1
1980
80-1. SUM OF THE YEAR
Consider the first sum to the right. It is possible to replace several of the numbers in this
problem by zeros in such a way that the total
becomes 1980. One such solution to the problem is the second sum.
How many solutions are there altogether and
what are they?
111
333
555
777
999
——–
——–
101
303
500
077
999
——–
1980
——–
80-2. EXASPERATING
This exasperatingly uninteresting and extraordinarily unconvincing paragraph musters fourteen 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 b2 c = ad
Which are they?
80-3. ROUND THE BEND
a. Two circles are drawn with the same centre. The
radius of the smaller one is r metres, the larger one s
metres. Show that the length of the longest straight line
which can be drawn
entirely in the ring bounded by the
√
two circles is 2 s2 − r2 metres.
b. Two people are carrying a long thin flagpole along
a corridor of constant width. The corridor is straight to
begin with but soon bends through 90o with the inside
wall following an arc of a circle of radius four metres.
Then it becomes straight again. Show that if the flagpole
is six metres long, is not bent in any way, and is carried
horizontally then it will not go round the bend unless the
corridor is more than one metre wide.
c. Now suppose that the corridor is actually three metres wide and that the flagpole is 11 43 metres long. Is it
then possible for the people to carry it round the bend,
keeping it straight and horizontal? Give your reasoning.
2
A
80-4. TRIANGULATION
A, B, C and D are points in space each one metre in
distance from each of the others. The points E, F, G, H, I
and J are the mid-points of AB, AC, BC, BD and CD,
respectively. There are over one hundred triangles with
vertices chosen from these ten points. The triangle AIJ is
one such.
Exactly how many such triangles are there, and how
B
many of them are equilateral?
G
F
E
H
C
J
I
80-5. DEGREES OF FREEDOM
A and D are fixed points in the plane, four centimetres apart. The points B and C are free to move
in the plane, provided that B is at all times two centimetres from A and that C is always three centimetres from both B and from D.
Suppose that B rotates at a steady rate about A
in a counterclockwise direction. What then happens
to the point C?
80-6. THEY ARE MAGIC!
The squares that are being shown to us here are examples of magic squares of order 4. Each square is a 4 × 4
arrangement of the numbers from 0 to 15 such that each
row, each column and each of the two diagonals sum to
30. Pairs of numbers that add up to 15 are said to be
partners in the square.
The partners in such a square can be arranged in many
different ways. The partners in the top square are in the
same row: the end two in any row are partners and also
the middle two. Make a diagram to show the partners for
the other square.
How many magic squares are there of each of these two
types that have either the numbers 0, 8, 10, 12 in some
order or other, or the numbers 0, 4, 12, 14 in some order or
other, down the diagonal that runs from the top left-hand
corner to the bottom right-hand corner of the square?
3
D
1981
81-1. A CRACKER
The equations below are written in code such that each digit shown represents some other
digit. Break the code, given that each of the following is true in ordinary base ten arithmetic:
8+7
50 + 9
= 62;
= 54;
5+3
11 × 1
= 5;
= 55;
12 + 8
0−9
= 23
= 1
Give 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 so
on, two at a time, until ten lines are completed. The example shows
what happens when you start with 2 and 3. You then glance at the
column and ‘immediately’ give the total of all ten numbers – the
trick being that you simply multiply the seventh number by 11 in
your head. (The answer in the example is 374.) Explain why this
trick always works.
81-3. STOCK QUESTION
In a sweet shop there are some boxes of chocolates at £5 each, some boxes at £1 each and
some chocolate bars at 10p each. Taking stock
one day, the shopkeeper noted that he had exactly 100 of these items in total and that, curiously, their total value was £100.
It is almost possible to deduce from this how
many he had of each kind. What in fact are all
the possibilities?
4
81-4. LEFTOVERS
When a certain positive number, (say N ), is divided by 3 it leaves a remainder of 1. When
N is divided by 5 the remainder is 3, and when N is divided by 7 the remainder is 5. Find
the three smallest numbers which obey all these conditions. Show that the sum of your three
numbers is exactly equal to 3 times one of them.
81-5. A LIKELY STORY!
In Incredibilia the unit of currency
is the incredible pound, I£, and cars
can be bought and sold only on April
1st each year. I want to replace
my Xavier-Jagworth Supercharger 5 12 ,
commonly called the XJS5 21 , by a new
XJS5 12 costing I£1000. From the table
below it can be seen that a one-yearold XJS5 21 can be bought for I£600,
a two-year-old XJS5 12 for I£450, etc.,
whereas the running costs of I£50 in
the first year, I£70 in the second year
and so on.
Show that the average cost per year of replacing my XJS5 12 every 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 2
inches.
They are stacked lengthways overlapping each
other as shown. The amount of overlap between
each domino and the one above it may vary as
you go up the stack. What is the greatest possible 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. The
difference between my first two digits equals the difference between my last two digits. My
hundreds 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 into
which you can divide a cube of ice-cream by four
straight cuts of a knife? A ’straight cut’ need not
be at right angles to a face, but the knife must
not twist. Give a clear diagram to justify your
answer.
82-3. BOTTLE STOPPER
A woman was travelling up a steadily
flowing river in a small boat fitted with
a constant speed outboard motor. Accidentally, a bottle dropped out of the boat
into the water. Fifteen minutes later, the
woman realized her loss, rapidly turned
around and started back downstream. She
eventually caught up with the bottle after
it had floated two miles. How fast was the
river flowing?
6
82-4. MISSING NUMBERS
An ’exclusive’ road has no numbers of the houses
(just names). It was decided to number them. One
side was numbered continuously with odd numbers
starting from 3. The first building on the other side
was a pair of semi-detached houses, numbered 2 and
4, but somewhere on that side there was a gap where
houses had still to be built, and allowance was made
for 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 less
than the other side. When the gap is filled, there will
be 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 an
expression for the fraction 17
19 as 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 cm
and length a 2 cm. A sheet of A4 paper is
folded as shown so that the bottom right-hand
corner touches the left-hand edge x cm from the
bottom, forming a straight crease of length c cm
running from the bottom edge to the right-hand
edge.
Explain why x cannot be greater than a.
Use Pythagoras’s theorem to find x in terms
of a when the crease just reaches the top righthand corner of the paper.
In fact the length of the crease c always satisfies the formula
c2 =
(x2 + a2 )3
,
4a2 x2
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 shortest
possible crease. This may be done by using any appropriate method, for example by plotting a
graph 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 denominators
are all different one-digit numbers. If you add 1 to each of our numerators, we are equal. We
both contain a digit that is a multiple of 4. Neither of us contains a digit that is a multiple of
3. What are our names?
83-2. KEEP IT DARK!
The Vino family has two cupboards for its
wine bottles, a small cupboard and one much
larger. They keep the small cupboard for daily
access to the wine and, when it is empty, they
transfer to it bottles from the main stock held
in the large cupboard. Being very fussy, they
do not like their wine to be exposed to the light
more than twelve times, including both the time
they buy it and the time they drink it. If they
drink one bottle each day, how often does the
Vino family need to buy wine?
