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?

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

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

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.

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

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

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?

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

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

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. . . .

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.

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. . . .

In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

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...

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.

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?

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

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

When is it impossible to make number sandwiches?

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

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

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

Can you rearrange the cards to make a series of correct mathematical statements?

What is the largest number of intersection points that a triangle and a quadrilateral can have?

How many noughts are at the end of these giant numbers?

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

This is the second article on right-angled triangles whose edge lengths are whole numbers.

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

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

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

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

Four jewellers share their stock. Can you work out the relative values of their gems?

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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

This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .

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

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

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.

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?