Find out how we can describe the "symmetries" of this triangle and investigate some combinations of rotating and flipping it.

This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

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!

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

What happens to these capital letters when they are rotated through one half turn, or flipped sideways and from top to bottom?

Find a way to cut a 4 by 4 square into only two pieces, then rejoin the two pieces to make an L shape 6 units high.

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

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?

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. . . .

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

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

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.

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

Cut a square of paper into three pieces as shown. Now,can you use the 3 pieces to make a large triangle, a parallelogram and the square again?

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?

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

Explore the effect of reflecting in two intersecting mirror lines.

Explore the effect of combining enlargements.

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.

These grids are filled according to some rules - can you complete them?

Why not challenge a friend to play this transformation game?

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.

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. . . .

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. . . .

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

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

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

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

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

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