1

198080-1. SUM OF THE YEAR

Consider the first sum to the right. It is pos-sible to replace several of the numbers in thisproblem by zeros in such a way that the totalbecomes 1980. One such solution to the prob-lem is the second sum.

How many solutions are there altogether andwhat are they?

111 101333 303555 500777 077999 999

——– ——–1980

——– ——–

80-2. EXASPERATING

This exasperatingly uninteresting and extraordinarily unconvincing paragraph musters four-teen words noticeably diversified in length.From it one can select four words of a, b, c and d letters, so that, at the same time,

a2 = bd and b2c = ad

Which are they?

80-3. ROUND THE BEND

a. Two circles are drawn with the same centre. Theradius of the smaller one is r metres, the larger one smetres. Show that the length of the longest straight linewhich can be drawn entirely in the ring bounded by thetwo circles is 2

√s2 − r2 metres.

b. Two people are carrying a long thin flagpole alonga corridor of constant width. The corridor is straight tobegin with but soon bends through 90o with the insidewall following an arc of a circle of radius four metres.Then it becomes straight again. Show that if the flagpoleis six metres long, is not bent in any way, and is carriedhorizontally then it will not go round the bend unless thecorridor is more than one metre wide.

c. Now suppose that the corridor is actually three me-tres wide and that the flagpole is 11 3

4metres long. Is it

then possible for the people to carry it round the bend,keeping it straight and horizontal? Give your reasoning.

2

80-4. TRIANGULATION

A,B,C and D are points in space each one metre indistance from each of the others. The points E,F,G,H, Iand J are the mid-points of AB,AC,BC,BD and CD,respectively. There are over one hundred triangles withvertices chosen from these ten points. The triangle AIJ isone such.Exactly how many such triangles are there, and how

many of them are equilateral?

A

B

CD

E F G

HI

J

80-5. DEGREES OF FREEDOM

A and D are fixed points in the plane, four cen-timetres apart. The points B and C are free to movein the plane, provided that B is at all times two cen-timetres from A and that C is always three centime-tres from both B and from D.

Suppose that B rotates at a steady rate about Ain a counterclockwise direction. What then happensto the point C?

80-6. THEY ARE MAGIC!

The squares that are being shown to us here are exam-ples of magic squares of order 4. Each square is a 4 × 4arrangement of the numbers from 0 to 15 such that eachrow, each column and each of the two diagonals sum to30. Pairs of numbers that add up to 15 are said to bepartners in the square.The partners in such a square can be arranged in many

different ways. The partners in the top square are in thesame row: the end two in any row are partners and alsothe middle two. Make a diagram to show the partners forthe other square.How many magic squares are there of each of these two

types that have either the numbers 0, 8, 10, 12 in someorder or other, or the numbers 0, 4, 12, 14 in some order orother, down the diagonal that runs from the top left-handcorner to the bottom right-hand corner of the square?

3

1981

81-1. A CRACKER

The equations below are written in code such that each digit shown represents some otherdigit. Break the code, given that each of the following is true in ordinary base ten arithmetic:

8 + 7 = 62; 5 + 3 = 5; 12 + 8 = 23

50 + 9 = 54; 11× 1 = 55; 0− 9 = 1Give some indication of how you got your first three or four digits.

81-2. PARTY TRICK

Ask someone to write down any two numbers one under the other;then write down the sum of these two numbers underneath and soon, two at a time, until ten lines are completed. The example showswhat happens when you start with 2 and 3. You then glance at thecolumn and ‘immediately’ give the total of all ten numbers – thetrick being that you simply multiply the seventh number by 11 inyour head. (The answer in the example is 374.) Explain why thistrick always works.

81-3. STOCK QUESTION

In a sweet shop there are some boxes of choco-lates at £5 each, some boxes at £1 each andsome chocolate bars at 10p each. Taking stockone day, the shopkeeper noted that he had ex-actly 100 of these items in total and that, curi-ously, their total value was £100.It is almost possible to deduce from this how

many he had of each kind. What in fact are allthe possibilities?

4

81-4. LEFTOVERS

When a certain positive number, (say N), is divided by 3 it leaves a remainder of 1. WhenN is divided by 5 the remainder is 3, and when N is divided by 7 the remainder is 5. Findthe three smallest numbers which obey all these conditions. Show that the sum of your threenumbers is exactly equal to 3 times one of them.

81-5. A LIKELY STORY!

In Incredibilia the unit of currencyis the incredible pound, I£, and carscan be bought and sold only on April1st each year. I want to replacemy Xavier-Jagworth Supercharger 5 1

2,

commonly called the XJS5 1

2, by a new

XJS51

2costing I£1000. From the table

below it can be seen that a one-year-old XJS51

2can be bought for I£600,

a two-year-old XJS5 1

2for I£450, etc.,

whereas the running costs of I£50 inthe first year, I£70 in the second yearand so on.

Show that the average cost per year of replacing my XJS5 1

2every 5 years is I£284. How

often should I replace my car so as to incur the least average annual cost?

81-6. CLIFF HANGER

You have ten dominoes each of length 2inches.They are stacked lengthways overlapping eachother as shown. The amount of overlap betweeneach domino and the one above it may vary asyou go up the stack. What is the greatest pos-sible size of ‘overhang’?Explain how you found your answer and why

you think it is correct.

5

1982

82-1. A QUESTION OF IDENTITY

I’m an odd number with three digits. All my digits are different and add up to 12. Thedifference between my first two digits equals the difference between my last two digits. Myhundreds digit is greater than the sum of the other two digits. Who am I?

82-2. COLD CUTS

What is the greatest number of pieces intowhich you can divide a cube of ice-cream by fourstraight cuts of a knife? A ’straight cut’ need notbe at right angles to a face, but the knife mustnot twist. Give a clear diagram to justify youranswer.

82-3. BOTTLE STOPPER

A woman was travelling up a steadilyflowing river in a small boat fitted witha constant speed outboard motor. Acci-dentally, a bottle dropped out of the boatinto the water. Fifteen minutes later, thewoman realized her loss, rapidly turnedaround and started back downstream. Sheeventually caught up with the bottle afterit had floated two miles. How fast was theriver flowing?

6

82-4. MISSING NUMBERS

An ’exclusive’ road has no numbers of the houses(just names). It was decided to number them. Oneside was numbered continuously with odd numbersstarting from 3. The first building on the other sidewas a pair of semi-detached houses, numbered 2 and4, but somewhere on that side there was a gap wherehouses had still to be built, and allowance was madefor their eventual numbering.Each digit cost £0.50. For example, the number

24 would cost £1.00. The total bill for the digits was£42.50 and the even-numbered side cost £5.50 lessthan the other side. When the gap is filled, there willbe exactly the same number of houses on each side.What is the number of the last odd-numbered house,and what are the missing numbers on the other side?

82-5. EGYPTIAN FRACTIONS

When hieroglyphics on a fragment of Egyptian papyrus were deciphered they proved to be anexpression for the fraction 17

19as the sum of four fractions, all different and each with numerator

1. Find such an expression, explaining the method used. If you can find another one (or two)as well, so much the better!

82-6 ORIGAMI ALGEBRA

A type of paper called A4 has width a cmand length a

√2 cm. A sheet of A4 paper is

folded as shown so that the bottom right-handcorner touches the left-hand edge x cm from thebottom, forming a straight crease of length c cmrunning from the bottom edge to the right-handedge.

Explain why x cannot be greater than a.

Use Pythagoras’s theorem to find x in termsof a when the crease just reaches the top right-hand corner of the paper.In fact the length of the crease c always satisfies the formula

c2 =(x2 + a2)3

4a2x2,

which is not too hard to prove using Pythagoras’s theorem.

Obtain the value of x2, expressed as a fraction of a2, which corresponds to the shortestpossible crease. This may be done by using any appropriate method, for example by plotting agraph or by direct experiment.

Finally, find the length in cm of this shortest crease as accurately as you can.

7

1983

83-1. PROPER SIMPLETONS

We are a pair of proper fractions, both in simplest terms. Our numerators and denominatorsare all different one-digit numbers. If you add 1 to each of our numerators, we are equal. Weboth contain a digit that is a multiple of 4. Neither of us contains a digit that is a multiple of3. What are our names?

83-2. KEEP IT DARK!

The Vino family has two cupboards for itswine bottles, a small cupboard and one muchlarger. They keep the small cupboard for dailyaccess to the wine and, when it is empty, theytransfer to it bottles from the main stock heldin the large cupboard. Being very fussy, theydo not like their wine to be exposed to the lightmore than twelve times, including both the timethey buy it and the time they drink it. If theydrink one bottle each day, how often does theVino family need to buy wine?

83-3. BLIND CORNERS

A game is played using a square board and somepieces. The board is divided into smaller squares (likea chessboard) but has two opposite corner squaresblocked off. The size of the board is the number ofsmall squares along a side. The illustration is of aboard of size 4.

The pieces consist of dominoes

and triads

.The object is to place the pieces any way up on the board so as to fill all the unblocked

squares. Any number of either shape may be used. Illustrate how this can be done for boardsof sizes 3, 4, 5 and 6. Explain how to do it for boards of any size. Can you see why it cannotbe done using only dominoes or triads?

