This article (the first of two) contains ideas for investigations. Space-time, the curvature of space and topology are introduced with some fascinating problems to explore.

How many different colours would be needed to colour these different patterns on a torus?

Is it possible to make an irregular polyhedron using only polygons of, say, six, seven and eight sides? The answer (rather surprisingly) is 'no', but how do we prove a statement like this?

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. . . .

What if the Earth's shape was a cube or a cone or a pyramid or a saddle ... See some curious worlds here.

This is the second of two articles and discusses problems relating to the curvature of space, shortest distances on surfaces, triangulations of surfaces and representation by graphs.

A personal investigation of Conway's Rational Tangles. What were the interesting questions that needed to be asked, and where did they lead?

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots, prime knots, crossing numbers and knot arithmetic.

How many different colours of paint would be needed to paint these pictures by numbers?

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.