger one out of clothor heavy plastic material. Youmight enlarge the pattern to makeeach measurement twice or threetimes the size of your paper tipi.

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Figure 48b

Figure 48c

DoorwayDoorway

Flaps

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Rodah and her husband lived inKenya, in East Africa. They hadjust bought a small plot of land inthe hills near Nairobi, the capital.The land had once belonged toEuropeans, who had taken thebest land for themselves and hadset up enormous farms. WhenKenya became independent fromGreat Britain in 1963, manyEuropeans left the country. Someof this land was divided into smallplots and sold to the people ofKenya.Rodah and her husband, with

the help of their friends, builtthree small rectangularbuildings—one for cooking, onefor eating, and one for sleeping.When everything was finished,Rodah invited her grandmother tovisit for a few weeks.“Do you live in a house with

corners?” her grandmother asked.“Yes, we do,” replied Rodah. Of

course, a rectangular house hascorners.

Her grandmother shook herhead sadly. “I’m sorry, but I can’tvisit you. I have always lived in around house with a center pole.There is no pole to support yourhouse. Besides, I would get lost insuch a house.”Rodah’s grandmother had

grown up in a house built in thestyle of the Kamba people ofKenya. The trunks of young treeswere pushed into the ground in acircular arrangement. They weretied at the top to a thick post.Hoops of wood were placed on thetrees. Then the whole structurewas covered with thick layers ofgrasses, called thatch. A greatdeal of hard work went into thebuilding of a Kamba house.

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Round Houses in Kenya

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The Kikuyu people also live inKenya, not far from the Kamba.Their homes are circular, withwalls that are parallel. They arewhitewashed to gleam in thesunlight. The roof is shaped like acone and is covered with a thickthatch of grasses. This style iscalled a “cone-cylinder” house.Many different peoples build cone-cylinder houses. Some houses aretall and narrow, while others areshort and wide. The style dependsupon the materials that peoplefind in their environment, theclimate, the terrain (hilly or flatland), and the cultural andhistorical traditions of the people.You can make a model of such ahouse from paper. Figure 49a

MATERIALS• Several sheets of construction paper• Scissors• Tape• Compass or bowl• Pencil• Brown crayon or marker, or clumps ofreal grass and some glue

MAKING A CONE-CYLINDER HOUSE1.The walls of the house are inthe shape of a cylinder. Cut outa paper rectangle. Cut out adoorway. Tape the edgestogether to form a cylinder.Figure 49b

2.With the compass or the bowl,trace a large circle and cut it out.

3.Draw a radius in the circle. Cutalong the radius.

4.Slide one edge under the otherto form a cone. You can decidehow steep you would like theroof to be. It should cover thewalls and overhang a bit.

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Cone-Cylinder Houses inKenya

Figure 49a

Figure 49b

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5. Draw brown grass on the roof,or glue real grass to the roof.Tape the edges. Figure 49c

6.Tape the cone to the cylinder.

THINGS TO THINK ABOUT AND DOWould you feel comfortable in around house? Why or why not?You might enjoy reading The

Village of Round and SquareHouses by Ann Grifalconi. It takesplace in Cameroon, a country inWest Africa, and tells why somehouses are round and others aresquare.Experiment with several

different styles of cone-cylinderhouses. Figure out themeasurements that will give youthe style you have in mind. Some people build the walls of

their houses using sun-driedbricks that they make themselves.You can shape clay into smallbricks and build a model of sucha house. Place a cone-shapedthatched roof on the walls. Some people who used to build

round houses have now changedto a rectangular shape. Thesehouses often have metal roofs.Can you think of reasons for thesechanges?A few years ago the Kenya

International Conference Centrewas erected in Nairobi, the capitalof the country. The main buildingis thirty-two stories tall. Are yousurprised to learn that the shapeof the building is cylindrical? Nearit is a large, low building, alsocylindrical, with a roof in theshape of a cone. To complete theconference center is a statue ofJomo Kenyatta, a member of theKikuyu ethnic group and the firstpresident of Kenya after it wonindependence from Great Britainin 1963. What factors do youthink inspired the builders tochoose these shapes for thebuildings? Figure 49dLook around you. How many

straight lines, squares, andrectangles do you see? What is theshape of the room you are in? Areyou sitting at a rectangular tableor desk, on a chair with straightlegs? Many of the objects that wemanufacture have straight lines.

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Figure 49c

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The Tangram puzzle is fun for allages. It was first mentioned abouttwo hundred years ago in China,where it is played mainly bychildren.The puzzle consists of seven

pieces, called tans. Each piece is apolygon, a closed figure withstraight-line sides. There are fivetriangles of three different sizes,one square, and one parallel-ogram. With a good imaginationyou can form hundreds of figureswith these polygons.

MATERIALS• 6-inch (15cm) square of heavycardboard or Styrofoam

• Ruler• Pencil• Scissors

MAKING A TANGRAM1.Draw lines 11⁄2 inches apart todivide the cardboard orStyrofoam square into sixteensmall squares.

2.Draw the polygons as in thediagram. The numbers on thediagram tell you the order inwhich to draw the polygons.Figure 50a

3.Cut out the seven polygons.

THINGS TO THINK ABOUT AND DOYou will need to know the manyways the different figures fittogether. Compare the polygons.Which triangles have the samesize and shape?How many small triangles have

the same area as the mediumtriangle? As a large triangle? Asthe square? As the parallelogram?Try to cover each larger figurewith small triangles.Describe the angles of each

triangle, of the square, of theparallelogram. Each angle of a

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Figure 50a

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Tangram Polygons from China

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square measures 90 degrees.Compare the other angles with the90-degree angle.Here are several shapes and

figures made with tans. Copy eachfigure with your tans. Figure 50bInvent a tangram figure. Lay it

on a sheet of paper and trace it.Ask your friend to make atangram figure like the one youdrew on paper.

Read Grandfather Tang’s Storyby Ann Tompert. Grandfather tellshis granddaughter a story aboutthe fox fairies that can changetheir shape to avoid being caught.Write your own story andillustrate it with tangrams.

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Figure 50b

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Among the Seven Wonders of theWorld in ancient times was theGreat Pyramid in Egypt. It wasbuilt about 4,600 years ago forKing Khufu and is the largest ofthe three pyramids that still standon the Giza plateau. About2,300,000 stone blocks, weighinga total of six million tons, had tobe cut from the quarries andmoved into place. Inside thepyramid are several passages andchambers. Each face of thepyramid is a triangle, and thebase is a square measuring 756feet (230m) along the edge. Theheight, 481 feet, is about that of afifty-story modern building.It is estimated that one hundred

thousand workers needed morethan twenty years to carry out thejob. There were skilled architects,masons, accountants, scribes,stone cutters, and people who didthe heavy work of dragging thestones up steep ramps. Most ofthese people were farmers who

worked during the three monthsof the year when the Nile Riverflooded the plains and farmingwas not possible. What atremendous amount of skill wentinto the planning, organizing, andbuilding of a pyramid!The five largest pyramids were

built within one century. Later theEgyptians constructed greatcolumned temples, as did theGreeks many centuries later. You,too, can construct your ownpyramid. Figure 51a

MATERIALS• Several sheets of construction paper• Pencil• Ruler• Scissors• Tape

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The Pyramids of Ancient Egypt

Figure 51a

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MAKING A PYRAMIDStart with a small pyramid. Makea net like the one shown in thediagram. Measure carefully. Figure 51b1.Draw a three-inch (7.5cm)square.