83-3. BLIND CORNERS
A game is played using a square board and some
pieces. The board is divided into smaller squares (like
a chessboard) but has two opposite corner squares
blocked off. The size of the board is the number of
small squares along a side. The illustration is of a
board 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 boards
of sizes 3, 4, 5 and 6. Explain how to do it for boards of any size. Can you see why it cannot
be 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 does
not overlap any of the others already there and does not overhang the edge of the board. The
last player to be able to place a counter wins. If you played first, what would you do to ensure
that 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 in
the direction he came from. He does this by suddenly turning her front wheel, holding the handlebars at a constant angle and then suddenly
straightening the wheel again. Draw a sketch
showing the tracks of the front and rear wheels
during this manoeuvre, assuming that the bicycle stays vertical at all times and that the handlebars turn about a vertical axis. Calculate
the distance between the initial and final wheel
tracks, given that the distance between the centres of the wheels is one metre and the angle
through which the front wheel is turned is 30◦ .
83-6. SEEING STARS
Here is a good way to draw stars. Start with a circle and mark a number of points (say n points) equally
spaced around the circumference. Now join each point in
turn 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. How
many different stars of the unsplittable type can be drawn
with n = 7, 8, 9, 10, and 15? Make a table of your results.
Find the property which n and m must have for the corresponding star to split up into two or more smaller stars.
Use this result to decide how many unsplittable stars can
be drawn with n = 42.
9
1984
84-1. STRAIGHT TO THE POINT
In a game of darts, each dart thrown lands on the
dartboard and makes a score. If the dart lands on
the grey area, the score is equal to the number on
the outside of the pie-shaped piece. If the dart lands
on the outer ring, the number is doubled; on the inner
ring, tripled. If the dart lands in the outer bull’s eye
(the innermost white circle), it scores 25; in the bull’s
eye (centre of the board), 50. Find the lowest two
numbers, excluding 1 and 2, which are impossible to
score in a game using (i) one dart only, (ii) two darts
only 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 is
square and covered in identical square tiles. Rupert’s room contains N tiles, his mother’s room
N + 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 pan
of 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 necessary
to hide the spoon completely. You may ignore the
volume 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 each
of the pieces produced has sides with lengths in the same proportion as the sides of the original
rectangle. Show
√ that any rectangular piece of paper with this property must have its sides in
the 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 circular
orbits all in the same plane. They move around the star in the
same direction. Planet P1 takes one (earth!) year to complete
its orbit, planet P2 takes two years, planet P3 takes three years
and so on. At a particular moment the planets happen all to
lie on the same side of the star in a line passing through its
centre. Find how long it takes before each possible pair of
planets lines up again with the star on the same side for the
first time; tabulate your results. Check that the first line-up
occurs after 1 41 years. Use your table to help you find the time
when three of the planets line up again in this way. Can you
extend 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 at
a point A on the side of a straight river of width 0.2
km. His house is on the other side 1 km along the bank
from the point directly opposite A. He quickly jumps
into the water and swims in a straight line at 3 km/h
across to a point B on the other side. Exhausted by
the swim, he can manage to run at only 6 km/h from
B along the bank to his house. Show that, if the point
B is directly opposite A across the water, then it takes
Bond 14 min to get home from A.
In general, if B is x km from the house, the total
time taken is given by the formula
q
T = 4 1 + 25(1 − x)2 + 10x min.
Try to prove this is true either by using Pythagoras’s
Theorem or by trigonometry. By plotting a graph of
this formula for various values of x, or by any other
method, find as accurately as you can the shortest possible time for Bond to get home from A. In this case,
what is the angle between the line AB and the river
bank?
11
1985
85-1. FLIP START
Choose any two of the numbers 1 to 9. Add them together (answer A). Now make up a couple
of two-digit numbers by putting the original numbers next to each other either way around. Add
these two new numbers together (answer B). You should find that the quotient B/A is always
the same. Explain why this always works.
85-2. BICYCLE MADE FOR ONE
Alan and Bill are out cycling and Alan’s bicycle has broken down beyond repair when they
are 16 km from home. They decide that Alan
will start on foot and Bill will start on his bicycle. After some time Bill will leave his bicycle
beside the road and continue on foot, so that
when Alan reaches the bicycle he can mount it
and ride the rest of the distance. Alan walks
at 4 km per hour and rides at 10 km per hour,
while Bill walks at 5 km per hour and rides at
12 km per hour. For what length of time should
Bill ride the bicycle if they are both to arrive
home at the same time?
85-3. A CUTE CAKE
Another year has passed and Sheila has
been given a flat square birthday cake. How
can she cut it into 14 triangular-shaped
pieces so that no piece includes a complete
side of the cake and each piece has all three
angles acute? Illustrate your answer with a
clearly drawn scale diagram.
12
85-4. GRAZE ELEGY
Ms Gardner’s lawn is in the shape of an
equilateral triangle with each side of length
20 metres. In order to save mowing she
buys three sheep and tethers one to a post
at each corner, so that each sheep can graze
the lawn out to a distance of 10 metres from
its post. Also she plans to make a circular
flower bed in the middle of the lawn. What
is the diameter of the largest flower bed she
can safely have in this position, and how
much area of lawn will she then have left to
mow?
85-5. SMART ALEC
Alec likes smarties. He has a bag containing a mixture of green
ones and red ones and a pocketful of green ones. Reaching into the
bag he extracts two at random. If they are of the same colour he
eats them both and then puts one green smartie from his pocket
into the bag. However, if they are of different colours he eats the
green one and puts the red one back into the bag. This delicious
process 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 played
at different places around the country. As soon as the
games were over, the home-team managers telephoned
each other to share the news of the results. The calls
were made in sequence, so that one manager could pass
along news of another game to the next.
Suppose that three games had taken place. Show that
three separate calls were necessary before all three game
results could be known at each of the three fields. However, if five games had been played, show that at least six
separate calls would have had to be made to share the
results.
What is the least number of separate telephone calls
needed to share the results of seven games? Try to extend
your reasoning to find an answer for n games.
13
1986
86-1. TIME TO START
How many times during any twenty-four hour period are the ‘minutes’ hand and the ‘hours’ hand of
a clock exactly at right angles to each other? Calculate the time to the nearest second when this occurs
between 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 period
2000 to 2019 A.D. one of the days of the week is the first day of three of the months in only
one year. On which day of the week and in which year does this occur? [2000 A.D. was a Leap
Year.]
86-3. TURNING POINTS
The diagram shows a square with sides of length 4 cm
and an equilateral triangle ABC with sides of length 2
cm sitting inside it. To begin with, B is at one corner
of the square and BC lies along its bottom edge. The
triangle now starts to rotate about its corners, C, A, B in
turn and rolls without slipping around the inside of the
square. Calculate the total distance travelled by A when
the corners A, B and C have returned to their original
positions.
14
86-4. SMART ARTIST
A clever painter decides to create a mathematical
work of art. He divides a square canvas into nine
equal squares and paints the central square red. He
then divides each of the remaining eight squares into
nine equal squares, painting each of the eight central
squares so formed yellow. The remaining squares are
again each divided into nine, the centres this time
being painted blue. This process is continued using
a different colour for each new set of central squares
until just over half the original area of the canvas has
been covered with paint. How many different colours
have been used and how many central squares have
been painted?
