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

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?

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. . . .

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

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube so that the surface area of the remaining solid is the same as the surface area of the original?

Can you mentally fit the 7 SOMA pieces together to make a cube? Can you do it in more than one way?

What is the minimum number of squares a 13 by 13 square can be dissected into?

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?

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

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

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.

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.

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

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

Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?

What can you see? What do you notice? What questions can you ask?

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. . . .

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?

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

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?

A huge wheel is rolling past your window. What do you see?

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?

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

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?

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?

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

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

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

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.

This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

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?

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

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

A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start. How many Hamiltonian circuits can you find in these graphs?

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?

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

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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

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

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

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

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

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

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