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

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. . . .

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

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?

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. . . .

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

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

Three frogs started jumping randomly over any adjacent frog. Is it possible for them to finish up in the same order they started?

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

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

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

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

Can you find the areas of the trapezia in this sequence?

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. . . .

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. . . .

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

Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

This is the second of two articles and discusses problems relating to the curvature of space, shortest distances on surfaces, triangulations of surfaces and representation by graphs.

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.

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

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. . . .

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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?

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

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.

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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.

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

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

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.

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

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. . . .

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

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 think that mathematical proof is really clearcut and universal then you should read this article.

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

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

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

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?

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

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

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

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.