2.Find the midpoint of each side.3.Draw a 21⁄2-inch (6.5cm) lineperpendicular (at right angles)to each side of the square.

4.Draw the two sides of eachtriangle.

5.Cut out the net.6.Fold the net so that the tips ofthe triangles meet at the vertex(tip) of the pyramid.

7.Tape the sides of the trianglestogether.

Now that you have made a model,try using cardboard or Styrofoamto make a larger pyramid. Youmight multiply the measurementsabove by two, three, or four,depending upon how large apyramid you want to build. Youmay have to cut out each triangleseparately and then tape thewhole structure together.

THINGS TO THINK ABOUT AND DOWhy do you think the ancientEgyptians built such impressivestructures?Find pictures of the Egyptian

pyramids and of the columnedtemples that they built later,especially the temple of thepharaoh Rameses II.Read Anne Millard’s book

Pyramids: Egyptian, Nubian,Mayan, Aztec, Modern. As thebook title tells you, many peoplehave built pyramids. Among themost recent pyramids is theTransamerica building in SanFrancisco, made to withstand thefrequent earthquakes in thatregion. It is 853 feet (260m) high.

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Figure 51b

21/2 inches3-inch base

Vertex

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The Parthenon is one of the mostbeautiful buildings of all time. Itwas built in Athens almost 2,500years ago as a temple to the Greekgoddess Athena. It has forty-sixcolumns altogether, and manymarble sculptures. Sadly, thislovely building is in terrible shapedue to the effects of pollution andwar. Many of its sculptures, whichare now called the Elgin marbles,were removed to the BritishMuseum.This classical Greek style of

architecture was very popular inthe United States in the early daysof the republic. The Capitol inWashington, DC, and many otherpublic buildings are good examplesof this style.The diagram shows one side of

the Parthenon inside a rectangle.The ancient Greeks thought thisrectangle was one of the mostpleasing shapes because of theratio of the longer side to theshorter side. They called it the

golden ratio, and the shape wascalled the golden rectangle. Figure 52What is the golden ratio? To

find out, measure the long sideand the short side of therectangle. Divide the largernumber by the smaller number. Isthe answer close to 1.6? Thelonger side measures 1.6 timesthe shorter side.

THINGS TO DOMake a picture. Cut a sheet ofpaper in the shape of a goldenrectangle. You might make thelonger side eight inches (20cm)and the shorter side five inches(12.5cm), or else double thosedimensions. Draw anything youlike. Draw a border around thepicture like a frame. Do you likethe shape?Conduct a search. Try to find

objects that are in the shape ofthe golden rectangle.

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Figure 52

The Parthenon in Greece

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About one thousand years ago theAnasazi people built Pueblo Bonitoin Chaco Canyon, in the regionthat is now the state of NewMexico. The Anasazi are theancestors of the Pueblo Indians,Native Americans of the South-west. Until the nineteenth century,Pueblo Bonito was the tallestapartment building in the UnitedStates. It had five stories andabout six hundred rooms.The word pueblo is Spanish and

means both “people” and “village.”The Pueblo Indians still live inArizona and New Mexico andbuild their homes in the style oftheir ancestors. The buildingmaterials are bricks made ofadobe, a mixture of clay, water,and straw. Then the structure iscovered with smooth clay. Thisconstruction serves as airconditioning and is perfect for thehot climate of the region.

The Native Americans built onelevel at a time. To support the roofor ceiling of each level, they usedlogs, which you can see projectingfrom the building. The next levelwas laid on top of the one below,but with room for a landing orplatform. Ladders helped them toclimb to the upper levels. Build amodel pueblo home yourself.

MATERIALS• Several boxes of different sizes• Scissors• Black paper• Glue or tape• Twigs or small round sticks• Toothpicks and popsicle sticks

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Figure 53

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Pueblo Buildings in the U.S.A.

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MAKING A PUEBLO HOME1.Arrange the boxes as in thepicture. Figure 53

2.Working with each boxseparately, add doors andwindows. You may cut thewindows out or glue or tapepieces of black paper to eachbox to look like doors andwindows.

3.For the projecting logs, makesmall holes in the boxes, insertthe round sticks, and glue ortape them in place.

4.Glue or tape the boxes together.5.To make a ladder, glue or tapetoothpicks to two popsiclesticks.

THINGS TO THINK ABOUT AND DOWhat shape is most important inthe construction of the Pueblohome?Read about the Pueblo people in

The Pueblo by Charlotte and DavidYue.

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Designs & Symmetryhat is symmetry? Look atyour face in a mirror. Theleft half of your face is

pretty much like the right side,except in reverse. Your left handand your right hand are usually aclose match, except that one isthe opposite of the other. Your feetare usually a close match, too.

We say that the face hassymmetry. The word symmetrycomes from the Greek language. Itmeans “same measure.” We canalso say that your body issymmetrical. Your two hands arepretty near symmetrical, and soare your two feet.Symmetrical designs are very

pleasing. They are often used assymbols or logos. A peace symbolhanging on a chain can be flippedover and it will look the same.Imagine that you draw a line downthe center of the symbol and thenfold the page along that line. Theleft half of the peace symbol wouldfit over the right half. The designhas line symmetry. Figure 54a

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Figure 54a

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This symbol stands for thetalons of an eagle. Figure 54b Seehow sharp they are! This is one ofthe symbols on adinkra cloth ofthe Asante people in Ghana, acountry in West Africa. Give this book a quarter turn.

The design looks the same. Now,turn it again, a quarter turn, andagain. The design looks the samein four different positions as youturn this book. The design hasturn symmetry. However, if youwere to fold this design in halfand then compare the two sides,they wouldn’t match. This designdoes not have line symmetry.The next few activities are about

symmetrical designs and figures.

Look around you. You will findmany objects and designs that aresymmetrical. Some have linesymmetry. Some have turnsymmetry. Some have both kindsof symmetry. Make a list or drawpictures of some of these objects.

