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Claudia Zaslavsky

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  • 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 Mens 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)

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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 recreationsJuvenile literature.[1. Mathematical recreations 2. Games.] I. Title.QA95.z37 1998793.7'4dc21

    Cover, interior design, and illustrations by Mel Kupfer

    Photo credits: p. 20D. W. Crowe; p. 102 courtesy of the Kenya Mission to the U. N.; pp. 21, 52, 133Sam 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 Yupik 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 . . . . . . . . . . . 8Picara, Native American . . . . . . . . . . . . . . . 109 Mens 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 Ki from Korea & China . . . . . . . . 32Mu Torere from New Zealand . . . . . . . . . . . . 34Pentalpha from Crete . . . . . . . . . . . . . . . . . . 36Kaooa from India . . . . . . . . . . . . . . . . . . . . . 38Awithlaknannai, Native American. . . . . . . . . 39Butterfly from Mozambique . . . . . . . . . . . . . 41Yot 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 Swallows Nest from Angola . . . . . . . . . . . . . . . . . . . . . . . 84

    The Chokwe Storytellers from Angola . . . . . . 85Decorations on the Walls from Angola . . . . . 87How the World Began from Angola . . . . . . . . 88Childrens 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. . . . . . . . . . . . . . . . . . . 122Yupik 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 worldAfrica, 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 Africanchildrens network that aEuropean scientist said wasimpossible. 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 activitydoesnt 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-checkingyou will know whetherthey are correct. See Chapter 10for the answers to some of theseactivities, but you probably wontneed them. Its 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 peoplegames 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. Its 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 importanthave fun!

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  • Math Games & Activitiesfrom around the

    World

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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.Thats 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

    1

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  • 3THREE-IN-A-ROW GAM

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    How did these diagrams come tobe on the roof of the Pharaohstemple? 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 meansmill 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 sameto 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 youdont 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. Thats 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 players 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 dont 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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  • 5Figure 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 adrawno winner or loser. Its 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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  • 7counter 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 drawno 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 players 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

    THREE-IN-A-ROW GAM

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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)

    8

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    Tsoro Yematatu TWO P L A Y E R S

    Figure 3

    fromZimbabwe

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  • 9DRAWING 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 cant 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 orPicara (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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    Picara 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 playedone 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 diagonaleight

    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 Picara 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 Picara is a bettergame? Why?Can you play the first version of

    Picara 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 Mens Morris, Five MensMorris, Nine Mens Morris, andTwelve Mens Morris. The numbertold you how many counters eachplayer used in the game.English pageants during the

    Renaissance used girls and boysas counters in Nine Mens 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 outGood 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 ships 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 callMlle. Germans call the samegame Mhle, 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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    9 Mens Morris from England TWO P L A Y E R S

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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 playerscounters from the board, with oneexception. You may not remove acounter from the other players 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.

    THREE-IN-A-ROW GAM

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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. Dont 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 playerscounters. 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 anopen 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 Mens 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 MensMorris (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 anerenchi. 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 Lanka TWO 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 place

    a 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 movingstage of the game may remove oneof the opponents counters fromany position on the board.To Finish. The loser is the player

    or 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 MensMorris was popular in the NewEngland colonies more than twohundred years ago. Other three-in-a-row games played with twelvecounters for each player are: SamKi in China, Kon-tjil in Korea, DigDig in Malaysia, and Shah inSomalia.

