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

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

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

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

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

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.

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

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 Pythagoras Theorem using enlargements and scale factors.

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.

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

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.

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Which set of numbers that add to 10 have the largest product?

This is an interactivity in which you have to sort into the correct order the steps in the proof of the formula for the sum of a geometric series.

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.

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...

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

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum orf two or more cubes.

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

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?

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

Show that if three prime numbers, all greater than 3, form an arithmetic progression then the common difference is divisible by 6. What if one of the terms is 3?

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.

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

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?

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?

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

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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

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

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.

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

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.

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

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.

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

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?

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

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.

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

A introduction to how patterns can be deceiving, and what is and is not a proof.