# How many ways are there to set up the pieces?

AttilaTheHorn
Posers and Puzzles 17 Oct '07 11:34
1. AttilaTheHorn
Erro Ergo Sum
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.
2. 17 Oct '07 13:50
Originally posted by AttilaTheHorn
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 ...[text shortened]... ng of the game.

That’s approximately equal to the number of seconds there are in 6,595 years.
The rules of Chess state that a unique position consists of "The same type and color of pieces on the same squares, with the same player to move, with the same moves possible." (see rules for three fold repition)

Also,althogh the rules of Chess do not distinguish between one white rook and another; they do distinguish between the squares a1, a8, h1, and h8: so reflections and rotaions of the pieces on the board will result in positions that are not legal.

there is therefore one and only one legal starting posion.

(this is of course ignoring the possibility of thematic tournaments)
3. AThousandYoung
All My Soldiers...
17 Oct '07 14:35
Originally posted by preachingforjesus
The rules of Chess state that a unique position consists of "The same type and color of pieces on the same squares, with the same player to move, with the same moves possible." (see rules for three fold repition)

Also,althogh the rules of Chess do not distinguish between one white rook and another; they do distinguish between the squares ...[text shortened]... tarting posion.

(this is of course ignoring the possibility of thematic tournaments)
He's counting it as two different setups if you swap the physical Pawns.
4. AttilaTheHorn
Erro Ergo Sum
17 Oct '07 15:32
Of course, the way I'm answering the question is the way a mathematician would answer it. Chess is a very useful subject for dealing with such things.
5. joe shmo
Strange Egg
23 Oct '07 01:42
Originally posted by AttilaTheHorn
Of course, the way I'm answering the question is the way a mathematician would answer it. Chess is a very useful subject for dealing with such things.
That all depends on how setting up the board is defined, you mean that there are that many different ways only if an individual pieces could be labled and swaped, but there is only one way to set up a chess board for a legal game.
6. 23 Oct '07 10:17
Originally posted by AttilaTheHorn
Of course, the way I'm answering the question is the way a mathematician would answer it. Chess is a very useful subject for dealing with such things.
That depends on the mathematician. A mathematician working on set theory would tell you that {1,2} is the same as {2,1}.

Richard