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?

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

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?

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

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?

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

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

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?

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?

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

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?

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

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?

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. . . .

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?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

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

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

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

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

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?

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

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

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

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

Can you cut up a square in the way shown and make the pieces into a triangle?

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?

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

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

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?

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?

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

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?

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?

Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?

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

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

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

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?

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

Watch this animation. What do you see? Can you explain why this happens?

Can you use the interactive to complete the tangrams in the shape of butterflies?

Can you fit the tangram pieces into the outline of this goat and giraffe?

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

An activity centred around observations of dots and how we visualise number arrangement patterns.

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

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