86-5. WEIGHT FOR IT!
John, who works for a security firm, has to
deliver forty parcels weighing 1, 2, 3, 4, ...,, 40
kg respectively to different addresses. He has
to check the weight of each parcel before delivery using a large pair of scales and a number of
standard weights. Each parcel is placed in turn
on the scales and balanced against the weights
which can be put in either or both scale pans as
necessary. Find the minimum number of standard weights and their values in kg which are
needed to check all forty parcels. Make a list
showing the particularly combination of standard weights used in each case.
86-6. PRESTIDIGITATION
Show that any number which consists of nine different
digits 1 to 9 in any order is divisible by 9. Find such a
nine-digit number in which the first two left-hand digits
form a number divisible by 2, the first three left-hand
digits form a number divisible by 3, the first four form
one divisible by 4 and so on. Give a concise but complete
account 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 the
size 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 Bad
there is a shop selling strings of sausages called Wurst.
There are two qualities of Bad (good and bad) and two
qualities of Wurst (best and worst). In a good Bad, every
fourth sausage in succession along each string is worst
Wurst and the rest are best. In a bad Bad, the sausages
are alternately best Wurst and worst Wurst. I stopped
at a Bad and bought a string of Wurst of which three
sausages turned out to be best Wurst. Later that day I
went back to the same shop and bought the same number
of sausages again. What are the possible numbers of best
Wurst sausages I could have received this time?
87-3. SPLITTING HEADACHE
Eighteen dominoes, each measuring two
inches by one inch, are put together to form a
square. Show that, no matter how the dominoes
are laid, the square can always be separated into
two rectangular parts by a straight line parallel
to one of the sides (without breaking the dominoes!).
16
87-4. RING THE CHANGES
A street on a new housing estate contains sixteen houses, consecutively numbered from 1 to
16. All the houses have telephones installed; these also are numbered consecutively, in the same
order. It is then noticed that, in each case, the telephone number is divisible by the number of
the house.. A new occupant moves into No. 13 and, being superstitious, changes the number of
the house to 17 but retains the original telephone number. It is then found that the telephone
number is still divisible by the number of the house. Given that the telephone numbers have
seven digits, what is the telephone number for No. 17?
87-5. SQUARE ROUTE
A square playground is bounded by four walls each of
length 30 metres. A gym teacher spaces his class out
along one wall and then tells Cynthia, who is standing
at the midpoint of the wall, to run as fast as she can
touching the other three walls in turn and back to her
place. Draw a sketch showing Cynthia’s shortest route
around the playground, calculate the total distance she
runs and explain why any other route would be longer.
The teacher now asks the others to try and beat Cynthia’s time but Peter, starting from a position further
along the wall, objects that the race is unfair because
he has further to run. Is this true? Justify your answer.
87-6. ROUND TRIP?
John lives at a house situated at H on the
edge of a circular lake with centre O and radius
100 metres. His neighbour Fred also lives by
the edge of the lake, at F , such that the angle
HOF is 60o . Given that John can swim at
a maximum speed of 12 metre per second, show
that it takes at least 200 seconds for him to swim
directly across to Fred’s house. At what speed
would he have to walk around the edge of the
lake to reach Fred’s house in the same time?
John is a good swimmer, but he can walk only
1 21 times faster than he can swim. One night on
the way home, he reached the edge of the lake
at a point J diametrically opposite his house
and saw that it was on fire. He had three possible ways of getting home: (i) by walking as
fast as possible all the way around the edge, (ii)
by swimming directly across, or (iii) by walking around to some point P and then swimming
across to his house from P . Using graphical
methods or otherwise find which route John had
to 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 try
to share out the coins between them they find one coin left over. Their discussion of what to
do with the extra coin becomes so animated that soon only twenty pirates remain capable of
making an effective claim on the hoard! However, when these twenty try to share out the coins
between them they again find one left over. Another fight breaks out leaving eleven pirates who
happily discover that they can now divide the coins equally with none left over. What is the
minimum 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 coated
on top and all around the sides with a thin layer
of chocolate. The cake has edges of length 7
inches, 8 inches and 9 inches and is an inch thick.
Draw a diagram showing how I can divide it in
two with one straight vertical cut so that my
friend Kim and I get equal helpings of cake and
chocolate.
88-3. THE HASTY PASTER
Wendy decides to decorate her house. She wishes to paper a wall 96 inches high and 147
inches long. The wallpaper is 21 inches wide and the pattern repeats itself vertically every 18
inches. The pattern at a point on the left edge of the paper matches the pattern on the right
edge at a point 3 inches higher up. What is the shortest total length of wallpaper Wendy needs
to buy in order to cover the wall with vertical sheets without the pattern mismatching at the
adjacent edges?
18
88-4. CHEW IT OVER
Clarence the caterpillar is browsing on a cabbage
in Farmer Fermat’s vegetable garden which is rectangular in shape. A passing moth tells Clarence
that he is situated 5 metres from one corner of the
garden, 14 metres from the opposite corner and
10 metres from another corner. Unfortunately the
moth flies away before Clarence can ask for the distance to the fourth corner, but after a thoughtful
munch he remembers Pythagoras’s Theorem and
his face brightens. How far is he from the fourth
corner of the garden?
88-5. GENERAL SOLUTION
Years ago a desert fort occupied by troops of the Foreign
Legion lay under siege. The fort was square in shape with 8
defensive positions: one at each corner and one in the middle
of each side.
The fort commander General Issimo knew that the enemy
would not charge as long as they could see 15 active defenders
on each side, so with 40 troops under his command, he stationed 5 in each defensive position. When one of his men was
wounded he rearranged the rest so that the enemy could still
see 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 prevent
a concerted attack. Reinforcements arrived just as the enemy
was about to charge. How many active defenders did they find
left in the fort?
88-6. TIGHT CORNER!
Bill and Ben move furniture. They have to carry
two large rectangular trunks along a straight corridor
of width 4 feet and out into the open air through a
doorway of width 3 feet in the side of the corridor.
The trunks must be kept upright with their top faces
horizontal. Show that the first trunk, which is 2 feet
wide and 5 feet long, can be carried out through the
doorway, but the second trunk, measuring 2 21 feet by
5 feet, will not go through. What is the area of the
top face of the largest trunk which Bill and Ben could
carry out through the doorway without tipping?
19
1989
89-1. OPENERS
The weekly newspaper Teacher’s Friend is
made up of a number of double sheets with a single sheet interleaved in the centre. When taken
apart, it is found that page numbers 26 and 46
occur on the same double sheet. What is the
number 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 3
metres from it. A ladder with one end resting on the horizontal ground beyond the wall can
reach to a maximum height of 7 metres up the house wall. How long is the ladder? What is the
minimum 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 diameters
8 inches, 12 inches and 16 inches, costing respectively £2, £3
and £4 each. Assuming that all pizzas have the same thickness
and you can buy them in half and quarter-sizes, what is the
cheapest way of feeding 14 hungry people so that each person
receives the equivalent of one 12 inch pizza?