Figure 54b

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Here is a grinning jack-o’-lantern,a face made by cutting the outershell of a pumpkin. If you draw aline down the center and fold thepaper on the line, one half of thejack-o’-lantern matches the otherhalf. The face has line symmetry.Figure 55aHalloween is a time when people

tell scary stories about ghosts andgoblins and dress up in fancycostumes. These customs go backmany centuries to early times inEngland, Scotland, and Ireland.The festival marked the beginningof winter, a scary time when itmight be hard to get food and tokeep warm. It was a time when thesun shone weakly or not at all anddarkness came early in the day.The British colonists brought theHalloween customs to America.Their Native American neighborstaught them to grow pumpkins.

You can make a mask thatlooks like a jack-o’-lantern. Youcan wear it on Halloween or atany other time.

MATERIALS• Sheet of plain paper• Sheet of orange construction paper• Compass or bowl about 8 inches(20cm) in diameter

• Scissors with sharp points• Pencil• Black marker or crayon• Tape• 2 lengths of string, about 12 inches(30cm) each

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Figure 55a

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Masks and Faces fromthe U.S.A.

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MAKING THE MASKFirst practice with plain paper tomake sure you have the correctmeasurements so that the maskwill fit you. Then make anothermask with the orange construc-tion paper. If you don’t haveorange, color white paper to lookpumpkin-colored.1.Using a compass or a bowl,draw a large circle about eightinches (20cm) in diameter. Cutit out.

2.Mark circles for the two eyesand a triangle for the nose. Cutout the eyes and the nose. Tomake sure that the mask issymmetrical, fold the paper inhalf and match the two halves.Are the eyes and nose in theright places to fit your face? Ifnot, correct them.

3.Draw the mouth. Cut along theline between the teeth.

4.Use a black marker to draw theteeth. Add any other scaryfeatures you like.

5.Place tape on each side of themask. Make a hole through thetape and tie one end of thestring on each side. Figure 55b

THINGS TO THINK ABOUT AND DOThe Pilgrims ate pumpkins attheir first Thanksgiving. This wasa new food for them. How did theylearn about pumpkins? Do youthink they made jack-o’-lanterns?Make a mask or draw a picture ofa sad jack-o’-lantern. How willyou change any part of the face?Is the face symmetrical?

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Figure 55b

Fold Line

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Here are two Native Americanmasks. One is an Aztec jade maskfrom Mexico. Figure 56a The other isa painted wood mask of theKwakiutl people from the PacificNorthwest. Figure 56bFrom about 1300 until 1520,

the Aztecs controlled a large partof the land that is now Mexico.The capital was Tenochtitlan, alarge city with a magnificent GreatTemple and a Royal Palace. WhenCortes with his band of Spanishconquerors came upon this city in1519, they were amazed. Theyhad never seen anything so largeand beautiful. Decorations on thestone buildings were in gold andprecious stones. Masks and otherstone carvings adorned them.

The Spaniards conquered thecity and all the Aztec lands. Morethan anything else, they wantedgold and precious gems to sendback to the rulers of Spain. Theybuilt Mexico City on the site ofTenochtitlan. In a short time, war,disease, slavery, and hunger hadwiped out most of the NativeAmerican population.This jade mask was carved by

an Aztec artist. Notice thesymmetry of the face and theplugs in the ears. Wealthy Aztecsliked to wear earplugs made ofgold or silver.

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Figure 56a

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Native American Masks

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The Kwakiutl people live nearthe Pacific Ocean in the land thatis now Canada. Their wonderfulwood carvings appear on thehouse posts and totem poles. Onspecial occasions they wear masksthat may have been in theirfamilies for many years. Thesemasks are made of wood that hasbeen steamed and bent intoshape.Some Kwakiutl masks may hold

a surprise. A mask may look likea serpent, but when the wearerpulls a string to open it, a fierceman’s face appears inside. Anactor telling a story would wearthis type of double mask.

THINGS TO DOCopy the Aztec mask and color itaccording to your own ideas. Youmay want to make an Aztec maskthat you can wear. Copy the Kwakiutl mask. The

mask is colored blue, with redeyes and mouth and blackeyebrows. Some parts may be thecolor of natural wood. Make up astory to go with the picture of themask.

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Figure 56b

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Here are three examples ofbaskets woven by Hopi women.The Hopi live in northeasternArizona. Their name means“people of peace.” Their ancestors,the Anasazi, wove such basketsmore than 1,500 years ago. Howdo we know? Baskets have beenfound in the ruins of Anasazihomes. Now Hopi women weavebaskets to hold bread and otherfoods. The most beautiful basketsare awarded to the winners of footraces. Each basket has a different kind

of symmetry. We will look for linesymmetry and turn symmetry.A design has line symmetry if

you can fold it on a line so thatthe two halves match. Somedesigns have more than one lineof symmetry.

A design has turn symmetry if it looks the same in more thanone different position as you turnit. Suppose you want to hang thisbasket on the wall. Figure 57a Canyou turn the basket so that itappears the same when you hangit from a different point? Howmany such points are there?When you examine these

baskets, remember that thedesigns in objects made by handmay not be as perfect as thosemade by machine.

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Hopi Flat Baskets

Figure 57a

NativeAmerican

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Figure 57a. This design looks likea flower with four petals. It hasfour lines of symmetry. It also hasturn symmetry, because it looksthe same in four differentpositions as you turn it.Figure 57b. This design is made

up of circles. It has many lines ofsymmetry, an infinite number ofthem. “Infinite” means you cankeep counting forever and nevercome to the end. The basket alsolooks the same in an infinitenumber of positions as you turn it.Figure 57c. This design may stand

for streaks of lightning. It has noline symmetry. It does have fourdifferent positions of turnsymmetry.

THINGS TO DOLook at the Hopi baskets. Find thelines of symmetry. Figure out theturn symmetry.Draw several circles. Make up a

design for each one, similar to thedesigns that the Hopi womenmade in their baskets. Try for adifferent kind of symmetry in eachdesign. Color your designs.

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Figure 57c

Figure 57b

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You may have done papercuttingalready. Perhaps you cut outfigures like these—a heart and apetal or a leaf. You probably foldeda sheet of paper in half before youcut. One half matches the otherhalf of the figure. Each figure hasline symmetry. Figure 58aThis beautiful design comes

from the German people whosettled in Pennsylvania manyyears ago. They are often called“Pennsylvania Dutch.” Thispattern is called “Love.” We knowthat a heart is a symbol for love.That’s why we draw and cut outhearts for Valentine’s Day. ThePennsylvania Dutch called such asymbol a hex. They hoped thatsuch symbols would protect themfrom harm.

To make a copy of the Lovepattern, you will cut out heartsand petals. Then you will gluethem to a large white circle. TheLove pattern has six petals. Youmay want to design your Lovepattern with only three or fourpetals. Figure 58b

MATERIALS• Several large sheets of white paper• Large bowl or a compass• Pencil• Red and green crayons or markers• Scissors• Red paper and green paper (optional)• Glue

DRAWING A HEX1.Place the bowl on a sheet ofwhite paper. Trace a circlearound it. (If you have acompass, use it to draw thecircle.) Draw a red and greenborder inside the circle. Cut outthe circle.

