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.

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. . . .

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

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.

This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.

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?

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...

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

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.

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?

Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . .

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?

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?

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

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?

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

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

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?

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?

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?

From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How. . . .

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

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

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

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

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

What can you say about the angles on opposite vertices of any cyclic quadrilateral? Working on the building blocks will give you insights that may help you to explain what is special about them.

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

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

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

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

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.

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

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

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?

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.

Here are some examples of 'cons', and see if you can figure out where the trick is.

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

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

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

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.