    THREE-IN-A-ROW GAM

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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 days 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 opponents counters fromthe board. This is called eatingthe 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 theopponents 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 ahorse 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 opponents row only if theopponent has no other counterson the board.Neither player may remove a

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

    THREE-IN-A-ROW GAM

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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 AmericaseasternBrazil, Suriname, and theCaribbean islandswhere 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 fortransferring. 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

    M

    Figure 9a

    2

    Oware game board

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    MANKALA: BOARD GAM

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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, calledendpot, 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 10bTo Move. Player One picks up all

    the 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 Twos 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 made

    from the opponents side of theboard. If the last bean in any movemakes a group of two in a cup onthe opponents side, the last playercaptures these two beans andplaces them into her endpot. Then,going backward, if the cup justbefore the previous one on theopponents 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 10dTo Finish. The game ends when

    one 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

    ES OF TRANSFERF igure 10d

    Player One is ready to move

    Player One has moved

    Player One has captured 4 beans

    Player Ones 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 Oware

    players 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 calledsowing the seeds. Some beansmay fall into the cups on theother players 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 11aMAN

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    Oware Game TWO P L A Y E R SThe Real from Ghana

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    To Capture. If the last beandropped into a cup on theopponents 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 11bTo Finish. If all the opponents

    cups 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 opponents 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

    ES OF TRANSFERF igure 11b

    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 players left.

    2.Players may make several lapsin one move.

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

    4.Each players 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 P L A Y E R S

    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 turns

    picking 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 opponents 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 opponents sideor into their own storage cup, it isthe opponents turn to move.

    To Capture. If the last bean fallsinto an empty cup on the playersside 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 opponents turn.Figure 12bTo Finish. When there are no

    beans left on a players 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

    ES OF TRANSFER

    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 13aTo Move. Player One picks up all

    the beans in any cup on his orher side and drops them, one byone, in each cup going around theboard. The player chooses thedirectioneither 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 players side,but he or she has not droppedany beans in the cups on theopponents side, the player movesagain. The Kikuyu say: You cantsteal the other persons cattleunless you cross into his land.That means you must drop beansin your opponents cups beforeyou may capture your opponentsbeans.If the last bean falls in an empty

    cup on the opponents side of theboard, it is the opponents turn.The game continues in this way,unless a player captures theopponents 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 a

    move, Player One has droppedbeans in the opponents cups, andthe last bean in Player Ones handfalls into an empty cup on his orher own side, then Player Onemay capture all the beans in theopponents 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 opponents 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 theplayers side of the board or withan empty cup on the opponentsside of the board. Figure 13b To FInish. The game is over

    when 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

    ES OF TRANSFER

    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 Ki, 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 HauKi, 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 gamereally morelike a puzzlethat 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

    3

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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 Picara (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 dont know whether it wentfrom Africa to Asia or in theopposite direction.

    Yot, 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 Ki (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

    32

    Figure 14a

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    Pong Hau Ki 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 players countersare on the two lower points andthe other players 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 drawno 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 Ki.

    MORE BOARD GAM

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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 whiteand 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

    34

    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 athree-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

    E

    B

    C

    H

    DF

    G

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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 thatisnt 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

    36

    Figure 16a

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    Pentalpha from Crete ONE P L A Y E R

    Figure 16b

    36

    36 36

    366"

    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. Saythree.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. MORE BOARD GAM

    ES

    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 cant 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 cant moveor when the vulture has capturedat least four crows.

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

    38

    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 SNativeAmerican

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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 opponents 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 opponentscounters, 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 opponents 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 opponentscounters.

    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 SfromMozambique

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  • Yot (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

    42

    Figure 20

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    Yot 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 turns

    moving 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 jump

    over an opponents counter intothe next space, if it is empty, andremove that counter from theboard. In addition, this playermay remove another of theopponents counters from theboard as a bonus.

    To Finish. The winner is theplayer who has captured all of theopponents 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

    GAMES

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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 isbalancedif its 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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    GAMES OF CHANCE

    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

    GAMES

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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?

    GAMES OF CHANCE

    First toss = 5 points

    Second toss = 4 pointsScore = 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 objectssticks, 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 couldnt 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.

    GAMES OF CHANCE

    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

    GAMES OF CHANCE

    F igure 23

    Figure 24

    #1 #2 #3 #4

    TWO O R MOREP L A Y E R S

    NativeAmericanStick Game

    Chapter 4 [44-59]_ Chapter 4 [44-59] 12/7/10 10:50 AM Page 51

  • 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 twe