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?

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

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?

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?

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?

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

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?

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.

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

Take it in turns to place a domino on the grid. One to be placed horizontally and the other vertically. Can you make it impossible for your opponent to play?

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

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

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

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

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

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?

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?

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

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5.

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

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

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

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

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

Which of these dice are right-handed and which are left-handed?

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

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

Make a cube out of straws and have a go at this practical challenge.

A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.

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

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!

Can you fit the tangram pieces into the outline of the child walking home from school?

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

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

Can you fit the tangram pieces into the outlines of these clocks?

Can you fit the tangram pieces into the outlines of these people?

Exploring and predicting folding, cutting and punching holes and making spirals.

Can you fit the tangram pieces into the outlines of the chairs?

Can you fit the tangram pieces into the outline of this shape. How would you describe it?

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?

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?

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?

Imagine a 3 by 3 by 3 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will have holes drilled through them?

Can you fit the tangram pieces into the outlines of the watering can and man in a boat?

Can you fit the tangram pieces into the outlines of the candle and sundial?

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