Question 1
Consider the number 32a35717b. It is possible to replace a and b with a (different)
digit so that the overall number is divisible by 72.
Find what these digits are, justifying why there is only one solution to this.
Question 2
My bike lock requires a four-digit code to unlock it. I always remember it because
a) it is a perfect square, and
b) it has the property that, when it is divided by 2, or 3, or 4, or 5, or 6, or 7, or
8, or 9 there is always a remainder of 1.
What is my four-digit code?
Question 3
A square is constructed inside a rectangle of side lengths a and b, with the square
touching the diagonal of the rectangle as shown in the diagram below.
If the square has side length h, prove that
1
1 1
= +
h a b
Question 4
(a) Numbers a, b and c have been written at the vertices of a triangle (one at each
vertex). They are such that each one is the average of its two neighbours.
Do all three numbers have to be the same?
(b) Numbers a1 , a2 , . . . , a10 have been written at the vertices of a decagon (one at
each vertex), such that each number is the average of its two neighbours.
Do all ten numbers have to be the same?
(c) An 8 × 8 chessboard has a number written on each square, so that each square
is the average of its neighbours (note that a square could have three neighbours
if it is in the corner, 5 if it is on the edge, or 8 if it’s not on the edge).
Do all 64 numbers have to be the same?
Question 5
I have 20 integers a1 , a2 , a3 , . . . , a20 , none of which is divisible by 5.
Show that if I sum up the 20th powers, of these, i.e.
20
20
20
a20
1 + a2 + a3 + . . . + a20
then the resulting number is divisible by 5.
Question 6
The midpoints of all three sides of a triangle have been marked. The original
triangle is then erased, leaving only the three marked points. How can the original
triangle be recreated using only a compass and straightedge?
Question 7
A white rook and a black bishop are on a nonstandard chessboard. They take
turns moving according to standard chess rules: the bishop moves diagonally any
number of squares, and the rook moves up, down or across any number of squares.
The rook moves first.
How should the rook move to capture the bishop if the size of the board is 3 × 10?
What if the board is 3 × 1000?
Question 8
I stayed in a hotel overnight and my hotel room was a 3 digit number. I thought that
the same number could be obtained by multiplying together all of the following:
i) one more than the first digit;
ii) one more than the second digit;
iii) the third digit.
Prove that I was mistaken.
Question 9
Consider the number line. Suppose that every integer point on it is coloured either
red or green. For example, the number line *could* be coloured like so (and
extending infinitely to the left and right).
−5 −4 −3 −2 −1
0
1
2
3
4
5
Prove that no matter how the number line is coloured, it is always possible to find
three integers such that the following conditions hold:
1. they all have the same colour, and
2. the middle one is the average of the other two (e.g. 1, 3, 5, or 5, 12, 19, or more
generally a − d, a, a + d for some integers a, d).
For example, in the above picture, the numbers −1, 2, 5 are all green and satisfy
both conditions. The numbers −4, 0, 4 are all red and also satisfy both conditions!
Question 10
A group of n people participate in a round-robin arm-wrestling tournament. Each
match ends in either a win or a loss (no draws!).
Explain how it is possible to label the players P1 , P2 , . . . , Pn in such a way that:
• P1 defeated P2
• P2 defeated P3
• etc. etc., and
• Pn−1 defeated Pn
but with no other restrictions. By “no other restrictions”, this means for example,
that P1 doesn’t have to have beaten P3 (only P2 ).
Question 11
The numbers 1 to 16 are written in a table as in the diagram on the left.
1
2
3
4
+1
-2
-3
+4
5
6
7
8
-5
+6
-7
+8
9
10
11
12
+9
-10
+11
-12
13
14
15
16
-13
+14 +15
-16
A plus sign or a minus sign is written in front of each number so that there are two
pluses and two minuses in each row and column. (For example, one possible way
of doing this is on the above right.)
Prove that the sum of the entries in the table is always 0.
Question 12 (Lights Out)
Suppose that n > 2 light bulbs are arranged in a row, numbered 1 to n. Under
each bulb is a button. Pressing the button will change the state of the bulb above
it (from on to off, or vice versa), and will also change the neighbours’ states (most
bulbs have two neighbours, but the bulbs on the end each only have one). The
bulbs start randomly (some on and some off).
The aim is to keep pressing the buttons until all the bulbs have been turned off,
i.e. to go “lights out”.
a) If n = 2, can we guarantee lights out? If yes, give a sequence of button presses
for each of the four starting states. If no, give a starting state which can’t go
lights out, and explain why.
b) If n = 3, can we guarantee lights out? If yes, give a sequence of button presses
for each of the eight starting states. If no, give a starting state which can’t go
lights out, and explain why.
c) If n = 4, can we guarantee lights out? If yes, give a sequence of button presses
for each of the sixteen starting states. If no, give a starting state which can’t
go lights out, and explain why.
d) Investigate general values of n, and (if you can) prove which cases we can/can’t
guarantee lights out.