Colouring maps: the four colour theorem

A primary-school lab (ages 8-12), very visual: everything is done by painting squares, triangles, honeycomb cells, numbers and countries with a click. Each room sets a colouring rule and asks the student to follow it, to count how many ways there are, or to decide whether it can be done at all.

The thread of the lab is the idea of a neighbour: two squares (or two countries) are neighbours if they share a side, not if they only touch at a point. That idea builds ever stricter rules, colours maps with four colours and, at the end, shows that a good colouring can prove that something is impossible, even when the problem says nothing about colours.

What students learn

• Recognise neighbours on a grid, on a triangle split into little triangles and on a honeycomb, and tell «sharing a side» from «touching at a point».
• Follow rules about neighbours (different colour, «three and one», «no orange touches another»), and see that a rule may leave many solutions, two or just one.
• Count in an orderly way (the hidden squares of a 4 × 4 board: 9 of 2 × 2 and 4 of 3 × 3).
• Look for a maximum and justify it: with 12 colours and two equal cells in every 2 × 2 square, the maximum is 11, and there is a reason why you cannot do better.
• Deduce forced colours (coloured numbers: the product of two different colours gets the third) and recognise a clash that proves a puzzle has no solution.
• Colour maps with few colours; understand why sometimes 2 or 3 are enough and sometimes 4 are needed (the flower with 5 petals), and meet the four colour theorem.
• Use the chessboard colouring as a tool: the beetles on the 5 × 5 board (13 against 12) and walks through every square that cannot start or end where asked.

Key mathematical ideas, by block

A · Neighbouring squares. With two colours and «neighbours of different colours» there are only two colourings: the chessboard and its opposite (on the triangle, the ones pointing up in one colour and the ones pointing down in the other). A square on the edge or in the corner has fewer neighbours, and that decides many rules: in «orange with 3 blue neighbours and blue with 1 orange neighbour» no orange can sit in a corner; on the 4 × 4 board there are two solutions, mirror images of each other, and on the triangle of 16, only one. To prove a maximum you need an example and a reason: nine 2 × 2 squares, each equal pair serves at most two, so 5 repeats are needed and 16 − 5 = 11 colours remain.

B · Coloured numbers. The rule makes colours «spread»: once 2 and 3 are known, 6 is forced, then 12, 18… Sometimes you reason backwards (if 9 were not blue, 18 could not be blue). With {2, 3, 4, 6, 12, 24} the forced colours clash: you do not need to try every way to know there is none.

C · Maps. Neighbours = a shared border. A pizza with 4 or 6 slices can be coloured with 2 colours; one with 5 cannot. The island needs 3. The flower with 5 petals needs 4: the centre uses up one colour and the petals, which form an odd ring, cannot alternate just two. The hard map with 20 countries can also be coloured with 4, like any map (the four colour theorem, proved in 1976 with the help of a computer).

D · Colouring to prove. The chessboard is not in the question: we paint it ourselves. Every step to a neighbour changes colour, so 13 beetles do not fit in 12 squares, a walk through all 25 squares of the 5 × 5 board must start on the colour that has 13, and on the 4 × 4 board you cannot visit every square going from one corner to the opposite one. The lab ends with a colour sudoku.

Numbers checked by brute force in scripts/verify_colorear.mjs.