Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

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.

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

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

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. . . .

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.

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.

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...

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

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

An article which gives an account of some properties of magic squares.

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

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

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

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. . . .

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

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 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!

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

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?

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.

Can you rearrange the cards to make a series of correct mathematical statements?

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?

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

What fractions can you divide the diagonal of a square into by simple folding?

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

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

This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .

Have a go at being mathematically negative, by negating these statements.

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.

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

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

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

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?

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?

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. . . .

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

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.