Enigma 1662: Red face?
4 December 2011
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From New Scientist #2828, 3rd September 2011 [link]
Our local craft shop stocks an ornament made of a piece of carved wood. Its surface consists of a number of faces, all being different colours but the same-sized regular polygon, and with the same number of faces meeting at each vertex. I bought a number of these ornaments, all absolutely identical, and placed them on my mantelpiece.
My nephew saw them and noted that three of them had the same colour face resting on the mantelpiece and that in all other cases the colours were different. “That’s an unnatural set-up,” he commented, and he arranged them so that all the colours resting on the surface were different.
However, he was embarrassed when I proved to him that the previous situation was more natural because, if the ornaments were placed at random, then there was 50 per cent more chance of having three the same (and the rest different) than having them all different.
How many ornaments did I have, and what shape were they?