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?

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

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 is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.

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

Imagine a 4 by 4 by 4 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will not have holes drilled through them?

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

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

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.

Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!

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

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?

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

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

A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?

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

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?

Can you split each of the shapes below in half so that the two parts are exactly the same?

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. . . .

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

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.

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?

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

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

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?

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

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

Make one big triangle so the numbers that touch on the small triangles add to 10.

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

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

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!

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 ...

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

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

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

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?

A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

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

Billy's class had a robot called Fred who could draw with chalk held underneath him. What shapes did the pupils make Fred draw?

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

This second article in the series refers to research about levels of development of spatial thinking and the possible influence of instruction.

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

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

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?

What shape is made when you fold using this crease pattern? Can you make a ring design?

Can you visualise what shape this piece of paper will make when it is folded?