8

83-4. COVER UP

This is a game for two players in which each has a large pile of identical circular counters.They take turns to place one at a time on a rectangular board, so that each new counter doesnot overlap any of the others already there and does not overhang the edge of the board. Thelast player to be able to place a counter wins. If you played first, what would you do to ensurethat you win? Give reasons for your answer.

83-5. ON YER BIKE!

A bicycle is being ridden in a straight line,when the rider does a U-turn and goes back inthe direction he came from. He does this by sud-denly turning her front wheel, holding the han-dlebars at a constant angle and then suddenlystraightening the wheel again. Draw a sketchshowing the tracks of the front and rear wheelsduring this manoeuvre, assuming that the bicy-cle stays vertical at all times and that the han-dlebars turn about a vertical axis. Calculatethe distance between the initial and final wheeltracks, given that the distance between the cen-tres of the wheels is one metre and the anglethrough which the front wheel is turned is 30◦ .

83-6. SEEING STARS

Here is a good way to draw stars. Start with a cir-cle and mark a number of points (say n points) equallyspaced around the circumference. Now join each point inturn to the one situated m points further on, as shown.Notice that the third star consists of two like the first star,but the second star cannot be split into smaller ones. Howmany different stars of the unsplittable type can be drawnwith n = 7, 8, 9, 10, and 15? Make a table of your results.Find the property which n and m must have for the cor-responding star to split up into two or more smaller stars.Use this result to decide how many unsplittable stars canbe drawn with n = 42.

9

1984

84-1. STRAIGHT TO THE POINT

In a game of darts, each dart thrown lands on thedartboard and makes a score. If the dart lands onthe grey area, the score is equal to the number onthe outside of the pie-shaped piece. If the dart landson the outer ring, the number is doubled; on the innerring, tripled. If the dart lands in the outer bull’s eye(the innermost white circle), it scores 25; in the bull’seye (centre of the board), 50. Find the lowest twonumbers, excluding 1 and 2, which are impossible toscore in a game using (i) one dart only, (ii) two dartsonly and (iii) three darts.

84-2. THE THREE SQUARES

Mr and Mrs Bear live with their son Rupert in a house in which the floor of every room issquare and covered in identical square tiles. Rupert’s room contains N tiles, his mother’s roomN + 99 tiles, and his father’s room N + 200 tiles. What is N?

84-3. HIDDEN DEPTHS

Mrs Bear is making porridge in a cylindrical panof diameter 24 cm. The spoon she is using is 26 cm.long. It accidentally falls and sinks into the porridge.Calculate the minimum volume of porridge necessaryto hide the spoon completely. You may ignore thevolume of the spoon itself.

10

84-4. ORIGAMO RATIO

If you cut a piece of A4 paper in half, with the cut parallel to the shorter sides, then eachof the pieces produced has sides with lengths in the same proportion as the sides of the originalrectangle. Show that any rectangular piece of paper with this property must have its sides inthe ratio 1 :

√2.

You are now given a piece of paper and told that its sides are in the ratio 1 :√2. Describe,

using diagrams, how you could check this by folding the paper.

84-5. ORBITERS

A certain star has five planets revolving around it in circularorbits all in the same plane. They move around the star in thesame direction. Planet P1 takes one (earth!) year to completeits orbit, planet P2 takes two years, planet P3 takes three yearsand so on. At a particular moment the planets happen all tolie on the same side of the star in a line passing through itscentre. Find how long it takes before each possible pair ofplanets lines up again with the star on the same side for thefirst time; tabulate your results. Check that the first line-upoccurs after 1 1

4years. Use your table to help you find the time

when three of the planets line up again in this way. Can youextend this to lines of four and lines of five planets?

84-6. HOME JAMES!

James Bond has to get home in a hurry! He is ata point A on the side of a straight river of width 0.2km. His house is on the other side 1 km along the bankfrom the point directly opposite A. He quickly jumpsinto the water and swims in a straight line at 3 km/hacross to a point B on the other side. Exhausted bythe swim, he can manage to run at only 6 km/h fromB along the bank to his house. Show that, if the pointB is directly opposite A across the water, then it takesBond 14 min to get home from A.

In general, if B is x km from the house, the totaltime taken is given by the formula

T = 4√

1 + 25(1− x)2 + 10x min.

Try to prove this is true either by using Pythagoras’sTheorem or by trigonometry. By plotting a graph ofthis formula for various values of x, or by any othermethod, find as accurately as you can the shortest pos-sible time for Bond to get home from A. In this case,what is the angle between the line AB and the riverbank?

11

1985

85-1. FLIP START

Choose any two of the numbers 1 to 9. Add them together (answer A). Now make up a coupleof two-digit numbers by putting the original numbers next to each other either way around. Addthese two new numbers together (answer B). You should find that the quotient B/A is alwaysthe same. Explain why this always works.

85-2. BICYCLE MADE FOR ONE

Alan and Bill are out cycling and Alan’s bi-cycle has broken down beyond repair when theyare 16 km from home. They decide that Alanwill start on foot and Bill will start on his bicy-cle. After some time Bill will leave his bicyclebeside the road and continue on foot, so thatwhen Alan reaches the bicycle he can mount itand ride the rest of the distance. Alan walksat 4 km per hour and rides at 10 km per hour,while Bill walks at 5 km per hour and rides at12 km per hour. For what length of time shouldBill ride the bicycle if they are both to arrivehome at the same time?

85-3. A CUTE CAKE

Another year has passed and Sheila hasbeen given a flat square birthday cake. Howcan she cut it into 14 triangular-shapedpieces so that no piece includes a completeside of the cake and each piece has all threeangles acute? Illustrate your answer with aclearly drawn scale diagram.

12

85-4. GRAZE ELEGY

Ms Gardner’s lawn is in the shape of anequilateral triangle with each side of length20 metres. In order to save mowing shebuys three sheep and tethers one to a postat each corner, so that each sheep can grazethe lawn out to a distance of 10 metres fromits post. Also she plans to make a circularflower bed in the middle of the lawn. Whatis the diameter of the largest flower bed shecan safely have in this position, and howmuch area of lawn will she then have left tomow?

85-5. SMART ALEC

Alec likes smarties. He has a bag containing a mixture of greenones and red ones and a pocketful of green ones. Reaching into thebag he extracts two at random. If they are of the same colour heeats them both and then puts one green smartie from his pocketinto the bag. However, if they are of different colours he eats thegreen one and puts the red one back into the bag. This deliciousprocess is repeated until there is only one smartie left in the bag.How do the original contents of the bag determine the colour of

the last smartie?

85-6. I WANT RESULTS

On Saturday last week several soccer games were playedat different places around the country. As soon as thegames were over, the home-team managers telephonedeach other to share the news of the results. The callswere made in sequence, so that one manager could passalong news of another game to the next.Suppose that three games had taken place. Show that

three separate calls were necessary before all three gameresults could be known at each of the three fields. How-ever, if five games had been played, show that at least sixseparate calls would have had to be made to share theresults.What is the least number of separate telephone calls

needed to share the results of seven games? Try to extendyour reasoning to find an answer for n games.

13

1986

86-1. TIME TO START

How many times during any twenty-four hour pe-riod are the ‘minutes’ hand and the ‘hours’ hand ofa clock exactly at right angles to each other? Calcu-late the time to the nearest second when this occursbetween 2.15 p.m. and 2.45 p.m.

86-2. CALENDAYS

Show that in any given year three of the months begin on the same day of the week.In this year (1986) January 1st was a Wednesday and the first day of three of the twelve

months fell on a Saturday. In what year does this next happen? During the twenty-year period2000 to 2019 A.D. one of the days of the week is the first day of three of the months in onlyone year. On which day of the week and in which year does this occur? [2000 A.D. was a LeapYear.]

86-3. TURNING POINTS

The diagram shows a square with sides of length 4 cmand an equilateral triangle ABC with sides of length 2cm sitting inside it. To begin with, B is at one cornerof the square and BC lies along its bottom edge. Thetriangle now starts to rotate about its corners, C, A, B inturn and rolls without slipping around the inside of thesquare. Calculate the total distance travelled by A whenthe corners A, B and C have returned to their originalpositions.

14

86-4. SMART ARTIST

A clever painter decides to create a mathematicalwork of art. He divides a square canvas into nineequal squares and paints the central square red. Hethen divides each of the remaining eight squares intonine equal squares, painting each of the eight centralsquares so formed yellow. The remaining squares areagain each divided into nine, the centres this timebeing painted blue. This process is continued usinga different colour for each new set of central squaresuntil just over half the original area of the canvas hasbeen covered with paint. How many different colourshave been used and how many central squares havebeen painted?

86-5. WEIGHT FOR IT!

John, who works for a security firm, has todeliver forty parcels weighing 1, 2, 3, 4, ...,, 40kg respectively to different addresses. He hasto check the weight of each parcel before deliv-ery using a large pair of scales and a number ofstandard weights. Each parcel is placed in turnon the scales and balanced against the weightswhich can be put in either or both scale pans asnecessary. Find the minimum number of stan-dard weights and their values in kg which areneeded to check all forty parcels. Make a listshowing the particularly combination of stan-dard weights used in each case.

