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?

Overlaying pentominoes can produce some effective patterns. Why not use LOGO to try out some of the ideas suggested here?

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

Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers?

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

There are thirteen axes of rotational symmetry of a unit cube. Describe them all. What is the average length of the parts of the axes of symmetry which lie inside the cube?

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

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?

Triangle ABC has equilateral triangles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. What can you prove about the triangle PQR?

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

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

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?

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?

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.

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

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

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...

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.

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 find triangles on a 9-point circle? Can you work out their angles?

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

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.

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

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. . . .

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

A game in which players take it in turns to choose a number. Can you block your opponent?

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

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

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?

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.

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

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?

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

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

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.

An Excel spreadsheet with an investigation.

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

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

Use an Excel spreadsheet to explore long multiplication.

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

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

Find the frequency distribution for ordinary English, and use it to help you crack the code.

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.

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?

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

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