20
89-4. A BURNING QUESTION
There are four Sundays in Advent and four Advent candles
on the altar. On Advent Sunday one candle is lit during Evensong and extinguished after the service. On the second Sunday,
two candles are alight during Evensong. On the third Sunday
three candles burn on the altar, and on the fourth Sunday all
four are alight during the service. Assuming that each candle
burns down 1cm during Evensong, is it possible to choose the
order in which the candles are lit to ensure that all four have
burnt down by exactly the same amount before Christmas?
Can this be achieved during Lent with five candles and five
Sundays?
89-5. GREEN LIGHT
Main street is a straight road 5 km. long. There are traffic lights at each end and at intervals
of 1 km. in between. The lights have only two colours, red (stop) and green (go). They all
change 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 in
colour down the road. Show, by means of a diagram, that Herbie can travel the whole length
of the street at constant speed without being stopped by the lights. For what range of constant
speeds 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 a
point Q on a straight road which runs Northwards from Q to her camp at C. She sets out from
P in a direction θ o North of West, walking in a straight line across the moor towards the road
at 3 km. per hour.
C
N o rth
5 k m /h r
R o a d
R
Q
x
3 k
m
1 2 k m
M o o rla n d
/h
q
r
P
E a st
When Sarah reaches the road
she is able to maintain a speed of
5 km. per hour back to the camp.
Assuming that the distance QC is
x, find an expression involving x
and θ for the total time of travel
from P to C. Hence, by plotting graphs or otherwise determine Sarah’s quickest route home
in 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, abcdef
x 4 = efabcd and abcdef x 5 = fabcde. Now calculate abcdef x 6 and abcdef x 7: why is the
pattern of these numbers so different?
90-2. A SWITCH IN TIME
When Tim collects his clock from the clock mender’s
it shows the correct time, so he doesn’t realise that the
fingers have been replaced in the wrong order: the hour
hand has been fixed to the minute hand spindle and vice
versa.
Later on at home Tim notices the time on the clock is
wrong and takes it back, but the clock mender then points
out that the clock is right again. What are the possible
time intervals that can have elapsed between Tim’s two
visits to the clock mender?
90-3 TUNNEL VISION
The Queensway tunnel under the river Mersey has four traffic lanes, a fast and slow lane in each direction. Cars in the fast
lane travel at 55 km per hour and are 25 metres apart. Cars
in the slow lane travel at 35 km per hour and are 20 metres
apart.
When we drive through the tunnel my brother and I play a
game of counting the cars coming the other way. I count the
cars in the fast lane, while my brother counts those in the slow
lane.
Which of us counts the most cars if we ourselves are travelling in the fast lane? Would the answer be the same if we were
driving in the slow lane?
90-4. HIGH AND DRY
During Angela’s flight to Pepsiland she completely filled
her cylindrical glass of radius 3 cm with 330 cubic cms of
coke. She drank some of it and then the ’plane started to
descend. The glass tilted at an angle of 30 to the vertical.
None of the precious liquid was spilled, but only just! How
much did Angela drink?
22
90-5. NIL RETURN
One hundred factorial (written ’100!’) is a very large number formed by multiplying together
all the numbers from 1 to 100: 1 x 2 x 3 x 4 ..98 x 99 x 100. How many zeros occur at the end
of 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 a
cylindrical 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? How
many 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 900
College Avenue to find the ages of its occupants. The owner gives them his own age and says
that three other people live there. The youngest is at least 3 years old and the product of their
ages (three different whole numbers) is the same as the number of the house.
The visitors ask for more information to which the
owner replies that he will tell Chris the age of the middle
person. He whispers this number to Chris who says aloud
that he is still unable to determine the ages of the other
two people. The owner then announces that he will tell
Polly the sum of the ages of the eldest of the three and
one of the other two. He whispers this number to Polly
but she openly admits that she also is still unable to figure
out the three ages.
The owner asks each in turn: Chris says that he cannot
find the ages from the information he possess; Polly says
she cannot either from what she has heard. They both
stand there for a while pondering. Then Chris repeats
that he still cannot work out the ages; Polly says she
cannot do so yet. After more thought, Polly sees that
Christ 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 this
job”, he remarked to his daughter Clarie. ”This bottle, for instance, cost me £3. I have to add
20% to get my mark-up price, then a further 5% of this mark-up price to allow for local income
tax. On top of this goes 15% of the total so far for VAT. Finally I give 10% discount on all the
wine sold, which makes the price £3.91 per bottle. Now I have to go through all this again for
every 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 converted
cost 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% and
discounted 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 a
sheepdog but not a terrier, I also have a poodle. I either
have both a poodle and a terrier or neither. If I have
a poodle then I also have a sheepdog.” What breed or
breeds 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 be
a quarter of FOURTH. Find two possible values for FOURTH.
1 ,1
2 ,1
3 ,1
4 ,1
...
1 ,2 1 ,3
1 ,4
2 ,2
2 ,3 2 ,4
4 ,2
4 ,3
3 ,2
...
3 ,3 3 ,4
...
...
...
...
...
... ...
4 ,4
91-4. COORDINATION
A game is played using a board on which a rectangular
array of squares is drawn. Each square is labelled by two
’coordinates’: the number of the row and the number of
the column in which it lies, as shown. The score associated
with any given square is the highest common factor of its
coordinates. The game consists of moving from the top
left-hand corner of the board to the bottom right-hand
corner one square at a time to the right or downwards
(diagonal moves are not allowed), adding up the scores
of the square as you go. Find the route which gives the
maximum total score - and say what this score is - if the
board has (i) 9 rows and 9 columns, (ii) 8 rows and 16
columns.
24
91-5. ESCAPE AID
A prison has 100 convicts housed in 100 cells, which are
numbered from 1 to 100, with each prisoner having a cell
to himself. The prison has 100 warders.
Every year the warders have a party to celebrate the
governor’s birthday at which they all drink too much. At
the height of the festivities the first warder unlocks every
cell from cell number 1 to cell number 100, the second
warder then locks every second cell (2, 4, 6, 8, ?), the
third warder goes to every third cell (3, 6, 9, ?) and locks
it (if it is unlocked) and unlocks it (if it is locked); this
continues with the k’th warder visiting cells k, 2k, 3k, ?
and locking them if they are unlocked and unlocking them
if they are locked. After the last warder has finish his tour
of the cells all the warders fall asleep.
How many prisoners can escape after the 100th and last
warder has gone to sleep?
91-6. KNIGHT’S GAMBIT
During the 1939/45 war when property was cheap, the
wealthy entrepreneur Sir Grabal D’Enclosedland acquired
a large triangular field in Cornwall with a fence running
all around its perimeter. The field had two edges each
of length 800 metres and the third edge of length 1000
metres.
Recently, fearing death was at hand following a severe
attack of gout, Sir Grabal decided to give the land at
once to his two sons. In order to ameliorate his imminent
interview with the Almighty, he divided the field as fairly
as possible with a straight fence, so that each part not only
had the same area but the same length of fence enclosing
it.
