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

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?

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.

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?

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?

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

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 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?

Why not challenge a friend to play this transformation game?

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

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

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

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

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?

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. . . .

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

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

Explore the effect of reflecting in two parallel mirror lines.

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

Explore the effect of reflecting in two intersecting mirror lines.

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

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.

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.

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

Explore the effect of combining enlargements.

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. . . .

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

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. . . .

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