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Figure 58a

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Love Pattern U.S.A.Pennsylvania

Dutchfromthe

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2.Figure out how large to makethe hearts so that they will fit inthe circle. Fold a sheet of paperto make a pattern for thehearts. Trace the pattern tomake seven hearts. Color themred and cut them out. If youhave red paper, you can use itfor the hearts.

3.Make a pattern for the petals.Trace it to make six petals.Color them green and cut themout. If you have green paper,you can use it for the petals.

4.Divide the circle into six equalparts, called sectors. You can dothis by folding the circle in half,then in thirds. If you don’t wantto fold this circle, make a smallcircle of scrap paper and fold it.Then mark the divisions on thelarge circle. (If you have acompass, see page 56 fordirections for dividing the circleinto six equal parts withoutfolding.)

5.Glue the hearts and petals tothe white paper.

THINGS TO THINK ABOUTLook at the Love pattern for linesymmetry. Imagine that you candraw a line and fold the patternon that line so that one half fitsover the other half. Now find sixdifferent fold lines. (Pretend thatthe small heart in the center isnot there.) Look again at the Love pattern,

this time for turn symmetry. Turnthis book so that the patternlooks the same in anotherposition. Find six different turnpositions. (Don’t pay attention tothe small heart in the center.)

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Figure 58b

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The Chinese and Japanese peoplehave a long history of cuttingcomplicated designs in paper. Infact, archaeologists discovered aChinese papercut that was madeabout 1,500 years ago. Figure 59aIn the old days warriors would

decorate their armor with thesedesigns. Now the designs appearas decorations in books, onpeople’s homes, and in manyother places. Paper cutouts areoften used in making cartoonmovies.In Japan this art is called mon-

kiri, the art of folding and cuttingpaper to make designs. You canmake a beautiful cutout in thestyle of mon-kiri. You might wantto give it to a family member or afriend for a birthday or valentine.

MATERIALS• 5-inch (12.5cm) circle of white paper• Pencil• Scissors• 6-inch (15cm) circle of blackconstruction paper

• Glue• 8-inch (20cm) square of redconstruction paper

MAKING A MON-KIRI CUTOUT1.Fold the white circle in half,then in half again. Figure 59b

2.Along the folded edges drawseveral designs that will be easyto cut out.

3.If you like, draw a borderpattern along the rim of thecircle.

4.Cut out the design and theborder pattern. Figure 59c

5.Glue the white circle to the blackcircle so that the black border iseven all around.

6.Glue the black circle to the redsquare so that the margins areeven all around. Figure 59d120

Figure 59a

Figure 59b

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1 2

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THINGS TO THINK ABOUT AND DOLook at the mon-kiri you made.Imagine that you can draw a lineand fold it so that one half fitsover the other half. Can you drawtwo different fold lines? These arelines of symmetry.Place a finger on the center of

the mon-kiri. Give the circle a halfturn. Now it looks the same aswhen you started. The patternlooks the same in two differentpositions. Fold another white circle as

directed above. Then fold it again,so that you have eight layers ofpaper. Draw and cut out designs.Before you open the circle, canyou imagine how many times eachdesign will appear? Experiment with squares and

circles of paper folded in differentways. Draw and cut out thedesigns. Before you open thepaper, try to imagine how it willlook. Then open the paper. Didyou imagine it correctly?Look at the patterns you made.

How many lines of symmetry canyou find in each pattern? In howmany different positions does thepattern look the same when youturn it about the center?

Another form of Japanesepapercutting is called origami.Read the book by Eleanor Coerr,Sadako and the Thousand PaperCranes, a true story about a littleJapanese girl who was poisonedby radiation when an atom bombwas dropped on her city duringWorld War II. She and her friendsbelieved that if they folded athousand paper cranes, she wouldbecome well. Sadly, she diedbefore they had finished the task.

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Figure 59d

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Repeating Patternshen you walk barefoot inwet sand, your feet makea pattern—left, right, left,

right. Figure 60aIn the food market, a row of

cans, all alike, sitting on a shelf,makes a repeated pattern. Figure 60bThis border pattern is on a

woven bag of the Ojibwa people,Native Americans of the GreatPlains. The design is repeated overand over again, first one way andthen upside down. Figure 60cA design may be repeated to

cover a whole surface. An exampleis this carved stonework in atemple in Mexico. The Zapotecpeople built it more than onethousand years ago. Figure 60d

Some repeated patterns occur innature. Other patterns are madeby people. Some repeated patternsare made as decorations. Othersare useful, like the bricks in awall and the tiles on a bathroomfloor. Those described in the nextactivities are patterns peoplemade for beauty.Find other examples of repeated

patterns along a line or row. Lookinside your home and outside.Large buildings usually have rowsof windows, all the same size andshape. Make a list or drawpictures of the examples you find.Look for examples of designsrepeated over a whole surface.Make a list or draw pictures ofsome examples. Patterns in clothand on tile floors are just a fewexamples that you might find inyour home and your closet.

WFigure 60a

Figure 60b

Figure 60c

Figure 60d

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The Yup’ik people live in southwestAlaska. They are one branch of thepeoples called Eskimo or Inuit.Southwest Alaska is not as cold asthe northern Arctic region. TheYup’ik people have never lived inigloos. Years ago they lived inhouses that were partly belowground level. Today their homesare just like those of most peoplein the United States. They dressjust as you do, go to similarschools, and use computers.In cold weather they wear coats,

called parkas, made of animalskins. Women decorate the parkaswith borders of a repeated pattern.Each family has its own pattern,which is handed down to thechildren and grandchildren. Theymake the border by sewing smallpieces of skin to a strip of skin in acontrasting color. The patterns arecalled tumaqcat, which means“cutout patterns.” Here are some ofthe patterns they used. Figure 61a

Each pattern has a name in the Yup’ik language. The endingof the name tells you that it is“pretend” —“pretend windows,”for example. The thick straightlines are “sled tracks.” Mary, a young Yup’ik woman,

is very proud of the parka hergrandmother made many yearsago. The border has this design,repeated all along the bottom ofthe parka. The design has fourwindows inside a bigger windowand mountains on both sides.Think how many tiny pieces hergrandmother had to cut and sew!Every piece had to be placedexactly right. Figure 61b

Figure 61a

Yup’ik Eskimo fromAlaska

Border Patterns

Boxes

Windows

Wanting to be tall

Braid

Mountains and rivers

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MAKING A YUP’IK PARKA BORDER Here are three different ways tomake the border. Before you workwith the materials, sketch thedesign and the pattern on paper.Measure carefully to make sure itfits together. You can make righttriangles by cutting squares inhalf along the diagonal.

