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Enigmas & puzzles

By Ian Stewart   August 2006

Desperately daubing dominoes

Sanguimenta was sitting at the kitchen table with a box of dominoes and four tubes of paint—red, yellow, green, and blue.

“Not often I see you relaxing,” said her sister Nonsequita.

“Relaxing? I’m not relaxing! I’m doing some research on the four-colour theorem!”

“What’s that?”

“Any map on a flat sheet of paper can be coloured, using at most four colours, so that adjacent countries have different colours,” said Sanguimenta. “The countries can be any shape you like. It was first proposed in 1852 but remained unproved until 1976. It’s easy to see that four colours are necessary by using curved boundaries. But I’ve been wondering what happens if every country is shaped like a domino.”

“A domino is just a rectangle, right?”

“Yes. With the long side twice the length of the short side. You can make ‘domino maps’ by laying dominoes of the same size out flat. I was wondering whether any domino map actually needs four colours.”

“And what did you decide?”

Sanguimenta picked up six painted dominoes—two yellow, two red, one green and one blue. They were still a bit wet and the paint got all over her hands and the table, but she seemed not to notice.

“If I take these six dominoes and arrange them like this,” she said, “then you can see that no two adjacent dominoes have the same colour. But you can also prove that it’s impossible to colour the same map with only three colours.”

What does Sanguimenta’s map look like?


Scroll down for the answer



The answer

The map looks like this
(or similar to it).

Here’s why it can’t be done with three colours. If domino one is the first colour, then two must be different. So three and four must both be the third colour—but that would mean both five and six have to be the first colour.

The winner was I Crann from London

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