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

Use this interactivity to sort out the steps of the proof of the formula for the sum of an arithmetic series. The 'thermometer' will tell you how you are doing

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 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. . . .

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

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

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

Which of these triangular jigsaws are impossible to finish?

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. . . .

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.

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.

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

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

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.

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

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

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

Can you work through these direct proofs, using our interactive proof sorters?

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

Can you correctly order the steps in the proof of the formula for the sum of a geometric series?

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.)

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

What can you say about the common difference of an AP where every term is prime?

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

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

Mark a point P inside a closed curve. Is it always possible to find two points that lie on the curve, such that P is the mid point of the line joining these two points?

Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

Can you work out where the blue-and-red brick roads end?

Prove Pythagoras' Theorem using enlargements and scale factors.

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

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

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

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

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

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

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

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.

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.

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.

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

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

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 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. . . .

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.