Cut-and-paste m ethod. Chooseone Yup’ik pattern to copy. Cutmany polygons—squares ortriangles or parallelograms—ofconstruction paper. Glue them toa strip of paper of a contrastingcolor. Dark brown and light brownare good colors, as are black andwhite.Tracing m ethod. Cut one poly-

gon of heavy paper or Styrofoamto use as a pattern. Then tracethis polygon as many times asnecessary on a strip of construc-tion paper. Color your border.Sew -and-paste m ethod. Cut

many polygons of felt and sew orglue them to a strip of cloth toform the border.

After you have made severalYup’ik borders, invent your ownrepeated pattern for a border,using polygons. First sketch it onpaper. Give your pattern a name.Decide what you will decorate.

THINGS TO THINK ABOUTTry to imagine how each patterngot its name. Name the different types of

polygons (shapes with straightsides) that you see in the borderpatterns. Which patterns have squares?

Find different sizes of squares inone border. How many differentsizes of squares can you find inMary’s grandmother’s parka?Which patterns have triangles?Which are white and which areblack? Which patterns haveparallelograms?

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Figure 61b

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This belt, now in the New YorkState Museum in Albany, is madeof thousands of wampum beadscarefully sewed onto a strip ofskin more than six feet long. It iscalled the Washington CovenantBelt because it sealed a treaty, orcovenant, of peace between theSix Nations of the Iroquois andthe thirteen colonies. It was acustom among some of the Nativepeoples to seal a treaty by makinga wampum belt. It was regardedas a permanent record of theagreement. Figure 62

But the peace between theIroquois and the thirteen colonieswas broken. Here are the words ofChief Hendrick of the Mohawks,one of the Six Nations: “We willsend up a Belt of Wampum to ourBrothers the other Five Nations toacquaint them the CovenantChain is broken between you andus.” He was speaking to GovernorGeorge Clinton and the council ofNew York in 1753. The New Yorkcolonial government had takeneight hundred thousand acres ofMohawk land. That was the end ofthe friendship between the NewYork colony and the Mohawks. REPEATIN

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The Covenant Belt Native American

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THINGS TO THINK ABOUTLook at the Washington CovenantBelt. It shows a repeated patternof people. Do you see thirteen tallpeople? Why are there thirteen?What other figures are in the belt,and what do you think they standfor? Examine the figures for sym-

metry. Is each person symmetri-cal? Does the left side match theright side? Is the house near thecenter symmetrical? Notice that a small person has

the same shape as a large person.We say that the two figures aresimilar—they have the same shapebut different sizes.Each wampum bead measures

about 1⁄10 by 1⁄3 of an inch. Onesquare inch of the belt wouldcontain about thirty beads. Thebelt is about six inches wide andmore than six feet long. Estimatehow many beads are in the belt.Then check your estimate with theanswers in Chapter 10.

THINGS TO DODesign a belt to seal a treaty ofpeace with a friend or familymember. Draw the belt and colorit. What designs will you put onthe belt? Will it have a repeatedpattern?If you have a computer, try to

design a treaty belt on thecomputer.

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The Kuba live along the KasaiRiver in Congo (formerly Zaire), acountry in Central Africa. Theybecame famous centuries ago fortheir wonderful embroidered clothand wood carvings. In the BritishMuseum is a carving of their kingShyaam aMbul aNgoong, whoruled in the early seventeenthcentury. In front of him is a gameboard for mankala. He iscelebrated for bringing thepeaceful arts to his people. Hebrought new crops, the arts ofcarving and weaving, and themankala game they called lela.Here are two border patterns

from the art of the Kuba people.Figure 63a

THINGS TO THINK ABOUTLook at each pattern. What is theshape of the design? Notice howthe design is repeated again andagain to make the pattern.Look at the cloth pattern.

Suppose you want to make astamp of the design. What is thesmallest design you need to cutinto the stamp? Here are threechoices. Figure 63bLook at the cloth pattern again.

Now you are going to make astencil. You can use it to trace thedesign many times. You can flipthe stencil over to the other side ifyou like. Which of the three isyour choice for the smallestdesign to cut into the stamp?Does it make a difference whetheryou put your design on a stampor on a stencil that can be usedon both sides? Check yourconclusions with the answers inChapter 10.

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Figure 63a

Figure 63b

African Patterns from Congo

Carved wooden box

Embroidered cloth

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THINGS TO DOYou can make your own bordersfor place mats, book covers, andother kinds of decorations. Hereare some ways to do it.Tracing m ethod. Cut a design

out of heavy paper or plastic.Draw a long line on a sheet ofplain paper. Trace your design onthis line. Repeat it as many timesas you can. Then color the design.Figure 63c

Stam p m ethod. Draw a simpledesign to put on a stamp. Tomake a stamp, cut the design onthe flat side of half a potato or ona sponge. Ask a grown-up to helpyou. Draw a track (two parallellines) in which to stamp the de-sign. Decide how you are going torepeat the pattern along the track.Use washable ink or finger paint.Then make another track andrepeat the pattern in a differentway. Figure 63dStencil m ethod. Here are some

instructions for making a stencilfor a design that you will repeat tomake a border. Figure 63e

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Figure 63d

Figure 63c

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MATERIALS• Rectangle of heavy paper or plastic,about 11⁄2 by 2 inches (31⁄2 by 5cm)

• Pencil• Scissors• Ruler• Sheet of plain paper• Crayons or markers

MAKING A BORDER STENCIL1.To make the stencil, draw asimple design on the rectangle.Cut it out.

2.On the plain paper, draw atrack as wide as the rectangle.Divide it into rectangles thesame size as the stencil.

3.Trace the design in pencil in thefirst rectangle on the track.

4.Decide how to place the designin the second rectangle. Youmight repeat the first rectangle,turn the stencil upside down, orflip it over. How many differentways can you find to make arepeated pattern with thisstencil? Try as many as youcan. Figure 63f

5. Color your border.

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Figure 63e

Figure 63f

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At the time of colonial Americaand the early days of the UnitedStates, a great deal of work, suchas weaving, sewing, and carpen-try, was done in the home. Inwealthy homes, servants andslaves did much of this work.Poorer people had to do their ownwork. Often these workers madevery artistic creations.One form of this art was

quilting. Women who were stuckin the house arranged to get outevery once in a while for a quiltingbee in a neighbor’s home. Theywould work together to make apatchwork quilt as a wedding giftor as a good-bye present to afamily moving out West. Eachwoman did her bit of the quilt.

