How Many Different Chess Positions Are There? (Solutions)

These are solutions to the math quiz for chess players. Some of this has previously been discussed on chess.com. The original quiz is on my blog.

Problem 1) What is the total number of different chess positions, including illegal positions?

To answer this, make the following assumptions.

• Use a standard chessboard labeled with algebraic notation.
• There are 12 different chessmen, with an unlimited number of each.
• Each individual square can have any one of the 12 chessmen, or it can be empty.
• Two positions are the same if they match on square a1, match on square a2, … , and match on square h8.
• Based on this definition, position A and position B are the same if they have the same chessmen in the same locations, even if one of the kings is allowed to castle in position A and not allowed to castle in position B.
• An empty chessboard counts as a position.

For example, one position could have 3 black kings, one black pawn and one white pawn, with the other squares empty.

Solution 1) There are 13⁶⁴ different chess positions.

Each square can have 12 different chessmen or be empty. There are 13 possibilities for each square, and there are 64 squares. Multiply the number of possibilities for each square to find the total number of different positions.

13⁶⁴ = 196053476430761073330659760423566015424403280004115787589590963842248961 = 1.96*10⁷¹

Problem 2) How many different positions are there with exactly one white king and exactly one black king on the board, including illegal positions?

Solution 2) There are 64*63*(11⁶²) positions with one white king and one black king.

There are 64 squares to place the white king on. That leaves 63 squares to place the black king on. The 62 remaining squares each have 11 different possibilities since they can have 10 different chessmen or be empty. Multiply the number of possibilities in each case to compute the total number of possibilities.

64*63*(11⁶²) = 148548066446350731075585726626009926340696949635420129892166819995072 = 1.49*10⁶⁸

Problem 3) Prove that the number of legal positions in standard chess is less than 0.1% of the total number of positions.

Solution 3) 64*63*(11⁶²)/(13⁶⁴) = .000758 < 0.001 = 0.1%

Every legal position contains exactly one white king and exactly one black king, so the total in problem 2 contains every legal position as well as many illegal positions. The total in problem 1 contains every possible position. Divide the total from problem 2 by the total from problem 1. The number of positions containing exactly one white king and exactly one black king is less than 0.1% of the total number of positions, and the number of legal positions is (much) smaller than that.

Problem 4) Are there more legal positions in chess960 or crazyhouse?

Solution 4) Crazyhouse has many more legal positions than chess960.

If you ignore castling privileges, then every legal position in chess960 is also a legal position in crazyhouse, but crazyhouse has many positions that are illegal in chess960.

Problem 5) What is the maximum number of light square bishops that white can legally have on the board at one time in crazyhouse?

Solution 5) White can have a maximum of 20 light square bishops in crazyhouse.

To do this, white needs to lose his dark square bishop and recapture it. White needs to capture all the black bishops and pawns. White needs to place all the bishops on light squares, and promote all 16 pawns on light squares, selecting a bishop each time.