Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . .

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

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

To avoid losing think of another very well known game where the patterns of play are similar.

This game challenges you to locate hidden triangles in The White Box by firing rays and observing where the rays exit the Box.

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.

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

An environment that enables you to investigate tessellations of regular polygons

Can you find a reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses?

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Match pairs of cards so that they have equivalent ratios.

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

The interactive diagram has two labelled points, A and B. It is designed to be used with the problem "Cushion Ball"

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

Here is a chance to play a fractions version of the classic Countdown Game.

A collection of our favourite pictorial problems, one for each day of Advent.

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

Can you give the coordinates of the vertices of the fifth point in the patterm on this 3D grid?

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

Six balls of various colours are randomly shaken into a trianglular arrangement. What is the probability of having at least one red in the corner?

Ask a friend to choose a number between 1 and 63. By identifying which of the six cards contains the number they are thinking of it is easy to tell them what the number is.

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.

A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?

Use Excel to explore multiplication of fractions.

There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you. . . .

You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.

Can you beat Piggy in this simple dice game? Can you figure out Piggy's strategy, and is there a better one?

A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red.

Can you spot the similarities between this game and other games you know? The aim is to choose 3 numbers that total 15.

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Which dilutions can you make using 10ml pipettes and 100ml measuring cylinders?

Can you fill in the mixed up numbers in this dilution calculation?

Which exact dilution ratios can you make using only 2 dilutions?

Can you work out which spinners were used to generate the frequency charts?

Can you break down this conversion process into logical steps?

Practise your skills of proportional reasoning with this interactive haemocytometer.

A simple spinner that is equally likely to land on Red or Black. Useful if tossing a coin, dropping it, and rummaging about on the floor have lost their appeal. Needs a modern browser; if IE then at. . . .