86-6. PRESTIDIGITATION

Show that any number which consists of nine differentdigits 1 to 9 in any order is divisible by 9. Find such anine-digit number in which the first two left-hand digitsform a number divisible by 2, the first three left-handdigits form a number divisible by 3, the first four formone divisible by 4 and so on. Give a concise but completeaccount of your investigations.

15

1987

87-1. INTO GEAR!

To make life simple, the zany fashion store Coates and Hatz sells its gear (coats and hats)in only three sizes, small, middling and roomy. Three of the store’s zaniest customers. Denzil,Dayglo and Dorrit, each decide to buy a new outfit. Denzil chooses a bigger coat than Dorrit,but a smaller hat than Dayglo. Both Dayglo’s coat and hat are bigger than Dorrit’s, but thesize of Dayglo’s hat matched that of Dorrit’s coat. Which sizes did Denzil buy?

87-2. FROM BAD TO WURST

Spas in Southern Germany are called Bads. In each Badthere is a shop selling strings of sausages called Wurst.There are two qualities of Bad (good and bad) and twoqualities of Wurst (best and worst). In a good Bad, everyfourth sausage in succession along each string is worstWurst and the rest are best. In a bad Bad, the sausagesare alternately best Wurst and worst Wurst. I stoppedat a Bad and bought a string of Wurst of which threesausages turned out to be best Wurst. Later that day Iwent back to the same shop and bought the same numberof sausages again. What are the possible numbers of bestWurst sausages I could have received this time?

87-3. SPLITTING HEADACHE

Eighteen dominoes, each measuring twoinches by one inch, are put together to form asquare. Show that, no matter how the dominoesare laid, the square can always be separated intotwo rectangular parts by a straight line parallelto one of the sides (without breaking the domi-noes!).

16

87-4. RING THE CHANGES

A street on a new housing estate contains sixteen houses, consecutively numbered from 1 to16. All the houses have telephones installed; these also are numbered consecutively, in the sameorder. It is then noticed that, in each case, the telephone number is divisible by the number ofthe house.. A new occupant moves into No. 13 and, being superstitious, changes the number ofthe house to 17 but retains the original telephone number. It is then found that the telephonenumber is still divisible by the number of the house. Given that the telephone numbers haveseven digits, what is the telephone number for No. 17?

87-5. SQUARE ROUTE

A square playground is bounded by four walls each oflength 30 metres. A gym teacher spaces his class outalong one wall and then tells Cynthia, who is standingat the midpoint of the wall, to run as fast as she cantouching the other three walls in turn and back to herplace. Draw a sketch showing Cynthia’s shortest routearound the playground, calculate the total distance sheruns and explain why any other route would be longer.

The teacher now asks the others to try and beat Cyn-thia’s time but Peter, starting from a position furtheralong the wall, objects that the race is unfair becausehe has further to run. Is this true? Justify your answer.

87-6. ROUND TRIP?

John lives at a house situated at H on theedge of a circular lake with centre O and radius100 metres. His neighbour Fred also lives bythe edge of the lake, at F , such that the angleHOF is 60o . Given that John can swim ata maximum speed of 1

2metre per second, show

that it takes at least 200 seconds for him to swimdirectly across to Fred’s house. At what speedwould he have to walk around the edge of thelake to reach Fred’s house in the same time?John is a good swimmer, but he can walk only

11

2times faster than he can swim. One night on

the way home, he reached the edge of the lakeat a point J diametrically opposite his houseand saw that it was on fire. He had three pos-sible ways of getting home: (i) by walking asfast as possible all the way around the edge, (ii)by swimming directly across, or (iii) by walk-ing around to some point P and then swimmingacross to his house from P . Using graphicalmethods or otherwise find which route John hadto take to get home in the least time.

17

1988

88-1. FOOL’S GOLD

A hoard of gold pieces comes into the possession of a band of thirty pirates. When they tryto share out the coins between them they find one coin left over. Their discussion of what todo with the extra coin becomes so animated that soon only twenty pirates remain capable ofmaking an effective claim on the hoard! However, when these twenty try to share out the coinsbetween them they again find one left over. Another fight breaks out leaving eleven pirates whohappily discover that they can now divide the coins equally with none left over. What is theminimum number of gold pieces which could have been in the hoard?

88-2. PIECE OF CAKE

I have a triangular slab of cake which is coatedon top and all around the sides with a thin layerof chocolate. The cake has edges of length 7inches, 8 inches and 9 inches and is an inch thick.Draw a diagram showing how I can divide it intwo with one straight vertical cut so that myfriend Kim and I get equal helpings of cake andchocolate.

88-3. THE HASTY PASTER

Wendy decides to decorate her house. She wishes to paper a wall 96 inches high and 147inches long. The wallpaper is 21 inches wide and the pattern repeats itself vertically every 18inches. The pattern at a point on the left edge of the paper matches the pattern on the rightedge at a point 3 inches higher up. What is the shortest total length of wallpaper Wendy needsto buy in order to cover the wall with vertical sheets without the pattern mismatching at theadjacent edges?

18

88-4. CHEW IT OVER

Clarence the caterpillar is browsing on a cabbagein Farmer Fermat’s vegetable garden which is rect-angular in shape. A passing moth tells Clarencethat he is situated 5 metres from one corner of thegarden, 14 metres from the opposite corner and10 metres from another corner. Unfortunately themoth flies away before Clarence can ask for the dis-tance to the fourth corner, but after a thoughtfulmunch he remembers Pythagoras’s Theorem andhis face brightens. How far is he from the fourthcorner of the garden?

88-5. GENERAL SOLUTION

Years ago a desert fort occupied by troops of the ForeignLegion lay under siege. The fort was square in shape with 8defensive positions: one at each corner and one in the middleof each side.The fort commander General Issimo knew that the enemy

would not charge as long as they could see 15 active defenderson each side, so with 40 troops under his command, he sta-tioned 5 in each defensive position. When one of his men waswounded he rearranged the rest so that the enemy could stillsee 15 on each side. How did he do this?Further casualties occurred. Explain how, as each man fell,

Issimo could rearrange his troops around the fort to preventa concerted attack. Reinforcements arrived just as the enemywas about to charge. How many active defenders did they findleft in the fort?

88-6. TIGHT CORNER!

Bill and Ben move furniture. They have to carrytwo large rectangular trunks along a straight corridorof width 4 feet and out into the open air through adoorway of width 3 feet in the side of the corridor.The trunks must be kept upright with their top faceshorizontal. Show that the first trunk, which is 2 feetwide and 5 feet long, can be carried out through thedoorway, but the second trunk, measuring 2 1

2feet by

5 feet, will not go through. What is the area of thetop face of the largest trunk which Bill and Ben couldcarry out through the doorway without tipping?

19

1989

89-1. OPENERS

The weekly newspaper Teacher’s Friend ismade up of a number of double sheets with a sin-gle sheet interleaved in the centre. When takenapart, it is found that page numbers 26 and 46occur on the same double sheet. What is thenumber of the back page?

89-2. OVER THE TOP

A vertical wall of height 3 metres runs parallel to the back of our house at a distance of 3metres from it. A ladder with one end resting on the horizontal ground beyond the wall canreach to a maximum height of 7 metres up the house wall. How long is the ladder? What is theminimum height that the ladder can reach up the house wall if one end remains on the ground?

89-3. PIZZA PI

At the Pizzarella, circular pizzas are sold with diameters8 inches, 12 inches and 16 inches, costing respectively £2, £3and £4 each. Assuming that all pizzas have the same thicknessand you can buy them in half and quarter-sizes, what is thecheapest way of feeding 14 hungry people so that each personreceives the equivalent of one 12 inch pizza?

20

89-4. A BURNING QUESTION

There are four Sundays in Advent and four Advent candleson the altar. On Advent Sunday one candle is lit during Even-song and extinguished after the service. On the second Sunday,two candles are alight during Evensong. On the third Sundaythree candles burn on the altar, and on the fourth Sunday allfour are alight during the service. Assuming that each candleburns down 1cm during Evensong, is it possible to choose theorder in which the candles are lit to ensure that all four haveburnt down by exactly the same amount before Christmas?Can this be achieved during Lent with five candles and fiveSundays?

89-5. GREEN LIGHT

Main street is a straight road 5 km. long. There are traffic lights at each end and at intervalsof 1 km. in between. The lights have only two colours, red (stop) and green (go). They allchange colour at the same time every minute. At the instant when Herbie enters Main Street,the first set of lights are green, the second set red, the third green and so on, alternating incolour down the road. Show, by means of a diagram, that Herbie can travel the whole lengthof the street at constant speed without being stopped by the lights. For what range of constantspeeds can he do this in under 15 minutes without breaking the speed limit of 70 km. per hour?

89-6. SALLY FORTH

Sarah, a potholer, climbs out of a pothole P situated in moorland 12 km. due East of apoint Q on a straight road which runs Northwards from Q to her camp at C. She sets out fromP in a direction θo North of West, walking in a straight line across the moor towards the roadat 3 km. per hour.

PQ

R

C

q1 2 k m

R oa d

N o r t h

E a s t

3 k m / h r

5 k m / h r

xM o o r l a n d

When Sarah reaches the roadshe is able to maintain a speed of5 km. per hour back to the camp.Assuming that the distance QC isx, find an expression involving xand θ for the total time of travelfrom P to C. Hence, by plot-ting graphs or otherwise deter-mine Sarah’s quickest route homein the cases (i) x = 12 km. and(ii) x = 6 km.

