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

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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?

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

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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?

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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?

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

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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?

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

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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?

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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?

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

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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?

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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.

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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?

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

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

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

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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?

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Seeing Squares game for an adult and child. Can you come up with a way of always winning this game?

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

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

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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?

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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. . . .

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

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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?

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

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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?

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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?

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

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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?

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

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Can you fit the tangram pieces into the outlines of the convex shapes?

This article looks at levels of geometric thinking and the types of activities required to develop this thinking.

Challenge Level

A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.

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

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

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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.

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Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?

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

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Can you fit the tangram pieces into the outlines of the camel and giraffe?

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

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Can you fit the tangram pieces into the silhouette of the junk?

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Can you fit the tangram pieces into the outlines of Mah Ling and Chi Wing?

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Can you fit the tangram pieces into the outline of the playing piece?

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

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

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