In this game, try not to colour two adjacent regions the same colour. Can you work out a strategy?

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

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

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

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

Make a mobius band and investigate its properties.

It might seem impossible but it is possible. How can you cut a playing card to make a hole big enough to walk through?

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. . . .

Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

This article for pupils describes the famous Konigsberg Bridge problem.

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. . . .

Did you know that ancient traditional mazes often tell a story? Remembering the story helps you to draw the maze.

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start. How many Hamiltonian circuits can you find in these graphs?

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

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?

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. . . .

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

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.

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

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.

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.

This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.