a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). . . .

Three dice are placed in a row. Find a way to turn each one so that the three numbers on top of the dice total the same as the three numbers on the front of the dice. Can you find all the ways to do. . . .

This article for the young and old talks about the origins of our number system and the important role zero has to play in it.

When you think of spies and secret agents, you probably wouldn’t think of mathematics. Some of the most famous code breakers in history have been mathematicians.

How many zeros are there at the end of the number which is the product of first hundred positive integers?

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?

How many different ways can you arrange the officers in a square?

This black box reveals random values of some important, but unusual, mathematical functions. Can you deduce the purpose of the black box?

Work with numbers big and small to estimate and calulate various quantities in biological contexts.