12 Oct 2013

Circle Patterns With Four Colours: Professor Pailyn's Mathematical Quiz PMQ40


Alice was waiting for her friend Brenda to arrive; she was late. Alice had prepared some games – Professor Pailyn seemed to have so many games, yet no children to actually play them. To pass the time, Alice was making patterns from a collection of circular plastic counters. Professor Pailyn walked in with a jug of what looked like wine but was most probably apple juice. Before Alice could make a sound, he walked out again. He quickly returned with some glasses.

“Professor, look, I can make patterns like this and I can do it with just three colours. Isn’t there some idea that you need four colours?”

“Yes, I think you mean the four colour map theorem.” He eased himself into a comfy armchair.”Take any plane, such as a sheet of paper, draw some regions – you can try to make them as complicated as you please. Then, if you try to colour each region in such a way that no two adjacent regions have the same colour, you find that only a maximum of four colours are needed.”


“Adjacent... you mean ‘next to each other’, right?”

“Yes, adjacent means they share a boundary, or at least part of a boundary. If regions meet at a point, that doesn’t count as adjacent; they have to meet along part of an edge.”

“But what about circles? I can make all sorts of patterns but I only ever need three colours. Look!”

Professor Pailyn stood up to look at Alice’s patterns spread out on the table. “Oh, I see what you’re trying to do. So ‘adjacent’ in this case means circles that are ‘touching’, and you want no two touching circles to be of the same colour.”

“Right!”

The Professor thought for a little while and a smile started to grow on his face. “Actually, there are configurations of coloured circles that do require four colours. Try to find one with the smallest number of circles.”

“Really?! You mean just like I’m trying to do? A pattern of touching circles, with any touching pair having different colours, but in such a way that four colours are absolutely necessary?”

“Yes. Let me see how long that’ll take you.” Alice frowned slightly. That was a challenge. She had long forgotten about Brenda. "And, one small hint: it doesn't have to be a repeating pattern, like you've been doing."

How many coloured circles are needed, and what is their arrangement, so that four different colours are necessary to ensure that no two touching circles have the same colour? You may submit a link to a diagram or explain the arrangement in words... or just state the minimum number of circles needed.

Have fun!

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Solution has now been published HERE.

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