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

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. . . .

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

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?

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. . . .

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

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.

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?

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

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. . . .

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. . . .

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.

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

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

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .

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.

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

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

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

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

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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 shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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?

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

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

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?

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?

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.

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.

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

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}.

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

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

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

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

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.

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.

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

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

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.

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

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.

Prove Pythagoras' Theorem using enlargements and scale factors.

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

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

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