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B
ALANCING
S
CALES
Problem
In order for the scales to balance the block on the right side must be 10 kg.
It is possible to use the same weights to weigh, for example, 4 kg.
In fact using 2 kg, 3 kg and 5 kg weights it is possible to weigh all but one value from
1 kg to 10 kg. What is that value?
(Note: You don't have to use all three weights.)
Solution
The following weights can be found, with the desired weight indicated in brackets.
Left Side  Right Side
3
2 (1)
2
(2)
3
(3)
2 5
3 (4)
5
(5)
3 5
2 (6)
2 5
(7)
3 5
(8)
2 3 5
(10)
And so it is not possible to weigh 9 kg.
What if you used 1 kg, 3 kg and 6 kg to weigh all the weights from 1 kg to 10 kg?
What about using 1 kg, 2 kg and 7 kg?
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If x + y + z = T, what must be special about x, y and z so that all the weights from 1
kg to T kg can be weighed?
(Hint: Try x + y = T to begin with.)
Problem ID: 92 (Dec 2002)     Difficulty: 1 Star     [
]
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B
EAM
O
F
L
IGHT
Problem
If a light beam is fired through the top left corner of a 2 by 3 rectangular prism block
at a 45 degrees towards the opposite wall, it will emerge from the top right corner.
If the light bounces around a 2 by 4 rectangle, it emerges from corner D.
If a beam of light is fired through the top left corner of a 2 by 50 rectangle, which
corner will it emerge from?
Solution
Consider the two diagrams.
It can be seen from this that rectangles measuring 2 2 2, 2 2 6, 2 2 10, 2 2 14, ... , 2 2 50,
that is, even non-multiples of 4, will finish in the bottom right corner, C.
Will the light ever return to A?
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Investigate other size grids, 3 3 k, 4 4 k, et cetera.
Problem ID: 63 (Jan 2002)     Difficulty: 1 Star     [
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B
IGGER
D
IGIT
Problem
For each of the numbers: 41, 83, 32, the first digit is greater in value than the second
digit.
How many 2-digit numbers have this property?
Solution
If we begin to list the numbers in groups: 10; 20,21; 30,31,32; 40,41,42,43; ... ;
90,91,92,93,94,95,96,97,98 ; we can see that the total number of 2-digit numbers,
for which the first digit is greater than the second digit, will be 1 + 2 + ... + 9 = 45.
How many 3-digit numbers exist for which the first digit is greater in value than both
the second digit and the third digit?
Can you generalise for n-digit numbers?
What about 3-digit numbers for which the first digit is greater than the sum of the
second and third digits?
Problem ID: 218 (30 Mar 2005)     Difficulty: 1 Star     [
]
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B
IRDS
A
ND
B
UNNIES
Problem
A cage contains birds and rabbits. There are sixteen heads and thirty-eight feet. How
many birds are there in the cage?
Solution
If we let the number of birds be represented by b and the number of rabbits be
represented by r then we get the following two equations:
b + r = 16 (1)
2b + 4r = 38 (2)
Dividing the second equation by two gives:
b + 2r = 19 (3)
If we now subtract equation (1) from equation (3) we get r = 3, and as b + r = 16 it
follows that the number of birds, b, must be 13.
What if you were told that the number of feet was equal to twice the number of
What if there were sixteen heads and twenty-eight feet?
Problem ID: 351 (17 Apr 2009)     Difficulty: 1 Star     [
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B
IRTHDAY
P
ARTY
Problem
At a birthday party, one-half drank only lemonade, one-third drank only cola, fifteen
people drank neither, and nobody drinks both.
How many people were at the party?
Solution
As 1/2 + 1/3 = 3/6 + 2/6 = 5/6, we know that 1/6 drank neither.
So there must have been 6  15 = 90 people at the party.
Problem ID: 274 (21 Apr 2006)     Difficulty: 1 Star     [
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