For many women, this was theonly means of artistic expression.One woman wrote: “It took menear twenty-five years to makethat quilt. My whole life is in thatquilt.” African American womensometimes bought their freedomfrom slavery with their fine quilts,which were often based on Africanpatterns. Native American womenalso made quilts to honor thearrival of a new baby or for someother special event.Today some quilters use the

computer to design their quilts.And men as well as women arenow quilting.A fine quilt is a result of careful

planning. Many patchwork quiltsare made up of a lot of squaressewed together. Each square, inturn, may be made up of severalshapes and colors. A quilt thathas no design or pattern is calleda “crazy quilt.” A hodgepodge ofthings with no order may also becalled a “crazy quilt.”130

Figure 64a

Figure 64b

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Patchwork Quilts from the U.S.A.

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For the squares, or patches,women and girls used leftover bitsof fabric and pieces of worn-outclothing. Nothing was wasted.Only a very special quilt or onemade for a wealthy home ratedlarge amounts of new material.Let’s design a simple quilt with

paper squares. You should cut outat least thirty-six squares.

MATERIALS• Pencil• Ruler• 2 or 3 sheets of white or light-coloredpaper

• Marker or crayon• Scissors

MAKING A PAPER QUILT1.Use the pencil and ruler todivide each sheet of paper intotwo-inch (5cm) squares. Youshould have at least thirty-sixsquares. Figure 64a

2.Draw the diagonals of all thesquares. Color each half ofeach square contrasting colors.Figure 64b

3.Cut out the squares.4.Arrange the squares in apleasing pattern to make a largequilt. Here are some ways tocombine the squares. You canrotate some squares a quarterturn or a half turn. Figure 64c

5.You may want to keep the bestpattern. Copy it on a sheet ofconstruction paper or glue thesquares to the sheet of paper.Draw a border around it as aframe, or glue it to a largersheet of a contrasting color.

You have just created atessellation. A tessellation is apattern that covers a surfacecompletely by combining certainshapes. Your tessellation is madeup of squares.

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Figure 64c

Figure 64d

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THINGS TO DOYou can make interesting three-by-three quilt blocks. Each blockwill have nine squares. You needtwo sheets of paper in twodifferent colors. Cut out ninesquares of one color and ninesquares of another color. Takefour squares of each color and cuteach square into two triangles.Figure 64dCombine the squares and

triangles to make a three-by-threequilt block. These are twotraditional quilt blocks. They arecalled “Shoo Fly” and “Ohio Star.”Figure 64eBoth have symmetry. Look at

each quilt block. Find four differentturn positions. Then find fourdifferent fold lines. (See page 116for a discussion of these terms.)Here is one fold line, a diagonal.Figure 64fSew a patchwork quilt cover out

of cloth. First plan it on paper.When you cut the fabric, allow anextra centimeter all around for theseams.

Read books about quilts. Somedescribe different kinds of quiltsin various cultures and how tomake them. The Quilt-BlockHistory of Pioneer Days, withProjects Kids Can Make, writtenfor children by Mary Cobb, givesinstructions in quilting. There arealso children’s storybooks thatdescribe the importance of quiltsin the lives of the characters. Oneexample is Deborah Hopkinson’sSweet Clara and the FreedomQuilt.

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Figure 64e

Fold

Figure 64f

Figure 64d

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This cloth is full of designs. Eachdesign is a symbol—it stands foran idea. The Asante (also calledAshanti) people of Ghana, in WestAfrica, call the cloth adinkra, aword that means “saying good-bye.” Long ago this was a specialcloth for funerals, but it is nowworn at any time. Figure 65aAn Asante cloth maker makes

stamps by cutting the designs intopieces of calabash, the hard shellof a fruit. He attaches handles tothese stamps and dips them inblack dye. The cloth is dividedinto rectangles. Each rectangle ofcloth has a different symbolstamped on it in neat rows.Here are some of the symbols

and their meaning. Figure 65b

MATERIALS• Sponges or potatoes sliced in half• Knife• Washable paint or dye (black oranother dark color)

• Shallow, flat pan• Several sheets of paper• Old newspapers• Length of white cloth

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Figure 65a

Figure 65b

Adinkra Cloth from Ghana

Circles: the king,or greatness

Heart: be brave Unity is strength

Talons of an eagle

Ram’s horns: wisdom and learning

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Figure 65c

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MAKING AN ADINKRA CLOTH1.Start with an easy symbol, likethe heart. Ask a grown-up tohelp you cut the design on thesponge or the cut half of thepotato. Figure 65c

2.Pour paint or dye into the pan.3.Place newspapers under a sheetof paper. Stamp the designs inseveral neat rows on the sheetof paper. If it’s hard to place thesymbols in a straight row, drawlines lightly in pencil on thepaper before you start. Figure 65d

4.Choose another symbol andfollow directions 1 through 3.

5.After you have practiced withsymbols on paper, try to workon cloth.

THINGS TO THINK ABOUT AND DOAs you stamp the design, youmust be careful that all thesymbols go the same way. Topsand bottoms must be lined up.Look at each symbol. Can youturn the stamp and still have thesymbol come out correct? In howmany different positions can youturn each stamp so that it stilllooks the same? In other words, inhow many positions is there turnsymmetry for each of thesesymbols?Look at the symbols: Heart,

Unity, Talons, Ram’s horns, andCircles.Suppose you cut each symbol

into a stencil and traced thedesign. Which symbols would notcome out correct if you flipped thestencil over before you traced thedesign? Check your conclusionswith the answers in Chapter 10.Make a real Adinkra cloth using

several different symbols. Designa set of symbols that stand forideas that are important to you.Decorate greeting cards and otherthings with these symbols.

Figure 65d

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This beautiful pattern is made ofcolored tiles inlaid on a flat sur-face. This form of inlaid art iscalled mosaic. The tiles cover theentire surface, with no overlapping.Imagine how well the artist had toplan to make everything fit. Noticethat the same design is repeatedseveral times. The repetition ofshapes to cover a surface is calleda tessellation. This picture showsonly a small section of a large wall.Islamic artists covered the walls ofmosques and palaces with suchmosaics. Figure 66The religion of Islam developed

in the Middle East in the seventhcentury. It spread eastward toIndia and other countries in Asia,and north and west to parts ofAfrica and to Spain and Turkey.Islamic artists used mainly

three types of decorations—geometric shapes, flowers, andcalligraphy (writing in Arabicscript). Some branches of theIslamic religion did not allow

pictures of human beings oranimals. Artists drew thegeometric patterns using only acompass and a straightedge (anunmarked ruler). Many mosaicswere made entirely of polygons,such as triangles, squares,parallelograms, and hexagons(six-sided shapes).

THINGS TO THINK ABOUTName the different polygons thatyou see in the pattern of inlaidwork. How many different sizesand shapes of triangles can youfind? How many different sizes ofsquares do you see? Can squareshave different shapes? Can youfind stars of different sizes? Howmany points are on each star?What other shapes do you see? A collection of pattern blocks is

ideal for making tessellations. Youcan buy a set in a school supplystore or from a catalog, or you canmake your own.

