What happens when you try and fit the triomino pieces into these two grids?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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?

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

Move just three of the circles so that the triangle faces in the opposite direction.

How will you go about finding all the jigsaw pieces that have one peg and one hole?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

What is the best way to shunt these carriages so that each train can continue its journey?

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

A toy has a regular tetrahedron, a cube and a base with triangular and square hollows. If you fit a shape into the correct hollow a bell rings. How many times does the bell ring in a complete game?

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

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?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

How many balls of modelling clay and how many straws does it take to make these skeleton shapes?

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?

What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour?

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. . . .

Can you find ways of joining cubes together so that 28 faces are visible?

What does the overlap of these two shapes look like? Try picturing it in your head and then use the interactivity to test your prediction.

Here are shadows of some 3D shapes. What shapes could have made them?

Try to picture these buildings of cubes in your head. Can you make them to check whether you had imagined them correctly?

Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

What is the greatest number of squares you can make by overlapping three squares?

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

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?

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

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?

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

Can you describe a piece of paper clearly enough for your partner to know which piece it is?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?

When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

Find your way through the grid starting at 2 and following these operations. What number do you end on?

Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.

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

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

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