Draw diagrams showing the different ways he could
have done this, calculating in each case the area of the
land given to each son and the length of the dividing fence.
25
1992
92-1. SOFT CENTRED
Terry was given a box of chocolates. Although she liked
chocolates, she was not greedy, so she decided to share her
chocolates and make them last. Her method of consumption was to eat one on the first day and give 10% of the
remainder away, to eat two on the second day and give
10% of the remainder away, to eat three on the third day
and give 10% of the remainder away, and continue in this
way until no chocolates were left.
How many chocolates were in the box and how many
days did they last?
92-2. THE LIE OF THE LAND
On a certain northern island all the inhabitants are either farmers or fishermen or fisherwomen. Before they are married, farmers always tell the truth, but fisherfolk always lie. After
marriage, however, their behaviour changes, farmers always lying and fisherfolk being truthful.
On overhearing the following conversation I immediately knew the occupations of Sara and
Robert and whether they were married or not.
Sara: Robert
Robert: Sara
Sara: Robert
Robert: Sara
is
is
is
is
not married yet.
a farmer
a fisherman
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 is
curious to know how fast the outer edge of the basket is
moving. She counts 52 teeth on the wheel she is turning.
These teeth mesh with 13 teeth at the centre of the lid
of the basket which has a diameter of 25 cm. She guesses
that her wheel turns about twice in a second so how fast,
in kilometres per hour, is the lettuce moving pressed up
against the side of the basket?
26
92-4. GOBSMACKED
Arkwright sold his gobstoppers in three different sized packets
only; small packets containing six gobstoppers, medium ones containing 9 and large ones containing 20, and he would never split
his packets open. When Ann asked for 55 gobstoppers he gave her
2 large, 1 medium and 1 small packet. For Billy’s order of 101 he
provided 4 large, 1 medium and 2 small packets. When Clare asked
for 19 he was unable to make that number. However, he foolishly
said that if she could tell him the largest number of gobstoppers
he could not supply without breaking open his packets, he would
give her twice that number free. After a few minutes’ thought she
worked 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 given
the responsibility for packing them in cubical boxes with a
side length of 60 cms. Fortunately, they had been very finely
graded and there were only two sizes, some with diameter 4
cms and others with diameter 5 cms, so he decided to pack
one size neatly in layers, with the same number in each layer,
in one box and do the same for the other size in a second
box. Remarkably, when all the apples were used up, both
boxes were full. How many apples were delivered? When
Arkwright came to pick up one he complained about his back
and asked which was the lighter. Granville did not know, but
can 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 10
cm long in the sand with his finger. This is the first step. Without lifting his finger, he draws
another line by turning left through a right angle. This time the line is 11 cm long. This is the
second step. He continues his squiral by turning left through a right angle and drawing a line 1
cm longer than the previous one. Thus, after two steps the length of the squiral is 21 cm and
after 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 nice
pattern of numbers as the step increases. Can you produce this numerical pattern? Use the
pattern 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 lines
parallel to the edges of the square find the total length of the longest squiral he can draw,
including the edge of the square.
27
1993
93-1. 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 pets
had 1, 2 or 3 children. They survey also found that no
family owned more than 2 pets and none had 2 pets of
the same kind. Furthermore, five of the families had no
children. How many families in the street had more than
3 children and how many rabbits had to share the family’s
affection with a dog?
93-2. TRUTH TABLE
When the table tennis tournament had finished, the five participants reported the results as follows.
Jackie: Rachel came second, I finished in third place
Mike: I came third. Sue came last
Rachel: 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 false
statement. Find the order of merit of the competitors.
93-3. PHYSICAL DIFFERENCE
Nuclear physicists have to find five numbers representing energies, but their experiment gives
only the differences between these numbers. They know that one number is 0 and the other four
are 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.
28
93-4. POINT TO POINT
Jane keeps her pony in a field shaped like a triangle
with each side 30 metres long. Since the pony is a good
jumper, he could escape from the field by jumping the
fence. To prevent this Jane keeps him tethered at the
centre of the field so that he can just reach the middle
of each side. What length of rope does she use and what
percentage 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 is
standing 10 metres above sea level and Brian is at the top of the Tower 150 metres above sea
level. 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 noticed
that, when the 6 o’clock news started on television, Tom’s
clock showed 5:57 p.m. Tom explained that his clock was
losing 7 minutes every hour, but that he had got used to
it.
Later in the same month Bill visited Tom again and
noticed that when the news started on the hour the clock
was showing the right time. ”I see that you had your
clock mended”, Bill remarked. ”No, I haven’t touched it”,
replied Tom. What day was it, and which news bulletin
were they watching?
29
1994
94-1. PRESSING PROBLEM
A journal ”Weekly Challenge” is published every Friday except Good Friday. Also it is not
published 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 hall
to his bedroom door. He knows that he can climb
one step or two steps at a time and wonders if he can
climb the stairs to bed in a different way every night
for a year. Make the decision for him, by finding how
many 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 table
top touches both walls and the feet are firmly on the ground. One point on the circumference
of the table, in the quarter circle between the two points of contact, is 10 cm from one wall and
5 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 triangles
with 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 sides
of 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 10
mile journey. The girls can walk at 2 mph and Tom can
jog at 4 mph. They also have a bicycle which only one
of them can use at a time. When riding, Sue and Sophie
can 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 and
riding in both directions is allowed, what is the shortest
time 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: one
set with tiles 4 x 4 cm square and the other set with tiles 5 x 5 cm square. She draws lots of
rectangles with one side 20 cm long and the other sides of various lengths. Obviously she cannot
tile the 20 cm by 1 cm, 20 cm by 2 cm, or the 20 cm by 3 cm rectangles, but she can tile the 20
cm by 4 cm and 20 cm by 5 cm rectangles with her square tiles. However, she fails again with
the 20 cm by 6 cm rectangle. Which of the larger 20 cm by n cm rectangles cannot be tiled
with her square tiles? Note that the idea is to use five of the smaller tiles or four of the larger
tiles to make a complete ‘row’ across the 20 cm, and to fill up the rectangle by such rows.
31
1995
95-1. EXPIRY DATE
Before Christmas Ken, a gullible entrepreneur, installed
in his office a new computer manufactured by the wellknown hardware firm Junior Computer Networks plc. The
specification for this machine states that, when it is running, the moving parts (disk drives and fan) last for 2000
hours, whereas the electronic components on the motherboard last for 2500. However, in the latter case this
lifetime is reduced by 2 hours every time the computer is
switched on and by 1 hour every time it is turned off. Ken
intends to use the computer from 9.00 a.m. to 5 p.m. every day from Monday to Friday inclusive, including Bank
Holidays, but not at weekends. Assuming that he switches
it on for the first time at 9.00 a.m. on January 2nd 1995,
calculate the date and exact time of day when you would
expect the computer to break down. Find also the exact
date and time when the computer will fail if Ken leaves it
running 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 regular
hexagon you can cut out of this piece of wood? Draw a picture of how you would do it. Is it
true 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 rectangles
exactly. However, she is only allowed one square tile of each size, but she can use as many
different 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 cm
she 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 bulb
which is suspended 1 metre from the centre of the ceiling. The height of the room
is 4 metres and its length is 12 metres. The shortest ray of light from the bulb to
the top of Fred’s head is, of course, 1 metre but what is the length of the ray of
light that bounces once from the wall before it meets Fred’s head? Another ray
of light bounces once from the ceiling then once from the wall before coming to
Fred’s head. What is the length of this ray?
