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

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

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.

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?

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

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?

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

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?

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 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?

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

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?

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. . . .

Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . .

Help the bee to build a stack of blocks far enough to save his friend trapped in the tower.

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

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

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. . . .

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 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 beat the computer in the challenging strategy game?

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.

Match pairs of cards so that they have equivalent ratios.

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

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

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

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

This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo!

On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?

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?

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

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

in how many ways can you place the numbers 1, 2, 3 … 9 in the nine regions of the Olympic Emblem (5 overlapping circles) so that the amount in each ring is the same?

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. . . .

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?

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.

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?

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

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?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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.

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 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?

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.

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

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