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Figure 66

Tessellations in Islamic Culture

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Follow in the footsteps of Islamicartists as you create wonderfulpatterns with polygons.

MATERIALS• Sheet of plain paper• Compass• Pencil• Ruler or straightedge• Scissors• Piece of cardboard or Styrofoam• Glue• 4 sheets of construction paper in 4different colors

MAKING A SET OF BLOCKS1.Mark a point on a sheet of plainpaper. Open the compass to aradius of about 11⁄2 inches (4cm).Place the point of the compasson the point you marked anddraw a circle. Draw a diameterfrom point A to point B. Figure 67a

2.Place the compass on point Aand draw two arcs that intersectthe circle. Do the same at pointB. Connect the six points on thecircle to form a hexagon. Cut itout. Glue it to the cardboard orStyrofoam and cut around it.Now you have a pattern to drawregular hexagons—all the sideshave equal length and eachangle measures 120 degrees.Figure 67b

3.Draw several hexagons on eachsheet of construction paper.We’ll call the colors of the paperyellow, red, green, and blue, butany colors are OK. Cut out theset of yellow hexagons.

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Figure 67a

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Figure 67b

Polygon Patterns Islamic

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4.Draw a diameter on each redhexagon. Cut out each hexagonand cut it in half. You now havea set of red trapezoids.

5.Draw all three diameters oneach green hexagon. Cut outeach hexagon and cut it insixths. You now have a set ofgreen equilateral triangles. Allthe sides are of equal lengthand the angles have equalmeasure—sixty degrees.

6. Draw three diameters on eachblue hexagon. Cut out eachhexagon and cut it in thirds.You now have a set of bluerhombuses (rhombi). Figure 67c

THINGS TO THINK ABOUT AND DOWork with each shape separately.Can you cover a surface with onlytriangles? With only rhombi? Withonly trapezoids? With onlyhexagons? We call this tessellatingthe plane.Try to tessellate the plane with

two shapes—for example, trianglesand hexagons. Can you form six-pointed stars? Islamic artistsstarted in one place and thenspread out from that point in alldirections. What other combina-tions of shapes will tessellate theplane? Figure 67dGlue your best tessellation to a

sheet of construction paper. If youuse commercial pattern blocks,trace the shapes on a sheet ofplain paper and color them tomake a beautiful mosaic.

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Cut

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Figure 67c

Figure 67d

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You can learn to draw tessel-lations using only a compass anda straightedge, like the Islamicartists. This tessellation is madeof equilateral triangles and regularhexagons. Figure 68a

MATERIALS• Large sheets of paper• Compass• Pencil• Ruler or straightedge• Colored markers or crayons

DRAWING A TESSELLATIONYou will start by constructing aregular hexagon using a compassand a straightedge. Then you willdraw the six equilateral trianglesto form a six-pointed star. Withthis star as the center, you canrepeat the design in all directions.1.Follow the directions on page 136to construct a hexagon. Draw theconstruction lines lightly inpencil. Figure 68b

2.Extend the lines that form thesides of the hexagon. Now youhave a six-pointed star. Figure 68c

3.Erase the circle, the hexagon,and the construction marks.Outline the star with a heavypencil line. Figure 68d

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Figure 68a

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Figure 68b

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4.Draw six hexagons around thestar. Here are directions to drawone hexagon. Use the sameopening (radius) of the compassas in Step number 1.a.Place the point of the compasson the tip of the star at pointA, and draw an arc at point C.

b.Place the point of the compasson the tip of the star at pointB, and draw an arc at point D.

c. Then draw arcs from points Cand D that intersect at pointE. Draw a line through pointsC and E. Draw another linethrough points D and E.Figure 68e

d.You now have a hexagon.Draw a heavy pencil line tooutline it.

e. Follow the directions above todraw five more hexagonsaround the star. You will findthat you have already drawnsome of the lines you need.You have also constructedmany more triangles.

5.Continue as far as you can.Color the stars in one color andthe hexagons in another color.Islamic artists liked to drawflowers in the center of eachshape.

Don’t be discouraged if yourpattern is a bit lopsided. This crafttakes a lot of patience. Keep at it,and you will have a beautiful workof art!

REPEATING PATTERN

S

B D

CA

K E

Figure 68c

Figure 68e

Figure 68d

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Selected Answersou will not find manyanswers in this section of thebook. That’s because most of

the activities are self-checking andyou are smart enough to figureout the reasons if somethingdoesn’t work out. For example, ifthe rows and columns of a magicsquare don’t add up correctly, youcan try to find your mistakerather than looking for an answerin the back of the book. That’s alot more fun, and it gives you afeeling of success.

CHAPTER 3Page 36, Pentalpha. This is reallya puzzle rather than a game. Onesolution is to place the counterson the points of the game board inalphabetical order, as shown inthis diagram. Begin with point Aand end with point I. Figure 69

CHAPTER 5Page 72, Rice Multiplies. On theeleventh day, the wise manreceives 1,024 grains. Multiply 2X 2 X 2 X . . . X 2 [10 factors]=210.On the twenty-first day, the

wise man should receive 220 grainsof rice. This number is the sameas 210 X 210 or 1,024 X 1,024. Toget a quick estimate, round thenumber 1,024 to 1,000. Thenmultiply 1,000 X 1,000 =1,000,000, or one million. Theexact answer is 1,048,576 grainsof rice.On the thirty-first day, the wise

man should receive 230 grains ofrice. This number is the same as210 X 210 X 210. The exact answer is1,073,741,824 grains. An easynumber that is close to the exactanswer is 1,000,000 X 1,000 =1,000,000,000, or one billiongrains of rice. The wise man didn’task for much, did he?

Y

10

G

H

E

C FA

I B

D

Figure 69

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CHAPTER 6Page 79, Crossing the River in theSea Islands. Jonah must makethree trips. He might have takenthe duck first, then gone back forthe fox and the corn.

Page 81, Crossing the River inLiberia. This is one way to get allthe objects across in seven trips.Another way is to interchange theleopard and the cassava leaves.Figure 70

Shore One Shore Two

Figure 70

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Page 82, Crossing the River withJealous Husbands. They can do itin nine trips if a wife is allowed tobe without her hus-band but withanother man as they transferbetween the boat and the shore.Figure 71a Otherwise they need atleast eleven trips. Figure 71b

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Figure 71a

Shore One Shore Two

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CHAPTER 9Page 125, The WashingtonCovenant Belt. There areapproximately 13,000 beads.

Page 127, African Patterns fromCongo. The smallest stamp hashalf of the design, while thesmallest stencil has one-quarterof the design. You can flip overthe stencil, but not the stamp. Ofcourse, it may be more convenientto put the complete design on thestamp or stencil.