21

1990

90-1. NO PROBLEM

Many six-digit numbers can be formed by rearranging the six different non-zero digits a, b,c, d, e and f. Find the values of these digits if abcdef x 2 = cdefab, abcdef x 3 = bcdefa, abcdefx 4 = efabcd and abcdef x 5 = fabcde. Now calculate abcdef x 6 and abcdef x 7: why is thepattern of these numbers so different?

90-2. A SWITCH IN TIME

When Tim collects his clock from the clock mender’sit shows the correct time, so he doesn’t realise that thefingers have been replaced in the wrong order: the hourhand has been fixed to the minute hand spindle and viceversa.Later on at home Tim notices the time on the clock is

wrong and takes it back, but the clock mender then pointsout that the clock is right again. What are the possibletime intervals that can have elapsed between Tim’s twovisits to the clock mender?

90-3 TUNNEL VISION

The Queensway tunnel under the river Mersey has four traf-fic lanes, a fast and slow lane in each direction. Cars in the fastlane travel at 55 km per hour and are 25 metres apart. Carsin the slow lane travel at 35 km per hour and are 20 metresapart.When we drive through the tunnel my brother and I play a

game of counting the cars coming the other way. I count thecars in the fast lane, while my brother counts those in the slowlane.Which of us counts the most cars if we ourselves are travel-

ling in the fast lane? Would the answer be the same if we weredriving in the slow lane?

90-4. HIGH AND DRY

During Angela’s flight to Pepsiland she completely filledher cylindrical glass of radius 3 cm with 330 cubic cms ofcoke. She drank some of it and then the ’plane started todescend. The glass tilted at an angle of 30 to the vertical.None of the precious liquid was spilled, but only just! Howmuch did Angela drink?

22

90-5. NIL RETURN

One hundred factorial (written ’100!’) is a very large number formed by multiplying togetherall the numbers from 1 to 100: 1 x 2 x 3 x 4 ..98 x 99 x 100. How many zeros occur at the endof 100! ? Explain why the digit just before all these zeros must be 4.

90-6. MARBLE GEOMETRY

How many spherical marbles each of diameter 2 cm can you place on the bottom of acylindrical container of diameter 6 cm?A second layer is started by placing another marble M so that it touches three of those in

the bottom layer. How far apart are the centres of these four marbles from each other? Howmany marbles can you place in this way to make up the second layer?Using Pythagoras’s theorem and trigonometry, or by any other method, find the height of

the centre of M from the floor of the container.

90-7. COMMON CENSUS

Polly, a public opinion pollster, and Chris, a census taker, together call at the house at 900College Avenue to find the ages of its occupants. The owner gives them his own age and saysthat three other people live there. The youngest is at least 3 years old and the product of theirages (three different whole numbers) is the same as the number of the house.

The visitors ask for more information to which theowner replies that he will tell Chris the age of the middleperson. He whispers this number to Chris who says aloudthat he is still unable to determine the ages of the othertwo people. The owner then announces that he will tellPolly the sum of the ages of the eldest of the three andone of the other two. He whispers this number to Pollybut she openly admits that she also is still unable to figureout the three ages.The owner asks each in turn: Chris says that he cannot

find the ages from the information he possess; Polly saysshe cannot either from what she has heard. They bothstand there for a while pondering. Then Chris repeatsthat he still cannot work out the ages; Polly says shecannot do so yet. After more thought, Polly sees thatChrist is still stuck and then she declares: ”Yes, of course!Now I know all three ages.”.What are the ages of the three other people living at

900 College Avenue?

23

1991

91-1. CLARIFICATION

Mr. Fyed was pricing the bottles of wine in his shop. ”It takes me such a long time to do thisjob”, he remarked to his daughter Clarie. ”This bottle, for instance, cost me £3. I have to add20% to get my mark-up price, then a further 5% of this mark-up price to allow for local incometax. On top of this goes 15% of the total so far for VAT. Finally I give 10% discount on all thewine sold, which makes the price £3.91 per bottle. Now I have to go through all this again forevery bottle of wine in the shop with a different cost price. If only there were a simpler way!”Clarie immediately produced her calculator and worked out a single number which convertedcost price to shelf price in one multiplication. Can you also clarify the problem in the same way?Find also the magic multiplier which Clarie found for spirits which are marked up by 25% anddiscounted by 15%, assuming that the rate of local tax and VAT remain the same.

91-2. PEDIGREE CHUMS

Sue is very fond of dogs and has at least one at home.When asked about it or them she replies, ”If I have asheepdog but not a terrier, I also have a poodle. I eitherhave both a poodle and a terrier or neither. If I havea poodle then I also have a sheepdog.” What breed orbreeds of dog does Sue keep at home?

91-3. QUARTERED

With capital letters representing digits 0 to 9, the six digit number OURTHF is found to bea quarter of FOURTH. Find two possible values for FOURTH.

1 , 1 1 , 2 1 , 3 1 , 42 , 1 2 , 2 2 , 3 2 , 43 , 1 3 , 2 3 , 3 3 , 44 , 1 4 , 2 4 , 3 4 , 4

. . .. . .. . .. . .. . .. . . . . . . . . . . .

91-4. COORDINATION

A game is played using a board on which a rectangulararray of squares is drawn. Each square is labelled by two’coordinates’: the number of the row and the number ofthe column in which it lies, as shown. The score associatedwith any given square is the highest common factor of itscoordinates. The game consists of moving from the topleft-hand corner of the board to the bottom right-handcorner one square at a time to the right or downwards(diagonal moves are not allowed), adding up the scoresof the square as you go. Find the route which gives themaximum total score - and say what this score is - if theboard has (i) 9 rows and 9 columns, (ii) 8 rows and 16columns.

24

91-5. ESCAPE AID

A prison has 100 convicts housed in 100 cells, which arenumbered from 1 to 100, with each prisoner having a cellto himself. The prison has 100 warders.

Every year the warders have a party to celebrate thegovernor’s birthday at which they all drink too much. Atthe height of the festivities the first warder unlocks everycell from cell number 1 to cell number 100, the secondwarder then locks every second cell (2, 4, 6, 8, ?), thethird warder goes to every third cell (3, 6, 9, ?) and locksit (if it is unlocked) and unlocks it (if it is locked); thiscontinues with the k’th warder visiting cells k, 2k, 3k, ?and locking them if they are unlocked and unlocking themif they are locked. After the last warder has finish his tourof the cells all the warders fall asleep.

How many prisoners can escape after the 100th and lastwarder has gone to sleep?

91-6. KNIGHT’S GAMBIT

During the 1939/45 war when property was cheap, thewealthy entrepreneur Sir Grabal D’Enclosedland acquireda large triangular field in Cornwall with a fence runningall around its perimeter. The field had two edges eachof length 800 metres and the third edge of length 1000metres.Recently, fearing death was at hand following a severe

attack of gout, Sir Grabal decided to give the land atonce to his two sons. In order to ameliorate his imminentinterview with the Almighty, he divided the field as fairlyas possible with a straight fence, so that each part not onlyhad the same area but the same length of fence enclosingit.Draw diagrams showing the different ways he could

have done this, calculating in each case the area of theland given to each son and the length of the dividing fence.

25

199292-1. SOFT CENTRED

Terry was given a box of chocolates. Although she likedchocolates, she was not greedy, so she decided to share herchocolates and make them last. Her method of consump-tion was to eat one on the first day and give 10% of theremainder away, to eat two on the second day and give10% of the remainder away, to eat three on the third dayand give 10% of the remainder away, and continue in thisway until no chocolates were left.

How many chocolates were in the box and how manydays did they last?

92-2. THE LIE OF THE LAND

On a certain northern island all the inhabitants are either farmers or fishermen or fisher-women. Before they are married, farmers always tell the truth, but fisherfolk always lie. Aftermarriage, however, their behaviour changes, farmers always lying and fisherfolk being truthful.

On overhearing the following conversation I immediately knew the occupations of Sara andRobert and whether they were married or not.

Sara: Robert is not married yet.Robert: Sara is a farmerSara: Robert is a fishermanRobert: Sara is a married woman

What did I deduce about Sara and Robert?

92-3. FAST FOOD

When using a salad spinner to dry the lettuce Amy iscurious to know how fast the outer edge of the basket ismoving. She counts 52 teeth on the wheel she is turning.These teeth mesh with 13 teeth at the centre of the lidof the basket which has a diameter of 25 cm. She guessesthat her wheel turns about twice in a second so how fast,in kilometres per hour, is the lettuce moving pressed upagainst the side of the basket?

26

92-4. GOBSMACKED

Arkwright sold his gobstoppers in three different sized packetsonly; small packets containing six gobstoppers, medium ones con-taining 9 and large ones containing 20, and he would never splithis packets open. When Ann asked for 55 gobstoppers he gave her2 large, 1 medium and 1 small packet. For Billy’s order of 101 heprovided 4 large, 1 medium and 2 small packets. When Clare askedfor 19 he was unable to make that number. However, he foolishlysaid that if she could tell him the largest number of gobstoppershe could not supply without breaking open his packets, he wouldgive her twice that number free. After a few minutes’ thought sheworked out the number, so how many free sweets did she get?

