How Many Different Chess Positions Are There? (Quiz)

Here is a math quiz for chess players. Some of this has previously been discussed on chess.com. The solutions are posted 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.

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

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

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

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