95-5. GREEK GODS
The Alphites are immortal creatures from planet
Alpha; each one produces 1 offspring every year but
4 offspring in even numbered years from 2 onwards.
The Betons from planet Beta, are also immortal and
each one produces 0 offspring in odd numbered years
from 1 onwards and 7 in even numbered years from
2 onwards.
A new planet is colonised in year 0 by 4 newborn
Alphites and 100 newborn Betons. After how many
years will Alphites outnumber Betons?
95-6. IN THE DARK
At the Senior Challenge Prize evening in ’95 you would
have seen a coloured computer printout of the eclipse of
the Sun in May 1994. Taking the images of the Sun and
Moon to be discs of equal radius and assuming the eclipse
takes 4 minutes from beginning to end (i.e. between the
two times when the discs are just touching), find what
percentage of the area of the Sun is visible after 1 minute.
How many seconds (approximately) from the start of the
eclipse will half the area of the Sun’s disc be covered by
the Moon? [You will need to play with your calculator for
trial 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 and
are called palindromic numbers. The number 17 is not palindromic, but if its digits are reversed
to give 71, then 17 + 71 = 88 which is palindromic so 1 reversal followed by addition created a
palindromic number. The number 19 needs 2 reversals and additions to become a palindromic
number 19 + 91 = 110, 110 + 011 = 121. The number 59 needs 3 reversals and additions to
become palindromic 59 + 95 = 154, 154 + 451 = 605, 605 + 506 = 1111. How many reversals
and 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 amounts
of money, but decided that each of them would make one true and one false statement and leave
it to those interested to work out a table showing the order in terms of the amount of money
raised.
“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 people
alight. Lime Street is the first station and the train stops
at 5 intermediate stations before arriving at Euston, the
seventh station. The number of people boarding the train
at the first six stations is inversely proportional to the
number of the station while the number alighting at the
last six stations is proportional to the number of the station. How many people were on the train when it left
Liverpool?
34
96-4. CORNER TO CORNER
Janet has a 6 by 8 metre rectangular patio covered by 48 1 metre square tiles. She walks
diagonally in a straight line from one corner to the opposite corner and finds that she crosses
12 of the 48 tiles. However, her garden path has 6 tiles forming a 1 by 6 metre rectangle and a
diagonal 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 pastry
in the shape of a rectangle 60 cm by 28 cm. With her pastry cutter
she cuts disks of diameter 5 cm from this rectangle. What is the
largest number of mince pies she can make? (Don’t forget that you
need 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 pies
she 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 the
maximum 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. The
red balls made a 2 row triangle with the white balls in the 3rd row. He wondered if it was
possible using equal numbers of red and white balls to make larger triangles with the similar
property of a triangle of red balls followed by rows of white balls. After some trials he found
that a triangle of 20 rows could be made from 105 red balls and 105 white balls. The first 14
rows 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 he
found 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 this
sequence.
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
1997
97-1. BEE LINE
A male bee - also called a drone - has only one parent,
its mother. A female bee has both mother and father. So
Dennis Drone has one parent (mother), two grandparents
(mother’s parents) and three greatgrandparents (two female 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. Move
the first figure to the end—that gives 562. Now multiply the first number by 10 and subtract the second
one 2560 − 562 = 1998. The answer is a multiple of
999. Always works!”
“Amazing,” said John’s sister Liz, who was rather
a whiz at Maths. “Let me see, that means that if we
take any three-figure multiple of 37 and shift the first
figure to the end, the result will still be a multiple
of 37. For example, 259 = 37 × 7 and 592 is also a
multiple 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 equal
squares of side 1 cm cut out of the corners. The four flaps
are folded upwards to make an open-topped box. What
is the volume of this box
Suppose 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 gives
the largest possible volume?
36
97-4. HAVING IT TAPED
Wally Walkman was wondering why tape cassettes in
his personal stereo rarely lasted more than 45 minutes a
side. He realised that there was a limit to how thin tapes
could be made and decided to try to measure the thickness
of one of his tapes. The tape was wound on a spool and
with a ruler he found that the outer radius was about 2.2
cm and the inner radius was about 1.1 cm. He then tried
to count the layers but soon gave up! So he listened to
the tape instead? .
By the time the tape finished 30 minutes later he had an inspiration. The cassette cover told
him that the speed was 4.76 cm per second. With a calculator he was soon able to estimate the
tape thickness. What do you reckon his answer was?
97-5. TRUTH TO TELL
Each Saturday Amanda came home and told her father
how many goals her favourite football team, Liverpool
United, had scored. For the first seven weeks these were
7, 3, 8, 4, 9, 5, 1 goals respectively. On the following
Saturday she said merely, “Well, they scored more goals
than two weeks ago, but not as many as seven weeks ago.”
What is the largest number of weeks that she could truthfully have said this for?
In fact no matter what the scores had been in the first seven weeks she could not have
truthfully 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 a
smaller one.)
97-6. ROUGH RIFFLES
An ordinary pack of 52 cards is arranged so that the
cards alternate red, black, red, black, and so on. About
half the pack, held face down, is dealt on to the table,
taking cards one at a time from the top of the pack and
putting them face down in a single pile. Then the remaining cards, and those in the pile on the table are riffleshuffled together—that is they are roughly interleaved.
Finally, from the top of the newly shuffled pack, cards are
taken 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 cards
are taken from the top of the shuffled pack? How many of these groups will contain exactly one
card of each suit?
37
1998
98-1. HOME MOVIE
A commuter has been in the habit of arriving at his suburban
station each evening at exactly five o’clock every working day of
his life. His wife has always met him at the station and driven him
home. One day, seized by a sudden sense of reckless adventure, he
takes an earlier train, arriving at the station at four. The weather
is pleasant, so instead of telephoning home he starts walking along
the route always taken by his wife. They met somewhere along the
way. He gets into the car and they drive home, arriving at their
house ten minutes earlier than usual. Assuming that his wife always
drives at a constant speed, and that on this occasion she left just in
time to meet the usual five o’clock train, how long did hubby walk
before he was picked up?
98-2. STONE AGES
Mr and Mrs Stone have five children, Ann, Ben, Clare, David and
Elaine. Ann’s age times Ben’s age equals 36; Ben’s age times Clare’s
age equals 9; Clare’s age times David’s age equals 8; David’s age
times Elaine’s age equals 24; Elaine’s age times Ann’s age equals
12. How old are the children?
98-3. STEPTOE AND SON
Mr Steptoe and his son make a 64 km journey, starting at 6
a.m., but have only one horse (which travels at a steady 8 km per
hour) which can only carry one person at a time. Mr Steptoe can
maintain a 3 km per hour walk and his son 4 km per hour. They
alternately ride and walk. Each one ties the horse after riding a
certain distance, then walks ahead leaving the horse for the other’s
arrival. At the half-way mark they come together and take halfan-hour’s rest for lunch and to feed the horse. Assuming that they
then repeat the same travelling pattern after lunch, when do they
arrive at their destination?
