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?

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?

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

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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

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 the shaded area of the semicircle is equal to the area of the inner circle.

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

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. . . .

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

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

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

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. . . .

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?

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?

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

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

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.

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

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.

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

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

Prove Pythagoras' Theorem using enlargements and scale factors.

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.

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.

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.

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.

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}.

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!

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.

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

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 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. . . .

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.

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 lengths of the sides of a quadrilateral whose vertices are on a unit circle?

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

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. . . .

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.

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

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.