92-5. BOXES OF COXES

After a load of apples was delivered Granville was giventhe responsibility for packing them in cubical boxes with aside length of 60 cms. Fortunately, they had been very finelygraded and there were only two sizes, some with diameter 4cms and others with diameter 5 cms, so he decided to packone size neatly in layers, with the same number in each layer,in one box and do the same for the other size in a secondbox. Remarkably, when all the apples were used up, bothboxes were full. How many apples were delivered? WhenArkwright came to pick up one he complained about his backand asked which was the lighter. Granville did not know, butcan you tell him?

92-6. SQUIRALS

Peter is doodling on a flat beach on the first day of his holiday. He draws a straight line 10cm long in the sand with his finger. This is the first step. Without lifting his finger, he drawsanother line by turning left through a right angle. This time the line is 11 cm long. This is thesecond step. He continues his squiral by turning left through a right angle and drawing a line 1cm longer than the previous one. Thus, after two steps the length of the squiral is 21 cm andafter three steps its length is 33 cm. How long is the squiral (i) after 54 steps and (ii) after 10

steps?Peter likes the number 19. He notices that if he produces a new number by adding 19 to

each step number and then divides the length of the squiral by this new number, he gets a nicepattern of numbers as the step increases. Can you produce this numerical pattern? Use thepattern to find the length of the squiral after n steps.

If he had only an area of sand 1 metre square in which to doodle and he draws the linesparallel to the edges of the square find the total length of the longest squiral he can draw,including the edge of the square.

27

199393-1. YAPPIE FAMILIES

Year 11 decided to carry out a pet survey in Fraser Street.In the 32 families living in the street there were 25 cats,19 dogs and 10 rabbits. All the families who owned petshad 1, 2 or 3 children. They survey also found that nofamily owned more than 2 pets and none had 2 pets ofthe same kind. Furthermore, five of the families had nochildren. How many families in the street had more than3 children and how many rabbits had to share the family’saffection with a dog?

93-2. TRUTH TABLE

When the table tennis tournament had finished, the five partici-pants reported the results as follows.

Jackie: Rachel came second, I finished in third placeMike: I came third. Sue came lastRachel: I finished as second. David came fourth.Sue: I am the winner. Mike came second.David: I was fourth. Jackie is the winner.

It turns out that each report contains one true and one falsestatement. Find the order of merit of the competitors.

93-3. PHYSICAL DIFFERENCE

Nuclear physicists have to find five numbers representing energies, but their experiment givesonly the differences between these numbers. They know that one number is 0 and the other fourare positive.

Suppose that their experiment gives

16, 15, 12, 9, 8, 7, 5, 3, 1

for the differences. Find two possible sets of the numbers they want.

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93-4. POINT TO POINT

Jane keeps her pony in a field shaped like a trianglewith each side 30 metres long. Since the pony is a goodjumper, he could escape from the field by jumping thefence. To prevent this Jane keeps him tethered at thecentre of the field so that he can just reach the middleof each side. What length of rope does she use and whatpercentage of the field can the pony graze?

93-5. SETTING TIME

Brian and Ann are watching the sunset at Blackpool on a calm summer’s evening. Ann isstanding 10 metres above sea level and Brian is at the top of the Tower 150 metres above sealevel. For how much longer can Brian see the sun after Ann sees it disappear below the horizon?You may assume that the circumference of the Earth is 40,000 kilometres.

93-6. TIME WASTING

Bill was visiting Tom on a Friday evening. He noticedthat, when the 6 o’clock news started on television, Tom’sclock showed 5:57 p.m. Tom explained that his clock waslosing 7 minutes every hour, but that he had got used toit.Later in the same month Bill visited Tom again and

noticed that when the news started on the hour the clockwas showing the right time. ”I see that you had yourclock mended”, Bill remarked. ”No, I haven’t touched it”,replied Tom. What day was it, and which news bulletinwere they watching?

29

1994

94-1. PRESSING PROBLEM

A journal ”Weekly Challenge” is published every Friday except Good Friday. Also it is notpublished in the week that Christmas Day or Boxing Day falls on a Friday.The first issue (number 1) is dated 3 January 1992. What is the date on which issue number

1000 will be published?Which issue will celebrate the journal’s 21st birthday?

94-2. STEP SEQUENCE

Ian notices that there are 13 stairs from the hallto his bedroom door. He knows that he can climbone step or two steps at a time and wonders if he canclimb the stairs to bed in a different way every nightfor a year. Make the decision for him, by finding howmany different ways there are?

94-3. CORNER TABLE

Arthur has a round table which just fits into a right-angled corner so that the horizontal tabletop touches both walls and the feet are firmly on the ground. One point on the circumferenceof the table, in the quarter circle between the two points of contact, is 10 cm from one wall and5 cm from the other. What is the diameter of the table?

30

94-4. CLUTCHING AT STRAWS

Janet is playing with a bundle of straws. Some of the straws are of length 1 cm, others 2 cm,3 cm, ? and so on. There are lots of straws of each size. Eventually she starts making triangleswith the ends of the straws being at the corners of a triangle.How many different triangles can she make if two of the straws used have lengths 4 cm and 2

cm respectively? (Note that triangles with sides 4 cm, 3 cm and 2 cm, and triangles with sidesof 4 cm, 2 cm and 3 cm must only be counted once.)If the longest straw is 5 cm long, what is the total number of different triangles that Janet

can make?

94-5. ON YER BIKE

Sue, Sophie and Tom all start together and go for a 10mile journey. The girls can walk at 2 mph and Tom canjog at 4 mph. They also have a bicycle which only oneof them can use at a time. When riding, Sue and Sophiecan travel at 12 mph, whereas Tom can pedal at 16 mph.Assuming that no time is lost getting on and off the bike,they all keep moving, the bike can be left unattended andriding in both directions is allowed, what is the shortesttime in which all three can finish the trip together?

94-6. COVER UP

Janet finishes experimenting with the straws and starts playing with two sets of tiles: oneset with tiles 4 x 4 cm square and the other set with tiles 5 x 5 cm square. She draws lots ofrectangles with one side 20 cm long and the other sides of various lengths. Obviously she cannottile the 20 cm by 1 cm, 20 cm by 2 cm, or the 20 cm by 3 cm rectangles, but she can tile the 20cm by 4 cm and 20 cm by 5 cm rectangles with her square tiles. However, she fails again withthe 20 cm by 6 cm rectangle. Which of the larger 20 cm by n cm rectangles cannot be tiledwith her square tiles? Note that the idea is to use five of the smaller tiles or four of the largertiles to make a complete ‘row’ across the 20 cm, and to fill up the rectangle by such rows.

31

199595-1. EXPIRY DATE

Before Christmas Ken, a gullible entrepreneur, installedin his office a new computer manufactured by the well-known hardware firm Junior Computer Networks plc. Thespecification for this machine states that, when it is run-ning, the moving parts (disk drives and fan) last for 2000hours, whereas the electronic components on the moth-erboard last for 2500. However, in the latter case thislifetime is reduced by 2 hours every time the computer isswitched on and by 1 hour every time it is turned off. Kenintends to use the computer from 9.00 a.m. to 5 p.m. ev-ery day from Monday to Friday inclusive, including BankHolidays, but not at weekends. Assuming that he switchesit on for the first time at 9.00 a.m. on January 2nd 1995,calculate the date and exact time of day when you wouldexpect the computer to break down. Find also the exactdate and time when the computer will fail if Ken leaves itrunning overnight during the week.

95-2. BEST SIX

Given a square piece of wood with sides 1 metre long, what is the area of the largest regularhexagon you can cut out of this piece of wood? Draw a picture of how you would do it. Is ittrue that the largest area is obtained by having the hexagon symmetrically placed?

95-3. SQUARING THE RECTANGLE

Janet decides to continue her career in tiling. She uses square tiles to try and cover rectanglesexactly. However, she is only allowed one square tile of each size, but she can use as manydifferent sizes as she wishes and there is no restriction on the length and width of the rectangle.After several trials she finds that with the 9 square tiles of sides

1, 4, 7, 8, 9, 10, 14, 15 and 18 cmshe can cover a rectangle.Find the length and width of this rectangle and draw the covering pattern.

32

95-4. LET THERE BE LIGHT

The top of Fred’s head is 2 metres from the floor. He stands under a light bulbwhich is suspended 1 metre from the centre of the ceiling. The height of the roomis 4 metres and its length is 12 metres. The shortest ray of light from the bulb tothe top of Fred’s head is, of course, 1 metre but what is the length of the ray oflight that bounces once from the wall before it meets Fred’s head? Another rayof light bounces once from the ceiling then once from the wall before coming toFred’s head. What is the length of this ray?

95-5. GREEK GODS

The Alphites are immortal creatures from planetAlpha; each one produces 1 offspring every year but4 offspring in even numbered years from 2 onwards.The Betons from planet Beta, are also immortal andeach one produces 0 offspring in odd numbered yearsfrom 1 onwards and 7 in even numbered years from2 onwards.A new planet is colonised in year 0 by 4 newborn

Alphites and 100 newborn Betons. After how manyyears will Alphites outnumber Betons?