38
98-4. IN THE CLEAR
Three women, Louise, Melanie and Nicola, go with their
husbands to a car boot sale to buy a variety of different objects. Their husbands’ names are Peter, Quentin
and Richard, though not necessarily in that order. Between then they buy everything available at the sale. By
a strange chance, the average price in pounds that each
person pays for her (or his) objects is the same as the
actual number of objects that she (or he) buys. Thus, if
Louise buys L objects, then they are at an average of £L
each, 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 two
players, the only rules being that the first player may play any card, but the number of diamonds
on each subsequent card played must be either a factor or a multiple of the number of diamonds
on the previous card. The first player unable to play a card loses. Who should win, the first
player or the second player—and how?
98-6. PIG AHOY
Poor old Jonah has been abandoned in a small dinghy
on the high seas by his shipmates on the Porky Pig. With
Jonah stood up in his dinghy in calm water his binoculars
are two metres above sea level. Given that the radius of
the Earth is approximately 6.3 × 10 metres, calculate how
far away the horizon appears from Jonah (in a straight
line) 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 Leg
Jake is sent up into the crow’s nest where his telescope is
15 metres above sea level. Jonah has constructed a flag in
the dinghy that is three metres above sea level. Black Leg
Jake can just see the top of Jonah’s flag on the horizon.
How far away is Jonah’s flag from Black Leg Jake (in a
straight line)?
39
1999
99-1. DIVINE ASSEMBLY
St. Divine’s High School wants to build a new assembly hall with exactly 528 seats, and
with each row having the same number of seats. There are two aisles. Each row must have 17
seats between the two aisles, with the remaining seats equally divided between the two sections
outside the aisles. How many rows should the hall have?
99-2. FOREVER EASTER
Mr Rabbit loves Easter so much that he cannot bear the thought of ever being without a
daily Easter treat. He decides, at Easter time,
to buy 365 little chocolate treats to last him for
the coming year. His local shop sells chocolate
Ants for 5p each, chocolate Bunnies for £1.60
each, and chocolate Chickens for 82p each. He
is not keen on the chocolate chickens, and buys
fewer of these than anything else. He finds that
he has spent exactly £300. How many items of
each type did he buy?
99-3. HECTOR’S HOUSE
Hector’s house lies between two bus stops, one of which lies
90 metres to the right and one 270 metres to the left. The bus
comes from the right and comes simultaneously into sight and
earshot at a point 90 metres further away from Hector’s house
than the right bus stop. Hector’s street is uphill to the right
and downhill to the left. Each day, Hector chooses one of the
two bus stops and walks towards it until he sees or hears the
bus, when he starts to run until he gets to his chosen bus stop.
Hector always walks at 2 metres per second, runs uphill at 3
metres per second and runs downhill at 5 metres per second.
The bus travels at 15 metres per second until it reaches the
first (right hand) stop where it waits for 8 seconds then travels
at 15 metres per second until it reaches the second (left hand)
stop here it again waits 8 seconds and then goes round a corner
out 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 out
18 minutes and 44 seconds later. There is no bus in sight and
Hector reckons it does not matter which stop he begins walking
to, since (with his usual strategy) he will catch the next bus
with the same amount of time to spare either way. How much
time does he have to spare when he catches the bus, and how
frequent are the buses?
40
99-4. GROTTY PAINTING
James Grot is an abstract painter, whose favourite colours are Green, Red, Orange and
Turquoise, but his most favourite is Red. He has in mind a painting which consists of two equal
squares against a Turquoise background, the first square being level (that is to say, with sides
parallel 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 the
overlap in James’s favourite colour, Red. James wants one corner of the tilted square to be at
the centre of the level square. He is not sure how much to tilt the tilted square to make the red
area 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 and
she can drive on circular arcs of radius at least 1 (in some suitable units). The figures shows
two ways she has found that she can start at the origin (which is clearly marked on the car park
tarmac) facing East and end up where she started but facing West, How far, in each case, does
she 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 straight
lines and circular arcs of radius at least 1. What is her shortest route now?
41
99-6. BLOCKBUSTERS
In the kingdom of Masochisto, they play a
two-person game with a bar of chocolate containing one poisoned square marked with an X.
Each player in turn must break the chocolate
along a horizontal or vertical line, not dividing
any of the component squares, eat one of the
two portions, and hand the remaining portion
to the other player. The loser is the one who
finally ends up with the poisoned square. For
example, if the bar is in the form
O X O
O O O
then the first player has a strategy to force a win, as follows. First, the first player breaks
the bar horizontally along the middle, eats the O O O and then gives the remaining
O X O to the second player. The second player can break this either as O X and
O 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 O
and 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.
42
2000
00-1.CUT AND COVER
Make a single cut to the wall of blocks in the
Figure, then rearrange the two pieces to make a
3 × 3 square of blocks.
00-2. SPINNING FREDDY
Fred went out for a day’s ride on his bike. After a third
of the total distance he stopped to have a rest and a bite
to eat. After another seventh of the total distance he
bought 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 from
2000 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 were
eleven people including me, and it turned out that everyone at the party had exactly three friends there.’
Jim snorted and said ’Impossible!’ Was he right?
43
00-6. TALL STORY
The Millennium Society has an ambitious plan to build
a Century Wall. Beginning on 1 January 2000 they have
every Saturday laid a new block and intend to keep on
doing so, one each Saturday. When finished the top of
the wall must be as shown in Question 1, but with many
more courses (layers), each course having two fewer blocks
than the course below. They want to finish by placing the
top block as near as possible to 31 December 2009. How
many blocks should they have in the bottom course? On
what date will they finish in 2099?
00-7. JIM AND JOAN
Jim said ‘I also went to a party last night...’ (‘Oh
no!’, said Joan under her breath)... and there were
16 people there, and each person had exactly three
friends. In fact I’ve drawn a diagram to illustrate
this, where the dots are people and the lines represent friendship.’ (‘Sounds pretty dotty to me’ said
Joan under her breath again.) ‘There was this game
where 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 by
dashed lines. The 45◦ slanting line is the path of a billiard ball starting
at the bottom left hand corner A. After five bounces it lands up in
another corner, which in this example is the bottom right hand corner
B. Try some other sizes of table, from 3 × 5 to 10 × 5, and count the
bounces. The ball always starts in the bottom left and ends up in one
of 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.
44
2001
01-1. PEN FRIENDS
Farmer Chris wants to make rectangular pens for her
chickens and pigs, the pens being of the same size, sharing
one side, as shown in the figure below. She has exactly 48
metres 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 wants
to 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 3
4 5 6
7 8 9
Joyce chose one card, but left it in place. Suppose it was the
8. Jim picked the cards up, with the column containing the chosen 8 picked up first. So after one deal and pickup the order is
2, 5, 8, 1, 4, 7, 3, 6, 9. The 8 is now in third place. Ian then dealt the
cards out in a square in the same way as before:
2 5 8
1 4 7
3 6 9
Again Jim picked up the cards with the column containing the chosen 8 picked first. After
the second deal and pickup the order is 8, 7, 9, 2, 1, 3, 5, 4, 6. The 8 has now come to first
place.
