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?

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

When dice land edge-up, we usually roll again. But what if we didn't...?

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Find all the ways to cut out a 'net' of six squares that can be folded into a cube.

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

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

Here are the six faces of a cube - in no particular order. Here are three views of the cube. Can you deduce where the faces are in relation to each other and record them on the net of this cube?

A 3x3x3 cube may be reduced to unit cubes in six saw cuts. If after every cut you can rearrange the pieces before cutting straight through, can you do it in fewer?

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

How can you paint the faces of these eight cubes so they can be put together to make a 2 x 2 x 2 cube that is green all over AND a 2 x 2 x 2 cube that is yellow all over?

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?

This is the first article in a series which aim to provide some insight into the way spatial thinking develops in children, and draw on a range of reported research. The focus of this article is the. . . .

Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?

Can you mark 4 points on a flat surface so that there are only two different distances between them?

Can you find a way of representing these arrangements of balls?

Can you make a 3x3 cube with these shapes made from small cubes?

This article explores ths history of theories about the shape of our planet. It is the first in a series of articles looking at the significance of geometric shapes in the history of astronomy.

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't. . . .

Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube?

Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.

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?

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?

Which of the following cubes can be made from these nets?

The second in a series of articles on visualising and modelling shapes in the history of astronomy.

The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

What is the shape of wrapping paper that you would need to completely wrap this model?

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

In the game of Noughts and Crosses there are 8 distinct winning lines. How many distinct winning lines are there in a game played on a 3 by 3 by 3 board, with 27 cells?

In this article for primary teachers, Fran describes her passion for paper folding as a springboard for mathematics.

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?

A useful visualising exercise which offers opportunities for discussion and generalising, and which could be used for thinking about the formulae needed for generating the results on a spreadsheet.

This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . .

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

Can you fit the tangram pieces into the outline of the sports car?

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

If you move the tiles around, can you make squares with different coloured edges?

This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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

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.

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

A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?

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

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

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