95-6. IN THE DARK

At the Senior Challenge Prize evening in ’95 you wouldhave seen a coloured computer printout of the eclipse ofthe Sun in May 1994. Taking the images of the Sun andMoon to be discs of equal radius and assuming the eclipsetakes 4 minutes from beginning to end (i.e. between thetwo times when the discs are just touching), find whatpercentage of the area of the Sun is visible after 1 minute.How many seconds (approximately) from the start of theeclipse will half the area of the Sun’s disc be covered bythe Moon? [You will need to play with your calculator fortrial and improvement to find this approximate answer.]

33

1996

96-1. PALINDROMIC NUMBERS

Certain numbers like 11, 121 and 414 are the same if the order of the digits is reversed andare called palindromic numbers. The number 17 is not palindromic, but if its digits are reversedto give 71, then 17 + 71 = 88 which is palindromic so 1 reversal followed by addition created apalindromic number. The number 19 needs 2 reversals and additions to become a palindromicnumber 19 + 91 = 110, 110 + 011 = 121. The number 59 needs 3 reversals and additions tobecome palindromic 59 + 95 = 154, 154 + 451 = 605, 605 + 506 = 1111. How many reversalsand additions do 68, 79 and 89 need?

96-2.GOOD FRIENDS

Six volunteers took part in a sponsored walk for charity. They all raised different amountsof money, but decided that each of them would make one true and one false statement and leaveit to those interested to work out a table showing the order in terms of the amount of moneyraised.

“Martin raised most” Ross said. “I was fifth.”“Ross is being modest, he was third.” Emily contradicted. “I was fourth.”“Kelly was third” Liz retorted. “I was second.”“Liz was first.” Neal declared. “I was fourth.”Martin said “I was worst, but Emily was second.”“Neal raised the least, whilst I was third” Kelly stated.

Work out the order from these statements.

96-3. JOURNEY’S END

A train leaves Liverpool for London where 343 peoplealight. Lime Street is the first station and the train stopsat 5 intermediate stations before arriving at Euston, theseventh station. The number of people boarding the trainat the first six stations is inversely proportional to thenumber of the station while the number alighting at thelast six stations is proportional to the number of the sta-tion. How many people were on the train when it leftLiverpool?

34

96-4. CORNER TO CORNER

Janet has a 6 by 8 metre rectangular patio covered by 48 1 metre square tiles. She walksdiagonally in a straight line from one corner to the opposite corner and finds that she crosses12 of the 48 tiles. However, her garden path has 6 tiles forming a 1 by 6 metre rectangle and adiagonal walk down the path has to cross all 6 tiles.Next door’s patio has 42 tiles in a 6 by 7 metre rectangle and a straight line diagonal walk

also crossed 12 tiles.Find out how many 1 metre square tiles she would cross for an m by n metre rectangular

patio.

96-5. PIE SERIES

Joanna is making mince pies. She starts with 3 mm thick pastryin the shape of a rectangle 60 cm by 28 cm. With her pastry cuttershe cuts disks of diameter 5 cm from this rectangle. What is thelargest number of mince pies she can make? (Don’t forget that youneed 2 disks for each mince pie.)After she had done this, Joanna then shapes the remainder of the

pastry and cuts more disks, each 3 mm thick and 5cm in diameter.If she keeps doing this, what is the maximum number of mince piesshe can make?Wendy prefers her pastry to be 2 mm thick, and 5 cm in diameter.

She starts with the same volume of pastry as Joanna, What is themaximum number of mince pies Wendy can make?

96-6. EVEN BREAK

Stephen arranged 3 red and 3 white snooker balls in the form of a triangle with 3 rows. Thered balls made a 2 row triangle with the white balls in the 3rd row. He wondered if it waspossible using equal numbers of red and white balls to make larger triangles with the similarproperty of a triangle of red balls followed by rows of white balls. After some trials he foundthat a triangle of 20 rows could be made from 105 red balls and 105 white balls. The first 14rows were red balls and the final 6 rows were white balls. This is because

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 = 105 = 15 + 16 + 17 + 18 + 19 + 20.

Pleased with his success he decided to search for more triangles with this property and hefound six. He tabulated them as follows:

Find the number of red rows and the number of rows in the triangle for the next one in thissequence.Find the ratio of the number of red rows to the total number of rows. Try to guess the

limiting value of this ratio if more and more triangles with this property were found.

35

199797-1. BEE LINE

A male bee - also called a drone - has only one parent,its mother. A female bee has both mother and father. SoDennis Drone has one parent (mother), two grandparents(mother’s parents) and three greatgrandparents (two fe-male and one male). Going back five more generations,how many ancestors does Dennis have?If you see the pattern giving the number of ancestors in

each generation, write it down for extra marks.

(Entirely by the way, OAB’s get Buzz Passes on the Bee Line!)

97-2. WHIZZY LIZZIE

“Hey,” said John, “I’ve just found out something!Take any three-figure number—let’s say 256. Movethe first figure to the end—that gives 562. Now mul-tiply the first number by 10 and subtract the secondone 2560 − 562 = 1998. The answer is a multiple of999. Always works!”“Amazing,” said John’s sister Liz, who was rather

a whiz at Maths. “Let me see, that means that if wetake any three-figure multiple of 37 and shift the firstfigure to the end, the result will still be a multipleof 37. For example, 259 = 37 × 7 and 592 is also amultiple of 37, in fact 37× 16.”“Huh?” said John. “What’s that got to do with

the 999 trick?” Can you explain both tricks?

97-3. IN A FLAP

A 6 cm × 6 cm square of cardboard has four equalsquares of side 1 cm cut out of the corners. The four flapsare folded upwards to make an open-topped box. Whatis the volume of this boxSuppose that instead a square of side x cm has been

removed. What would be the volume then?What is the difference of the two volumes? Which x givesthe largest possible volume?

36

97-4. HAVING IT TAPED

Wally Walkman was wondering why tape cassettes inhis personal stereo rarely lasted more than 45 minutes aside. He realised that there was a limit to how thin tapescould be made and decided to try to measure the thicknessof one of his tapes. The tape was wound on a spool andwith a ruler he found that the outer radius was about 2.2cm and the inner radius was about 1.1 cm. He then triedto count the layers but soon gave up! So he listened tothe tape instead? .

By the time the tape finished 30 minutes later he had an inspiration. The cassette cover toldhim that the speed was 4.76 cm per second. With a calculator he was soon able to estimate thetape thickness. What do you reckon his answer was?

97-5. TRUTH TO TELL

Each Saturday Amanda came home and told her fatherhow many goals her favourite football team, LiverpoolUnited, had scored. For the first seven weeks these were7, 3, 8, 4, 9, 5, 1 goals respectively. On the followingSaturday she said merely, “Well, they scored more goalsthan two weeks ago, but not as many as seven weeks ago.”What is the largest number of weeks that she could truth-fully have said this for?

In fact no matter what the scores had been in the first seven weeks she could not havetruthfully made the same statement for more weeks than she actually did. Explain why this is.(A diagram may help, using a dot for each week, and using arrows to join a bigger score to asmaller one.)

97-6. ROUGH RIFFLES

An ordinary pack of 52 cards is arranged so that thecards alternate red, black, red, black, and so on. Abouthalf the pack, held face down, is dealt on to the table,taking cards one at a time from the top of the pack andputting them face down in a single pile. Then the remain-ing cards, and those in the pile on the table are riffle-shuffled together—that is they are roughly interleaved.Finally, from the top of the newly shuffled pack, cards aretaken in pairs.

How many of these pairs will contain exactly one red and one black card (in either order)?Experiment with a pack of cards and explain the result.What happens if instead the cards are arranged in repeating suits, say clubs, diamonds,

hearts, spades, clubs, diamonds, hearts, spades, and so on, and at the end groups of four cardsare taken from the top of the shuffled pack? How many of these groups will contain exactly onecard of each suit?

37

199898-1. HOME MOVIE

A commuter has been in the habit of arriving at his suburbanstation each evening at exactly five o’clock every working day ofhis life. His wife has always met him at the station and driven himhome. One day, seized by a sudden sense of reckless adventure, hetakes an earlier train, arriving at the station at four. The weatheris pleasant, so instead of telephoning home he starts walking alongthe route always taken by his wife. They met somewhere along theway. He gets into the car and they drive home, arriving at theirhouse ten minutes earlier than usual. Assuming that his wife alwaysdrives at a constant speed, and that on this occasion she left just intime to meet the usual five o’clock train, how long did hubby walkbefore he was picked up?

98-2. STONE AGES

Mr and Mrs Stone have five children, Ann, Ben, Clare, David andElaine. Ann’s age times Ben’s age equals 36; Ben’s age times Clare’sage equals 9; Clare’s age times David’s age equals 8; David’s agetimes Elaine’s age equals 24; Elaine’s age times Ann’s age equals12. How old are the children?

98-3. STEPTOE AND SON

Mr Steptoe and his son make a 64 km journey, starting at 6a.m., but have only one horse (which travels at a steady 8 km perhour) which can only carry one person at a time. Mr Steptoe canmaintain a 3 km per hour walk and his son 4 km per hour. Theyalternately ride and walk. Each one ties the horse after riding acertain distance, then walks ahead leaving the horse for the other’sarrival. At the half-way mark they come together and take half-an-hour’s rest for lunch and to feed the horse. Assuming that theythen repeat the same travelling pattern after lunch, when do theyarrive at their destination?