Page 133, Adinkra Cloth. Turnsymmetry (including the startingposition): Heart, one; Unity, two;Talons, four; Ram’s horns, two;Circles, infinite.If you flipped the stencils, the

talons would curve in the oppositedirection, but the other symbolswould come out right.

Shore One Shore Two

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F igure 71b

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BOOKS FOR ADULTSAppleton, Leroy. American IndianDesign and Decoration. NewYork: Dover, 1971.

Ascher, Marcia. Ethnomathematics:A Multi-cultural View ofMathematical Ideas. Belmont,CA: Brooks/ Cole, 1991.

Bell, Robbie & Michael Cornelius.Board Games Round the World.New York: Cambridge UniversityPress, 1988.

Bourgoin, J. Arabic GeometricalPattern and Design. New York:Dover, 1973.

Centner, Th. L’enfant Africain etses Jeux. Elisabethville, Congo:CEPSI, 1963.

Crane, Louise. African Games ofStrategy. Urbana: University ofIllinois, 1982.

Culin, Stewart. Games of the NorthAmerican Indians, 1907. Reprint:Dover, 1975.

Dolber, Sam M. From Computationto Recreation. Atwater, CA:Author, 1980.

Gerdes, Paulus. GeometricalRecreations of Africa. Paris:Harmattan, 1997.

Grunfeld, Frederic, ed. Games ofthe World. New York: Ballantine,1975.

Kenschaft, Patricia C. Math Power:How to Help Your Child LoveMath, Even If You Don’t. Reading,MA: Addison-Wesley, 1997.

Krause, Marina. MulticulturalMathematics Materials. Reston,VA: National Council ofTeachers of Mathematics, 1983.

Lumpkin, Beatrice & DorothyStrong. Multicultural Science andMath Connections. Portland, ME:J. Weston Walch, 1995.

McConville, Robert. A History ofBoard Games. Palo Alto, CA:Creative Publications, 1974.

Neihardt, J. G. Black Elk Speaks.Lincoln, NE: University ofNebraska Press, 1961.

Russ, Laurence. Mankala Games.Algonac, MI: ReferencePublications, 1984.

UNICEF. The State of the World’sChildren (annual). OxfordUniversity Press.

Williams, Geoffrey. AfricanDesigns from TraditionalSources. New York: Dover, 1971.

Zaslavsky, Claudia. Africa Counts:Number and Pattern in AfricanCulture. Chicago: Lawrence HillBooks, 1973, 1979.

———. The Multicultural MathClassroom: Bringing in theWorld. Portsmouth, NH:Heinemann, 1996.

———. Multicultural Mathematics:Interdisciplinary Cooperative-Learning Activities. Portland,ME: J. Weston Walch, 1993.

———. Multicultural Math: Hands-On Activities from around theWorld. New York: ScholasticProfessional, 1994.

———. Preparing Young Childrenfor Math. New York: Schocken,1986.

———. Tic Tac Toe and OtherThree-in-a-Row Games. New York:Crowell, 1982.

Bibliography

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BOOKS FOR KIDSBirch, David. The King’s Chess-board. New York: Dial, 1988.Growth by doubling, in an oldtale from India, Persia, andChina.

Carlson, Laurie. More thanMoccasins: A Kid’s Activity Guideto Traditional North AmericanIndian Life. Chicago ReviewPress, 1994.

Cobb, Mary. The Quilt-BlockHistory of Pioneer Days, withProjects Kids Can Make.Brookfield, CT: The MillbrookPress, 1995.

Coerr, Eleanor. Sadako and theThousand Paper Cranes. NewYork: Putnam, 1993.

Corwin, J. H. African Crafts. NewYork: Franklin Watts, 1990.

D’Amato, Janet & Alex. IndianCrafts. Toronto: McLeod, 1968.

Ernst, Lisa Campbell. Sam John-son and the Blue Ribbon Quilt.New York: William Morrow,1983.

———. The Tangram Magician.New York: Harry Abrams, 1990.

Flournoy, Valerie. The PatchworkQuilt. New York: Dutton, 1985.

Graymont, Barbara. The Iroquois.New York: Chelsea, 1988.

Grifalconi, Ann. The Village ofRound and Square Houses.Boston: Little, Brown, 1987.

Hopkinson, Deborah. Sweet Claraand the Freedom Quilt. NewYork: Knopf, 1993.

Kohl, Herbert. Insides, Outsides,Loops, and Lines. New York:Freeman, 1995.

Millard, Anne. Pyramids: Egyptian,Nubian, Mayan, Aztec, Modern.New York: Kingfisher, 1996.

Neale, Robert. Origami Plain andSimple. New York: St. Martin’sGriffen, 1994.

Orlando, Louise. The MulticulturalGame Book. New York:Scholastic Professional, 1993.

Osofsky, Audrey. Dream Catcher.New York: Orchard Books,1992.

Pittman, Helena C. A Grain ofRice. New York: Hastings House,1986. Based on a Chinese folktale about growth by doubling.

Temko, Florence. TraditionalCrafts from Africa. Minneapolis:Lerner, 1996.

Tompert, Ann. Grandfather Tang’sStory. New York: Crown, 1990.About tangrams.

Yue, Charlotte & David. ThePueblo. Houghton Mifflin, 1986.

———. The Tipi. Boston:Houghton Mifflin, 1984.

Zaslavsky, Claudia. Tic Tac Toeand Other Three-in-a-RowGames. New York: Crowell,1982.

BIB

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A WORD ABOUT UNICEF

UNICEF, the United Nations Children’s Fund, is an organization that helpschildren around the world. They help fight hunger and disease among poor

and sick children, work to end child labor, expand opportunities foreducation, and deal with many other issues.

Children in this country can do many things. For example, with a treatmentcalled Oral Rehydration Therapy, or ORT for short, you can prevent a child’sdeath from severe diarrhea. ORT is a mixture of salt, sugar, and clean water.Can you guess the cost of one ORT treatment? It is just eight cents, muchless than the price of a candy bar. Some of the money you raise for UNICEFon Halloween goes for this treatment. ORT saves the lives of one millionchildren every year, but millions more die for lack of the treatment.

UNICEF is also concerned with child labor. More than 250 million childrenbetween the ages of five and fourteen are made to work long hours underterrible conditions, and do not have a chance to go to school. They makesoccer balls, sew clothing, and work on farms. This happens even in rich

countries like the United States.

You can get involved and help children less fortunate than yourself byasking a parent or teacher to contact the UNICEF office in your area orcontact the U.S. Committee for UNICEF, 333 East 38th Street, New York,NY 10016; telephone (212) 686-5522; fax (212) 779-1679. UNICEF will sendkits for fund-raising and other activities. Halloween is a great time to collect

funds for UNICEF while you’re out trick-or-treating with a parent orguardian. Always remember, children can make a difference.

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