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...

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

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

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total.

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. . . .

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

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?

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. . . .

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

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. . . .

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

Can you make sense of these three proofs of Pythagoras' Theorem?

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

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

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

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.

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

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

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

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}.

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

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

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?

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

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

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.

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. . . .

Prove Pythagoras' Theorem using enlargements and scale factors.

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

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

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

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

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

Can you explain why a sequence of operations always gives you perfect squares?

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

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

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.

Keep constructing triangles in the incircle of the previous triangle. What happens?

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. . . .

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. . . .

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

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.