38

98-4. IN THE CLEAR

Three women, Louise, Melanie and Nicola, go with theirhusbands to a car boot sale to buy a variety of differ-ent objects. Their husbands’ names are Peter, Quentinand Richard, though not necessarily in that order. Be-tween then they buy everything available at the sale. Bya strange chance, the average price in pounds that eachperson pays for her (or his) objects is the same as theactual number of objects that she (or he) buys. Thus, ifLouise buys L objects, then they are at an average of £Leach, and so she spends £L2 altogether.Louise buys 23 more objects than Quentin. Melanie

buys 11 more than Peter. Each of the three women spends£63 more than her husband. Who is married to whom?How many objects are there in total?

98-5. DIAMOND CUT DIAMOND

The ace to 10 of diamonds of a pack of cards are played alternately to a pile between twoplayers, the only rules being that the first player may play any card, but the number of diamondson each subsequent card played must be either a factor or a multiple of the number of diamondson the previous card. The first player unable to play a card loses. Who should win, the firstplayer or the second player—and how?

98-6. PIG AHOY

Poor old Jonah has been abandoned in a small dinghyon the high seas by his shipmates on the Porky Pig. WithJonah stood up in his dinghy in calm water his binocularsare two metres above sea level. Given that the radius ofthe Earth is approximately 6.3×10 metres, calculate howfar away the horizon appears from Jonah (in a straightline) assuming that it is a clear calm sunny day.Jonah’s shipmates feel guilty about abandoning him

and set sail to find him in the Porky Pig. Black LegJake is sent up into the crow’s nest where his telescope is15 metres above sea level. Jonah has constructed a flag inthe dinghy that is three metres above sea level. Black LegJake can just see the top of Jonah’s flag on the horizon.How far away is Jonah’s flag from Black Leg Jake (in astraight line)?

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1999

99-1. DIVINE ASSEMBLY

St. Divine’s High School wants to build a new assembly hall with exactly 528 seats, andwith each row having the same number of seats. There are two aisles. Each row must have 17seats between the two aisles, with the remaining seats equally divided between the two sectionsoutside the aisles. How many rows should the hall have?

99-2. FOREVER EASTER

Mr Rabbit loves Easter so much that he can-not bear the thought of ever being without adaily Easter treat. He decides, at Easter time,to buy 365 little chocolate treats to last him forthe coming year. His local shop sells chocolateAnts for 5p each, chocolate Bunnies for £1.60each, and chocolate Chickens for 82p each. Heis not keen on the chocolate chickens, and buysfewer of these than anything else. He finds thathe has spent exactly £300. How many items ofeach type did he buy?

99-3. HECTOR’S HOUSE

Hector’s house lies between two bus stops, one of which lies90 metres to the right and one 270 metres to the left. The buscomes from the right and comes simultaneously into sight andearshot at a point 90 metres further away from Hector’s housethan the right bus stop. Hector’s street is uphill to the rightand downhill to the left. Each day, Hector chooses one of thetwo bus stops and walks towards it until he sees or hears thebus, when he starts to run until he gets to his chosen bus stop.Hector always walks at 2 metres per second, runs uphill at 3metres per second and runs downhill at 5 metres per second.The bus travels at 15 metres per second until it reaches thefirst (right hand) stop where it waits for 8 seconds then travelsat 15 metres per second until it reaches the second (left hand)stop here it again waits 8 seconds and then goes round a cornerout of sight.Hector comes out of his house to see a bus just disappearing

on the left. He goes back inside, waits, then comes back out18 minutes and 44 seconds later. There is no bus in sight andHector reckons it does not matter which stop he begins walkingto, since (with his usual strategy) he will catch the next buswith the same amount of time to spare either way. How muchtime does he have to spare when he catches the bus, and howfrequent are the buses?

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99-4. GROTTY PAINTING

James Grot is an abstract painter, whose favourite colours are Green, Red, Orange andTurquoise, but his most favourite is Red. He has in mind a painting which consists of two equalsquares against a Turquoise background, the first square being level (that is to say, with sidesparallel to the sides of the outer frame) and the second square be at an angle, as in the diagram.The first square (the level square) is to be painted Green, the tilted square Orange and theoverlap in James’s favourite colour, Red. James wants one corner of the tilted square to be atthe centre of the level square. He is not sure how much to tilt the tilted square to make the redarea as large as possible. Can you help him?

99-5. ABOUT TURN

Fanny is learning to drive in a large empty carpark. So far she can drive straight ahead andshe can drive on circular arcs of radius at least 1 (in some suitable units). The figures showstwo ways she has found that she can start at the origin (which is clearly marked on the car parktarmac) facing East and end up where she started but facing West, How far, in each case, doesshe drive? ( You may of course ignore the length of the car.)What is the shortest route you can find for her to perform this move (from facing East at

the origin to facing West at the origin)? Later she learns to drive backwards too along straightlines and circular arcs of radius at least 1. What is her shortest route now?

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99-6. BLOCKBUSTERS

In the kingdom of Masochisto, they play atwo-person game with a bar of chocolate con-taining one poisoned square marked with an X.Each player in turn must break the chocolatealong a horizontal or vertical line, not dividingany of the component squares, eat one of thetwo portions, and hand the remaining portionto the other player. The loser is the one whofinally ends up with the poisoned square. Forexample, if the bar is in the form

O X OO O O

then the first player has a strategy to force a win, as follows. First, the first player breaksthe bar horizontally along the middle, eats the O O O and then gives the remainingO X O to the second player. The second player can break this either as O X andO or as O and X O . The first player then receives either O X or X O .Either way the first player can win the game by breaking down the middle, eating the Oand giving the X to the second player.

Who should win the game when the bar is two squares by two squares?

What if the bar is three squares, by three squares, or four squares by four squares?

In each case, you should consider all possible locations for the poisoned square.

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2000

00-1.CUT AND COVER

Make a single cut to the wall of blocks in theFigure, then rearrange the two pieces to make a3× 3 square of blocks.

00-2. SPINNING FREDDY

Fred went out for a day’s ride on his bike. After a thirdof the total distance he stopped to have a rest and a biteto eat. After another seventh of the total distance hebought himself a drink. One mile later he was halfway.How many miles was his day’s outing?

00-3. HIGH POWERED

Which is bigger, 23000 or 71000 ?How about 25978 and 72135 ?Show all your working!

00-4. ALL IN GOOD TIME

How many months will there be in the century from2000 to 2099 inclusive? How many complete weeks?[Reminder; most years have 365 days, but every fourth

year from 2000 to 2096 is a leap year, with 366 days.]

00-5. JOAN AND JIM

Joan said ’At a party I went to last night there wereeleven people including me, and it turned out that every-one at the party had exactly three friends there.’Jim snorted and said ’Impossible!’ Was he right?

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00-6. TALL STORY

The Millennium Society has an ambitious plan to builda Century Wall. Beginning on 1 January 2000 they haveevery Saturday laid a new block and intend to keep ondoing so, one each Saturday. When finished the top ofthe wall must be as shown in Question 1, but with manymore courses (layers), each course having two fewer blocksthan the course below. They want to finish by placing thetop block as near as possible to 31 December 2009. Howmany blocks should they have in the bottom course? Onwhat date will they finish in 2099?

00-7. JIM AND JOAN

Jim said ‘I also went to a party last night...’ (‘Ohno!’, said Joan under her breath)... and there were16 people there, and each person had exactly threefriends. In fact I’ve drawn a diagram to illustratethis, where the dots are people and the lines repre-sent friendship.’ (‘Sounds pretty dotty to me’ saidJoan under her breath again.) ‘There was this gamewhere we had to split into eight pairs of friends?’.‘Impossible!’ said Joan. Was she right?

00-8. MILLENNIARDS

The diagram shows a 2 × 5 billiard table, marked out in squares bydashed lines. The 45◦ slanting line is the path of a billiard ball startingat the bottom left hand corner A. After five bounces it lands up inanother corner, which in this example is the bottom right hand cornerB. Try some other sizes of table, from 3 × 5 to 10 × 5, and count thebounces. The ball always starts in the bottom left and ends up in oneof the other three corners.

Do you see any patterns in these numbers of bounces?What about other sizes of table?Can you find any general rules for the number of

bounces and also which corner the ball ends in?Give a rule, if you can, for finding all the sizes of billiard

tables which need 2000 bounces between start and finish.

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2001

01-1. PEN FRIENDS

Farmer Chris wants to make rectangular pens for herchickens and pigs, the pens being of the same size, sharingone side, as shown in the figure below. She has exactly 48metres of fencing. Suppose that b is 4 metres. What is a?What is the area of each pen?

01-2. SECOND THOUGHTS

Farmer Chris has second thoughts and wantsto make the area of each pen 48 square metres.What must a and b be then?

01-3. SQUARE DEALS

Ian dealt out nine cards numbered 1 to 9 as shown:

1 2 34 5 67 8 9

Joyce chose one card, but left it in place. Suppose it was the8. Jim picked the cards up, with the column containing the cho-sen 8 picked up first. So after one deal and pickup the order is2, 5, 8, 1, 4, 7, 3, 6, 9. The 8 is now in third place. Ian then dealt thecards out in a square in the same way as before:

2 5 81 4 73 6 9

Again Jim picked up the cards with the column containing the chosen 8 picked first. Afterthe second deal and pickup the order is 8, 7, 9, 2, 1, 3, 5, 4, 6. The 8 has now come to firstplace.They tried it again, Joyce choosing 5 this time, and started to make a table:

Complete the table. Do you notice anything interesting?

