(related to Problem: Setting The Board)

The White pawns may be arranged in $40,320$ ways, the White rooks in $2$ ways, the bishops in $2$ ways, and the knights in $2$ ways. Multiply these numbers together, and we find that the White pieces may be placed in $322,560$ different ways. The Black pieces may, of course, be placed in the same number of ways. Therefore the men may be set up in $322,560 \times 322,560 = 104,044,953,600$ ways. But the point that nearly everybody overlooks is that the board may be placed in two different ways for every arrangement. Therefore the answer is doubled and is $208,089,907,200$ different ways.

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Project Gutenberg

  1. Dudeney, H. E.: "Amusements in Mathematics", The Authors' Club, 1917

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