With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

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?

Explore the effect of combining enlargements.

Explore the effect of reflecting in two intersecting mirror lines.

A cylindrical helix is just a spiral on a cylinder, like an ordinary spring or the thread on a bolt. If I turn a left-handed helix over (top to bottom) does it become a right handed helix?

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

Explore the effect of reflecting in two parallel mirror lines.

Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.

Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . .

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

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

Draw all the possible distinct triangles on a 4 x 4 dotty grid. Convince me that you have all possible triangles.

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

How many different transformations can you find made up from combinations of R, S and their inverses? Can you be sure that you have found them all?

Sort the frieze patterns into seven pairs according to the way in which the motif is repeated.

Can you find a way to turn a rectangle into a square?

See the effects of some combined transformations on a shape. Can you describe what the individual transformations do?

Does changing the order of transformations always/sometimes/never produce the same transformation?

Why not challenge a friend to play this transformation game?

Jenny Murray describes the mathematical processes behind making patchwork in this article for students.

This article looks at the importance in mathematics of representing places and spaces mathematics. Many famous mathematicians have spent time working on problems that involve moving and mapping. . . .

This problem is based on the idea of building patterns using transformations.

How many differently shaped rectangles can you build using these equilateral and isosceles triangles? Can you make a square?

An introduction to groups using transformations, following on from the October 2006 Stage 3 problems.

Have you ever noticed how mathematical ideas are often used in patterns that we see all around us? This article describes the life of Escher who was a passionate believer that maths and art can be. . . .

Scientist Bryan Rickett has a vision of the future - and it is one in which self-parking cars prowl the tarmac plains, hunting down suitable parking spots and manoeuvring elegantly into them.

This task develops knowledge of transformation of graphs. By framing and asking questions a member of the team has to find out which mathematical function they have chosen.

A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.

Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.