How many different positions are there with 32 chessmen on the board?
How many different chess positions are there, with all 32 standard chessmen on the board, including illegal positions?
Start with a standard chessboard labeled using algebraic coordinates, and a standard set of 32 chessmen.
- Place each of the 32 chessmen on a square, so that no square contains more than one chessman.
- When counting positions, there are eight white pawns and eight black pawns on the board. None of the pawns are promoted.
- Every possible position is allowed, including illegal positions.
- Two positions are the same if they have the same type of chessmen on the same squares, and each square is labeled with algebraic coordinates.
- Since some of these positions are illegal, the ability to castle or capture en passant is ignored. Position A and position B are the same if they have the same chessmen on the same squares, even if it is possible to castle or capture en passant in position A, but not in position B.
To count the number of positions, number the chessmen from 1 to 32.
- Number 1 to 8 are white pawns.
- Number 9 and 10 are white knights.
- Number 11 and 12 are white bishops.
- Number 13 and 14 are white rooks.
- Number 15 is the white queen.
- Number 16 is the white king.
- Number 17 to 24 are black pawns.
- Number 25 and 26 are black knights.
- Number 27 and 28 are black bishops.
- Number 29 and 30 are black rooks.
- Number 31 is the black queen.
- Number 32 is the black king.
Place the chessmen on the board in order, from number 1 to 32.
- There are 64 squares to choose from for the 1st chessman.
- There are 63 squares remaining to choose from for the 2nd chessman.
- There are 62 squares remaining to choose from for the 3rd chessman.
- Continuing this process, there are 33 squares remaining to choose from for the 32nd chessman.
Let x be the total number of ways to place the 32 numbered chessmen on the board.
x = (64!) / (32!)
= 482219923991114978843459072919892677776312893440000000
= 4.82 (1053)
Some of these outcomes produce duplicate positions.
- The eight white pawns are interchangeable, so divide x by 8! to eliminate duplicate positions.
- The two white knights are interchangeable, so divide x by 2.
- The two white bishops are interchangeable, so divide x by 2.
- The two white rooks are interchangeable, so divide x by 2.
- The eight black pawns are interchangeable, so divide x by 8! to eliminate duplicate positions.
- The two black knights are interchangeable, so divide x by 2.
- The two black bishops are interchangeable, so divide x by 2.
- The two black rooks are interchangeable, so divide x by 2.
After eliminating duplicate positions, let y be the total number of different positions with all 32 standard chessmen on the board.
y = (64!) / [ (32!) (8!) (8!) (26) ]
= 4634726695587809641192045982323285670400000
= 4.63 (1042)
The picture shows some of the Lewis Chessmen.
