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

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?

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

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...

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

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

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

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

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

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

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

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

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. . . .

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

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 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. . . .

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, 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?

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

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. . . .

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?

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.

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

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

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

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

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

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.

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?

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

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?

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

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.

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

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

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?

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!

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

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?

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

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

When is it impossible to make number sandwiches?

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

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.

What is the largest number of intersection points that a triangle and a quadrilateral can have?