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01-4. A PENSIVE CHAT

Farmer Chris was chatting with her friend Farmer Lesliein the local pub over a pint. ‘I’ll bet you could get a biggerarea for each pen than 48 square metres if you tried,’said Leslie. ‘Nonsense!’ returned Chris, and she beganscribbling on a beer mat. Was Leslie right? Don’t forgetthat Chris has only 48 metres of fencing.

01-5. OH FOR A MOON!

Alan, Brenda, Charlotte and Denis need to cross a nar-row and precarious bridge in the dark. They have onlyone torch between them and it must be carried on everycrossing. They all walk at different speeds. Alan can crossthe bridge in 1 minutes, Brenda takes 2 minutes, Char-lotte is a lot slower at 5 minutes while Denis, who hashurt his foot, takes a full 10 minutes to cross. The bridgewill only hold two people at a time, and when two walktogether they must go at the speed of the slower person,so that both can use the torch.What is the fastest time in which all four can get across

the bridge?

01-6. GARDEN OFF CENTRE

The figure shows a circular flower bed of radius 360cm, and two strings at right-angles making four areas inwhich Farmer Chris’s husband, Pythagoras, is going toplant different colours of geraniums. The two strings areof length 560 cm and 640 cm. Find how far the placewhere the strings cross is from the centre of the circle.This is marked in the figure with a dot, or is it a flowerpot?

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01-7. SOME FUN!

Frankie and Charlie have a fun run. They start out from their own houses, and they runtowards each other at the same speed. One of them starts a few minutes earlier than theother. They meet 3 miles from Frankie’s house, just say ‘Hi!’, turn round and run home, wherethey turn round and run towards each other again!. They meet the second time 3 1

2miles from

Frankie’s house, turn round and do the same as before. Where do they meet for the third time?

01-8. BIG DEALS

The trick in Question 3 can be repeated with other numbers of cards, for example with 25cards, numbered 1 to 25 in a 5 × 5 array. A card is always chosen, and then after each ‘deal’the cards are picked up with the column containing the chosen card picked up first. What doyou find when you complete the table this time? Can you explain what is happening? Whathappens with 15 cards in an array with 5 rows (horizontal lines) and 3 columns? What aboutother numbers of rows and column?

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2002

02-1. SQUARES

The left-hand figure shows a 4× 4 square cut into four pieces by two lines which are at rightangles. On the right the four pieces have been rearranged to make a slightly larger square, witha square hole in the middle. Find the size of the square hole.

2

3

3

3

3

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2

02-2. A RUSSIAN TALE

Anton, Boris and Carla are gathering mushrooms in the forest.At the end of the day the number collected by Anton is 20% lessthan the number collected by Boris, while the number collected byCarla is 20% greater than the number collected by Boris. Antoncollects 300 mushrooms. How many does Carla collect?

00-3. HELIPAD

A helicopter landing pad is to be marked by a large equilateraltriangle, with sides of length 10 metres, divided into three parts asshown, by two lines parallel to the base. What is the total area ofthe landing pad?The three parts are to be painted black, white and red. As it

stands in the diagram the three areas are not equal. How shouldthe lines be spaced so that the black, white and red areas are equal?

02-4. THE PARTY’S OVER

Lavinia and her husband Gerald had a number of married couplesat their party and some single people as well. At the end of theparty everyone said goodbye to everyone else, except that, naturally,no husband said goodbye to his wife, and no wife said goodbye toher husband. If there had been two married couples besides Laviniaand Gerald, and two single people, how many goodbyes would havebeen said? How about one other couple and three singles?In fact 102 goodbyes were said. How many married couples besides Lavinia and Gerald were

there at the party?

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02-5. NOW FOR THE WASHING UP

’Gerald, choose a 2-figure number,’ said Lavinia, as they weredoing the washing up. ’Okay, 82,’ said Gerald, absent-mindedlyscrubbing the dishcloth with the washing-up brush. ’Now,’ con-tinued Lavinia, make another 2-figure number by taking the sepa-rate figures of your number away from 9.’ ’Hmmm,’ said Gerald,carefully washing the dishcloth with a dirty plate, ’9 − 8 = 1 and9− 2 = 7, so I get 17.’’Good,’ said Lavinia, drying the dishcloth with a tea-towel, now put the numbers together

and divide by 11.’ ’Hmmm, 8217/11 = 747’ returned Gerald, writing a quick calculation in thesoapsuds with the handle of a spoon. ”Finally,’ said Lavinia, subtract 9 and divide by 9 ... doyou notice anything?’ ’Well,’ said Gerald, ’747− 9 = 738 and 738/9 = 82 ... that’s the numberI started with!’Would this have happened, no matter which number Gerald had started with?

02-6. CAN SHE DO IT?

On a 4 × 6 board there are two black counters (Lou’s) and two white counters (Mike’s)arranged as in the diagram.

Lou and Mike in turn move either of their counters one square forwards, toward the oppositeend of the board. Lou starts. If, after any number of moves, a black counter lies between twowhite counters either horizontally or diagonally (as in these pictures) then the black counter iscaptured and taken off the board.

Can Lou get both her pieces from the top of the board to the bottom, or can Mike preventher?

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02-7. PENTAJIG

Colour this jigsaw with five colours in such a way that in each row, in each column and ineach of the five pentominoes comprising the jigsaw each of the five colours occurs once and onlyonce. (There is more than way of doing it.)

02-8. RECTANGLES

In Question 02-1, how big is the whole square after the pieces have been rearranged? Supposethat you take a rectangle, 12×8, and cut across it by two lines through the centre of the rectangle,as shown. 4 8

1

7

481

7

Explain why the two lines drawn through the centre are at right angles.By cutting along the lines and rearranging the shapes, how many different rectangles with

rectangular holes can you make? You are allowed to turn the pieces over if you wish. How manydifferent shapes of hole are there? (The shape of a rectangle is measured by dividing the longerside by the shorter side, so for example 2× 3 and 12× 8 rectangles have the same shape.) Areany of the holes the same shape as the original rectangle, which is measured by 12/8 = 3/2?What happens for other rectangles, always cutting along two lines through the centre that

meet at right angles?

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2003

03-1. CUISENAIRE

Suppose you have rods with lengths 1, 2, 4, 8. Then, apart from these lengths you can makequite a few others, for example 3 = 2+ 1, 5 = 4+ 1, 7 = 4+ 2+ 1. What in fact is the shortestwhole number length you cannot make with these four rods?

03-2. UNEXPECTED GUEST

Giovanni and Leporello invite four youngladies to a quiet evening of pizza andKaraoke. They cut the circular pizza intosix equal pieces and are about to start eat-ing when an unexpected guest arrives. Lep-orello cuts equal amounts off the six piecesto give to the guest so that all seven peo-ple have the same amount of pizza. Whatfraction of each piece did he cut off?

03-3. GOOD DOGS

‘Be good dogs,’ said Mr Sumhope as he left Fidoand Trusty to guard his house while he was out.When they were alone, the two dogs started totear the living room carpet into pieces. WhenFido chose a piece he tore it into four parts,and when Trusty chose a piece she tore it intoseven parts. Being good dogs, they never chosethe same piece at the same time. When MrSumhope returned he found 2003 pieces of car-pet. Were there any missing?

03-4. GARDENERS’ QUESTION TIME

Mrs Pythagoras is making a semicircular patio, as in the figure, with three squares exactly fittingas shown. These squares are to be covered with paving stones and the rest of the semicircle,outside the squares, is for planting flowers. Given that the diameter of the semicircle is 10m,how much area is left for planting?

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03-5. DOING THE SPLITS

Amanda and her brother Gareth were play-ing with six-figure numbers, splitting them inthe middle to make two three-figure numbers.Amanda found one six-figure number which wasexactly 7 times what she got by multiplying to-gether her two three-figure numbers. Find thissix-figure number if you can. (Hint: 1001 is di-visible by 7.)

Gareth looked for a six-figure number which ac-tually equalled what he got by multiplying to-gether his two three-figure numbers. Do youthink he succeeded?

03-6. RUNNING MATES

Two runners, Chris and Alex, each running at his own constant speed, leave their houses at thesame time, each running towards the other’s house. They meet 600m from Chris’s house butcontinue running, turning round as soon they they reach each other’s house and meeting again400m from Alex’s house. How far apart are the houses?

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03-7. SQUARE BASHING

Two clever painters Abe and Beth are playingthe following game. They start with a table hav-ing marked on it a 6 × 4 grid of squares. Theytake turns and at each turn must paint a squareof any size with the edges along the grid linesand not overlapping the previously painted area.

For example, the figure shows a possible posi-tion after Abe painted a 2× 2 square and Bethpainted a 1 × 1 square next to it and Abe thenpainted a 1× 1 square in the corner.

The person to paint the last piece of the table wins. Starting from scratch, who should win?

What about other sizes of grid such as 5×7 or 10×20? (You may assume that both dimensionsare even or both are odd.)

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