Many chess positions that one may easily set up on a chess board are impossible to achieve in a game of legal moves. For example, among the impossible situations would be:

• A position in which both kings are in check.
• A position in which there is are pawns on the first or on the last rank.
• A position with two white pawns on the same file, but black still has all his pieces.
• A position with a white bishop on the first rank, trapped by two white pawns on the second rank, but the bishop is not on c1 or f1.
• A position with two black-square white bishops and eight white pawns.

The logician Raymond Smullyan wrote a delightful book The Chess Mysteries of Sherlock Holmes: Fifty Tantalizing Problems of Chess Detection, containing many interesting chess detective stories, some involving positions that were impossible for sometimes very subtle reasons.

My question is:

Question. What proportion of the chess positions that one can set up on the board, using a legal collection of chess pieces, can actually arise in a legal chess game?

What I mean is that collection of pieces is legal, if it occurs in a position of a legal chess game, a game played according to the rules. This collection is somewhat broader than one might naively expect, since it is legally possible, for example, to have a king of each color with nine white queens, as white may have promoted all the pawns while all other pieces were captured. And other similarly strange collections of pieces are possible. So the collection of positions I am considering are those that can be obtained by messing up the pieces on the board from an actual legal game.

Of course it will be too difficult to get an exact answer, and I shall be satisfied merely with good bounds. The Wikipedia page on chess and mathematics mentions some numbers, including estimates on the number of legal positions, but the information there doesn't seem to answer this question. Perhaps those who are more familiar with that work can point to where this question is answered there.

I guess the answer must be a rather small proportion, because it seems that many legal chess positions can be easily transformed into many illegal ones, by placing both kings in check, by adding a pawn to the first rank (unless all pawns are already used), etc. Is this right, and can such an argument be used to make tight bounds?

I am here at the Mountain Lake Chess Camp, where we've been discussing the question, when one of the instructors mentioned the numerical bounds on the total number of chess positions, and the question arose whether this included impossible-to-achieve positions or not.

Many chess positions that one may easily set up on a chess board are impossible to achieve in a game of legal moves. For example, among the impossible situations would be:

• A position in which both kings are in check.
• A position in which there are pawns on the first or on the last rank.
• A position with two white pawns on the same file, but black still has all his pieces.
• A position with a white bishop on the first rank, trapped by two white pawns on the second rank, but the bishop is not on c1 or f1.
• A position with two black-square white bishops and eight white pawns.

The logician Raymond Smullyan wrote a delightful book The Chess Mysteries of Sherlock Holmes: Fifty Tantalizing Problems of Chess Detection, containing many interesting chess detective stories, some involving positions that were impossible for sometimes very subtle reasons.

My question is:

Question. What proportion of the chess positions that one can set up on the board, using a legal collection of chess pieces, can actually arise in a legal chess game?

What I mean is that collection of pieces is legal, if it occurs in a position of a legal chess game, a game played according to the rules. This collection is somewhat broader than one might naively expect, since it is legally possible, for example, to have a king of each color with nine white queens, as white may have promoted all the pawns while all other pieces were captured. And other similarly strange collections of pieces are possible. So the collection of positions I am considering are those that can be obtained by messing up the pieces on the board from an actual legal game.

Of course it will be too difficult to get an exact answer, and I shall be satisfied merely with good bounds. The Wikipedia page on chess and mathematics mentions some numbers, including estimates on the number of legal positions, but the information there doesn't seem to answer this question. Perhaps those who are more familiar with that work can point to where this question is answered there.

I guess the answer must be a rather small proportion, because it seems that many legal chess positions can be easily transformed into many illegal ones, by placing both kings in check, by adding a pawn to the first rank (unless all pawns are already used), etc. Is this right, and can such an argument be used to make tight bounds?

I am here at the Mountain Lake Chess Camp, where we've been discussing the question, when one of the instructors mentioned the numerical bounds on the total number of chess positions, and the question arose whether this included impossible-to-achieve positions or not.

