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

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

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

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?

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?

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

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?

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.

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

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

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

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

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?

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

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.

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.

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 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.

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.

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. . . .

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

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.

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.

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

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?

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

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

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.

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

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

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?

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?

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

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

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. . . .

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

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...

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

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.

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

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

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?

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.

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.

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.

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.

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?