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

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

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

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Nine cross country runners compete in a team competition in which there are three matches. If you were a judge how would you decide who would win?

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

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?

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

Kyle and his teacher disagree about his test score - who is right?

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?

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. . . .

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.

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.

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . .

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .

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

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?

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

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

Prove Pythagoras' Theorem using enlargements and scale factors.

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. . . .

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

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.

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.

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.

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

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

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

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?

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. . . .

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.

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

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.

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

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!

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

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...

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

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

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.

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

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. . . .