In how many ways can you fit all three pieces together to make shapes with line symmetry?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?

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 of the purposes and skills of visualising.

Sharon Walter, an NRICH teacher fellow, talks about her experiences of trying to embed NRICH tasks into her everyday practice.

Two sudokus in one. Challenge yourself to make the necessary connections.