17 Oct '07 11:34

How many ways are there to set up the chess pieces on a chessboard in a legal position for the start of the game?

Answer: Well, the answer is not one. OK, so you can turn the board around and set it up from the other side too. But the answer is not two either.

Considering the White pieces only, there are eight pawns, each of which can go on eight different squares. That makes 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320 ways to set up the White pawns. Then you can set up each rook, knight, and bishop on two different squares. Therefore, this makes 2³ = 2 x 2 x 2 = 8 different ways for them to be set up. The king and queen can each go on only one square each, so we don’t need to consider them. We now have 40,320 x 8 = 322,560 ways to set up White, but that’s only one side. The same number is also true for the Black pieces.

For each of the 322,560 ways to set up the pieces on one side, there are also the same number of ways to set up the other side too, thus making (322,560)² = 104,044,953,600 ways to set up the board, but we’re not finished yet. There is still the reflection obtained by putting one side on the other side, or exchanging sides. So we multiply that number by 2 and there are exactly 208,089,907,200 ways to set up a chessboard in a legal position at the beginning of the game.

That’s approximately equal to the number of seconds there are in 6,595 years.

Answer: Well, the answer is not one. OK, so you can turn the board around and set it up from the other side too. But the answer is not two either.

Considering the White pieces only, there are eight pawns, each of which can go on eight different squares. That makes 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320 ways to set up the White pawns. Then you can set up each rook, knight, and bishop on two different squares. Therefore, this makes 2³ = 2 x 2 x 2 = 8 different ways for them to be set up. The king and queen can each go on only one square each, so we don’t need to consider them. We now have 40,320 x 8 = 322,560 ways to set up White, but that’s only one side. The same number is also true for the Black pieces.

For each of the 322,560 ways to set up the pieces on one side, there are also the same number of ways to set up the other side too, thus making (322,560)² = 104,044,953,600 ways to set up the board, but we’re not finished yet. There is still the reflection obtained by putting one side on the other side, or exchanging sides. So we multiply that number by 2 and there are exactly 208,089,907,200 ways to set up a chessboard in a legal position at the beginning of the game.

That’s approximately equal to the number of seconds there are in 6,595 years.