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?

Imagine picking up a bow and some arrows and attempting to hit the target a few times. Can you work out the settings for the sight that give you the best chance of gaining a high score?

Carry out some time trials and gather some data to help you decide on the best training regime for your rowing crew.

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

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

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

7 balls are shaken in a container. You win if the two blue balls touch. What is the probability of winning?

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

Work out how to light up the single light. What's the rule?

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

Interactive game. Set your own level of challenge, practise your table skills and beat your previous best score.

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?

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

Meg and Mo still need to hang their marbles so that they balance, but this time the constraints are different. Use the interactivity to experiment and find out what they need to do.

A game for 1 person to play on screen. Practise your number bonds whilst improving your memory

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?

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

Can you locate the lost giraffe? Input coordinates to help you search and find the giraffe in the fewest guesses.

Meg and Mo need to hang their marbles so that they balance. Use the interactivity to experiment and find out what they need to do.

How many different triangles can you make which consist of the centre point and two of the points on the edge? Can you work out each of their angles?

Practise your diamond mining skills and your x,y coordination in this homage to Pacman.

Mo has left, but Meg is still experimenting. Use the interactivity to help you find out how she can alter her pouch of marbles and still keep the two pouches balanced.

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

Learn how to use the Shuffles interactivity by running through these tutorial demonstrations.

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

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.

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

A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.

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

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

Two engines, at opposite ends of a single track railway line, set off towards one another just as a fly, sitting on the front of one of the engines, sets off flying along the railway line...

Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.

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.

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.

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

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?

Two circles of equal radius touch at P. One circle is fixed whilst the other moves, rolling without slipping, all the way round. How many times does the moving coin revolve before returning to P?

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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.

This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.

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

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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.