How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct.

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

The sums of the squares of three related numbers is also a perfect square - can you explain why?

A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.

Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total.

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation?

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Which set of numbers that add to 10 have the largest product?

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .

Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

After some matches were played, most of the information in the table containing the results of the games was accidentally deleted. What was the score in each match played?

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

Can you fit Ls together to make larger versions of themselves?

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

Can you make sense of the three methods to work out the area of the kite in the square?

Four jewellers possessing respectively eight rubies, ten saphires, a hundred pearls and five diamonds, presented, each from his own stock, one apiece to the rest in token of regard; and they. . . .

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.