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Age 16 to 18
Challenge Level

Why do this problem?

This problem gives a fascinating insight into topology and various ideas from higher mathematics. It is ideal as a guided class activity or a solitary workout for the more able maths student. It will provide interesting insights to those interested in pursuing mathematics beyond school.

The ideas raised in this problem would make for interesting visual displays of mathematics.

Possible approach

The problem naturally splits into two parts. First it is crucial that students will need to understand how the square relates to the physical torus and then students can begin to concentrate on the colouring aspect.

A good way to test understanding of the first part is to ask students to give explanations of the setup to each other. Clear explanations will give evidence of understanding; students must realise that closed loops drawn on the torus will yield lines which pass through opposite points on the squares. You can draw some patterned squares which don't give nice pictures on the torus to reinforce the point that opposite sides on the squares are to be identified (some suggestions are shown in the key questions).

The second part of the problem is best approached practically, with students being encouraged to draw designs on squares and work through the possible colourings. There are two levels of sophistication that can be used. At one level students can draw designs on the square and try to work out the minimal colouring. (Note again that for a design to be valid, the lines must intersect the opposite sides of the square at the correct corresponding points.). At the highest level, students can try to create patterns with the specific properties of needing 5, 6 and 7 colourings.

Although the thinking level in this problem is quite high, the content level is relatively low; the problem could be attempted by younger students, perhaps in a maths club context.

The design possibilities for this task are interesting, and perhaps students could try to draw the designs from various squares onto tori, or vice versa.

Key questions

Do you understand how the squares relate to tori? Can you see why the images below don't give rise to closed loops on tori?

Possible extension

There are, in fact, no designs which need more than 7 colours to fill. Although the proof will be beyond students, interested students could research this idea on the internet. Alternatively, students could try to find as many topologically different patterns which require 7 colours to fill.

Possible support

You could first, or instead, try the problem Painting By Numbers which raises many of the interesting colouring ideas of this problem but without the problems raised by the topology of the torus.