Combinatorics: A Very Short Introduction by Robin Wilson

Author:Robin Wilson
Language: eng
Format: azw3, epub
Tags: Combinatorics, Mathematics
ISBN: 9780198723493
Publisher: Oxford University Press
Published: 2016-03-15T18:38:36.109965+00:00

It can be shown that the number of essentially different colourings is found by averaging these numbers, giving essentially different colourings, as before.

In a similar way, we can find the number of essentially different black-and-white colourings of an 8 × 8 chessboard, but here the number is so large that we can no longer list them explicitly:

Clockwise rotation through 90° or 270°: Here we can arbitrarily choose how to colour each of the top-left sixteen squares, and these must then be replicated in the other quarters of the board, so the number of colourings that remain unchanged in each case is 216.

Clockwise rotation through 180°: Here we can arbitrarily choose any colouring for the thirty-two squares in the top half of the board, and these must be replicated in the lower half, so the number of colourings that remain unchanged is 232.

Clockwise rotation through 360°: This is the same as leaving the square in its original position, so there are two choices of colour for each square and all 264 colourings remain unchanged.

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