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

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

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

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

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

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.

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?

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

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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

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?

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.

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?

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?

Can you explain the strategy for winning this game with any target?

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

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

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 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.

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

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.

This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.

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

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

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

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...

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. . . .

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. . . .

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.

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?

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.

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?

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

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?

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

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

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.

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

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

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.

Identical discs are flipped in the air. You win if all of the faces show the same colour. Can you calculate the probability of winning with n discs?

An activity based on the game 'Pelmanism'. Set your own level of challenge and beat your own previous best score.

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

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.