Many chess positions that one may easily set up on a chess board are impossible to achieve in a game of legal moves. For example, among the impossible situations would be:

• A position in which in which both kings are in check.
• A position in which there is are pawns on the first or on the last rank.
• A position with two white pawns on the same file, but black still has all his pieces.
• A position with a white bishop on the first rank, trapped by two white pawns on the second rank, but the bishop is not on c1 or f1.
• A position with two black-square white bishops and eight white pawns.

The logician Raymond Smullyan wrote a delightful book The Chess Mysteries of Sherlock Holmes: Fifty Tantalizing Problems of Chess Detection, containing many interesting chess detective stories, some involving positions that were impossible for sometimes very subtle reasons.

My question is:

Question. What proportion of the chess positions that one can set up on the board, using a legal collection of chess pieces, can actually arise in a legal chess game?

What I mean is that collection of pieces is legal, if it occurs in a position of a legal chess game, a game played according to the rules. This collection is somewhat broader than one might naively expect, since it is legally possible, for example, to have a king of each color with nine white queens, as white may have promoted all the pawns while all other pieces were captured. And other similarly strange collections of pieces are possible. So the collection of positions I am considering are those that can be obtained by messing up the pieces on the board from an actual legal game.

Of course it will be too difficult to get an exact answer, and I shall be satisfied merely with good bounds. The Wikipedia page on chess and mathematics mentions some numbers, including estimates on the number of legal positions, but the information there doesn't seem to answer this question. Perhaps those who are more familiar with that work can point to where this question is answered there.

I guess the answer must be a rather small proportion, because it seems that many legal chess positions can be easily transformed into many illegal ones, by placing both kings in check, by adding a pawn to the first rank (unless all pawns are already used), etc. Is this right, and can such an argument be used to make tight bounds?

I am here at the Mountain Lake Chess Camp, where we've been discussing the question, when one of the instructors mentioned the numerical bounds on the total number of chess positions, and the question arose whether this included impossible-to-achieve positions or not.

Many chess positions that one may easily set up on a chess board are impossible to achieve in a game of legal moves. For example, among the impossible situations would be:

• A position in which both kings are in check.
• A position in which there is are pawns on the first or on the last rank.
• A position with two white pawns on the same file, but black still has all his pieces.
• A position with a white bishop on the first rank, trapped by two white pawns on the second rank, but the bishop is not on c1 or f1.
• A position with two black-square white bishops and eight white pawns.

The logician Raymond Smullyan wrote a delightful book The Chess Mysteries of Sherlock Holmes: Fifty Tantalizing Problems of Chess Detection, containing many interesting chess detective stories, some involving positions that were impossible for sometimes very subtle reasons.

My question is:

Question. What proportion of the chess positions that one can set up on the board, using a legal collection of chess pieces, can actually arise in a legal chess game?

What I mean is that collection of pieces is legal, if it occurs in a position of a legal chess game, a game played according to the rules. This collection is somewhat broader than one might naively expect, since it is legally possible, for example, to have a king of each color with nine white queens, as white may have promoted all the pawns while all other pieces were captured. And other similarly strange collections of pieces are possible. So the collection of positions I am considering are those that can be obtained by messing up the pieces on the board from an actual legal game.

Of course it will be too difficult to get an exact answer, and I shall be satisfied merely with good bounds. The Wikipedia page on chess and mathematics mentions some numbers, including estimates on the number of legal positions, but the information there doesn't seem to answer this question. Perhaps those who are more familiar with that work can point to where this question is answered there.

I guess the answer must be a rather small proportion, because it seems that many legal chess positions can be easily transformed into many illegal ones, by placing both kings in check, by adding a pawn to the first rank (unless all pawns are already used), etc. Is this right, and can such an argument be used to make tight bounds?

I am here at the Mountain Lake Chess Camp, where we've been discussing the question, when one of the instructors mentioned the numerical bounds on the total number of chess positions, and the question arose whether this included impossible-to-achieve positions or not.