Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

Exchange the positions of the two sets of counters in the least possible number of moves

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

Can you fit the tangram pieces into the outline of Little Ming?

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface. . . .

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?

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

Can you work out what is wrong with the cogs on a UK 2 pound coin?

How many different triangles can you make on a circular pegboard that has nine pegs?

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

Can you make a 3x3 cube with these shapes made from small cubes?

A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start. How many Hamiltonian circuits can you find in these graphs?

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

Can you fit the tangram pieces into the outline of Granma T?

Make a flower design using the same shape made out of different sizes of paper.

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

Can you mark 4 points on a flat surface so that there are only two different distances between them?

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.

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

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

Can you fit the tangram pieces into the outline of these convex shapes?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

Can you fit the tangram pieces into the outline of this junk?

Can you fit the tangram pieces into the outline of the rocket?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

Can you fit the tangram pieces into the outline of this plaque design?

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

Can you fit the tangram pieces into the outline of this sports car?

Here's a simple way to make a Tangram without any measuring or ruling lines.