They tried it again, Joyce choosing 5 this time, and started to make a table:
Complete the table. Do you notice anything interesting?
45
01-4. A PENSIVE CHAT
Farmer Chris was chatting with her friend Farmer Leslie
in the local pub over a pint. ‘I’ll bet you could get a bigger
area for each pen than 48 square metres if you tried,’
said Leslie. ‘Nonsense!’ returned Chris, and she began
scribbling on a beer mat. Was Leslie right? Don’t forget
that Chris has only 48 metres of fencing.
01-5. OH FOR A MOON!
Alan, Brenda, Charlotte and Denis need to cross a narrow and precarious bridge in the dark. They have only
one torch between them and it must be carried on every
crossing. They all walk at different speeds. Alan can cross
the bridge in 1 minutes, Brenda takes 2 minutes, Charlotte is a lot slower at 5 minutes while Denis, who has
hurt his foot, takes a full 10 minutes to cross. The bridge
will only hold two people at a time, and when two walk
together 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 360
cm, and two strings at right-angles making four areas in
which Farmer Chris’s husband, Pythagoras, is going to
plant different colours of geraniums. The two strings are
of length 560 cm and 640 cm. Find how far the place
where the strings cross is from the centre of the circle.
This is marked in the figure with a dot, or is it a flower
pot?
46
01-7. SOME FUN!
Frankie and Charlie have a fun run. They start out from their own houses, and they run
towards each other at the same speed. One of them starts a few minutes earlier than the
other. They meet 3 miles from Frankie’s house, just say ‘Hi!’, turn round and run home, where
they turn round and run towards each other again!. They meet the second time 3 12 miles 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 25
cards, 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 do
you find when you complete the table this time? Can you explain what is happening? What
happens with 15 cards in an array with 5 rows (horizontal lines) and 3 columns? What about
other numbers of rows and column?
47
2002
02-1. SQUARES
The left-hand figure shows a 4 × 4 square cut into four pieces by two lines which are at right
angles. On the right the four pieces have been rearranged to make a slightly larger square, with
a square hole in the middle. Find the size of the square hole.
2
3
2
3
2
3
3
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% less
than the number collected by Boris, while the number collected by
Carla is 20% greater than the number collected by Boris. Anton
collects 300 mushrooms. How many does Carla collect?
00-3. HELIPAD
A helicopter landing pad is to be marked by a large equilateral
triangle, with sides of length 10 metres, divided into three parts as
shown, by two lines parallel to the base. What is the total area of
the 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 should
the 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 couples
at their party and some single people as well. At the end of the
party everyone said goodbye to everyone else, except that, naturally,
no husband said goodbye to his wife, and no wife said goodbye to
her husband. If there had been two married couples besides Lavinia
and Gerald, and two single people, how many goodbyes would have
been 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?
48
02-5. NOW FOR THE WASHING UP
’Gerald, choose a 2-figure number,’ said Lavinia, as they were
doing the washing up. ’Okay, 82,’ said Gerald, absent-mindedly
scrubbing the dishcloth with the washing-up brush. ’Now,’ continued Lavinia, make another 2-figure number by taking the separate figures of your number away from 9.’ ’Hmmm,’ said Gerald,
carefully washing the dishcloth with a dirty plate, ’9 − 8 = 1 and
9 − 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 the
soapsuds with the handle of a spoon. ”Finally,’ said Lavinia, subtract 9 and divide by 9 ... do
you notice anything?’ ’Well,’ said Gerald, ’747 − 9 = 738 and 738/9 = 82 ... that’s the number
I 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 opposite
end of the board. Lou starts. If, after any number of moves, a black counter lies between two
white counters either horizontally or diagonally (as in these pictures) then the black counter is
captured and taken off the board.
Can Lou get both her pieces from the top of the board to the bottom, or can Mike prevent
her?
49
02-7. PENTAJIG
Colour this jigsaw with five colours in such a way that in each row, in each column and in
each of the five pentominoes comprising the jigsaw each of the five colours occurs once and only
once. (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? Suppose
that 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
7
1
8
4
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 many
different shapes of hole are there? (The shape of a rectangle is measured by dividing the longer
side by the shorter side, so for example 2 × 3 and 12 × 8 rectangles have the same shape.) Are
any 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?
50
2003
03-1. CUISENAIRE
Suppose you have rods with lengths 1, 2, 4, 8. Then, apart from these lengths you can make
quite a few others, for example 3 = 2 + 1, 5 = 4 + 1, 7 = 4 + 2 + 1. What in fact is the shortest
whole number length you cannot make with these four rods?
03-2. UNEXPECTED GUEST
Giovanni and Leporello invite four young
ladies to a quiet evening of pizza and
Karaoke. They cut the circular pizza into
six equal pieces and are about to start eating when an unexpected guest arrives. Leporello cuts equal amounts off the six pieces
to give to the guest so that all seven people have the same amount of pizza. What
fraction of each piece did he cut off?
03-3. GOOD DOGS
‘Be good dogs,’ said Mr Sumhope as he left Fido
and Trusty to guard his house while he was out.
When they were alone, the two dogs started to
tear the living room carpet into pieces. When
Fido chose a piece he tore it into four parts,
and when Trusty chose a piece she tore it into
seven parts. Being good dogs, they never chose
the same piece at the same time. When Mr
Sumhope returned he found 2003 pieces of carpet. Were there any missing?
03-4. GARDENERS’ QUESTION TIME
Mrs Pythagoras is making a semicircular patio, as in the figure, with three squares exactly fitting
as 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?
51
03-5. DOING THE SPLITS
Amanda and her brother Gareth were playing with six-figure numbers, splitting them in
the middle to make two three-figure numbers.
Amanda found one six-figure number which was
exactly 7 times what she got by multiplying together her two three-figure numbers. Find this
six-figure number if you can. (Hint: 1001 is divisible by 7.)
Gareth looked for a six-figure number which actually equalled what he got by multiplying together his two three-figure numbers. Do you
think he succeeded?
03-6. RUNNING MATES
Two runners, Chris and Alex, each running at his own constant speed, leave their houses at the
same time, each running towards the other’s house. They meet 600m from Chris’s house but
continue running, turning round as soon they they reach each other’s house and meeting again
400m from Alex’s house. How far apart are the houses?
52
03-7. SQUARE BASHING
Two clever painters Abe and Beth are playing
the following game. They start with a table having marked on it a 6 × 4 grid of squares. They
take turns and at each turn must paint a square
of any size with the edges along the grid lines
and not overlapping the previously painted area.
For example, the figure shows a possible position after Abe painted a 2 × 2 square and Beth
painted a 1 × 1 square next to it and Abe then
painted 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 dimensions
are even or both are odd.)
53