[QUOTE=Christenson;257622]How much state information do we need beyond the positions of the pieces to distinguish all the configurations?

I think we need to keep track of whether four rooks can be castled, and whether 16 pawns have the chance to capture en passant.[/QUOTE]

Yes on the four rooks -- that's one bit each, give or take. The pawns are easier: at most one pawn on the board can be vulnerable, so that should take less than 16 bits to describe.

[QUOTE=Christenson;257622]I think we need to keep track of whether four rooks can be castled, and whether 16 pawns have the chance to capture en passant.[/QUOTE]
Castling is only allowed once per king.

[QUOTE=http://www.chessvariants.org/d.chess/castlefaq.html]When are you not allowed to castle?
There are a number of cases when castling is not permitted.
Your king has been moved earlier in the game.
The rook that castles has been moved earlier in the game.
There are pieces standing between your king and rook.
The king is in check.
The king moves through a square that is attacked by a piece of the opponent.
The king would be in check after castling.[/QUOTE]

and
[url]http://en.wikipedia.org/wiki/Castling[/url]

The issue was referred to earlier in post #33 plus William's post #41 which I initially dismissed but later acknowledged with apology.

As well as castling and En Passant ability, other features needed are:
- who is to move (a diagrammed position with White to move is not the same as the same diagram with Black to move)
- what moves have been played since the last pawn move or capture by either side.

The second one is explained by the fact that either player can claim a draw if about to move into a position which will have occurred at least 3 times, or one in which 50 moves have been played by both sides without pawn move or capture. The exact moves since last pawn move or capture (rather than just the number of them) are important in determining possible future repetitions of position. A pawn move or capture alters the position sufficiently so that repetition of a previous position is impossible.

HOWEVER: I would like to repeat my view, hinted at in post #33, that these niceties are somewhat incongruous because the upper and lower bounds which we can hope to calculate are going to be miserably weak. Perhaps only the consideration of which player is to move, which almost doubles the number of legal positions (only a minority of positions are legal with only one of the players to move) is really worth worrying about. Even then I don't think we can get anywhere near the type of accuracy-certainty which will make a factor of 2 important. Castling and En Passant make much less of a difference. Moves since last pawn move or capture a lot more, but then we're straining the definition of a chess position and trying to run before we can walk in terms of solving this very difficult problem. Sorry for my pessimism.:smile:

[QUOTE=Christenson;257622]Thank you Mr P-1, for pointing out a fundamental ambiguity in the thread title. I thought we were trying to count configurations, not positions, as defined above.

How much state information do we need beyond the positions of the pieces to distinguish all the configurations?

I think we need to keep track of whether four rooks can be castled, and whether 16 pawns have the chance to capture en passant.

@SM88: Can you tell me the intent of the word "positions" in the thread title?[/QUOTE]

my original thought to your question is. if you don't know by now why give advice or suggest I'm wrong on the amount in the first place?

[QUOTE=Uncwilly;257641]Castling is only allowed once per king.[/QUOTE]

Christenson is right, though -- we can just track whether each rook is allowed to castle, marking both as unable to castle once a castle occurs (even if with the other rook). Of course there are other circumstances under which the rook will be marked unable to castle.

[QUOTE=Brian-E;257648]

HOWEVER: I would like to repeat my view, hinted at in post #33, that these niceties are somewhat incongruous because the upper and lower bounds which we can hope to calculate are going to be miserably weak.
<snip>
Sorry for my pessimism.:smile:[/QUOTE]

I'm just as pessimistic about the ultimate bounds; professional effort had orders of magnitude between the provable upper bound and the best estimate.

However, I hope to create some mathematical precision in the discussion, and niceties like whether one pawn can take en passant, which side is to move, and whether a rook can be involved in a castle are an important part of that.

if I could remember my Wikipedia account information I'd suggest we make a page walking through them they have a setup of how to display them.

[QUOTE=science_man_88;257659]my original thought to your question is. if you don't know by now why give advice or suggest I'm wrong on the amount in the first place?[/QUOTE]
Because I *thought* I knew which was intended; however, you have a habit of writing with such ambiguity that we all live on quicksand. Many others here also clearly thought they knew. I also don't always need a great deal of precision to spot numbers that are mathematically hopeless and clearly wrong, and you have plenty of history to suggest that in any case.

Back to my questions:
1) What exactly do we mean when we count the number of legal positions with three pieces on the board? Are we ignoring en passant and castling when we count? What about symmetry?
2) With an exact definition in hand, what is the number of positions with three pieces on the board? How do you prove it?

I suggest we finish these before we get on to the more complicated problems of counting the number of legal positions with four pieces on the board through 31 pieces on the board, which we will need as part of counting all the legal positions.

I take legal to mean "achievable without violating the accepted rules of chess", by players who have reaching the specified position as a goal.

[QUOTE=Christenson;257733]Because I *thought* I knew which was intended; however, you have a habit of writing with such ambiguity that we all live on quicksand. Many others here also clearly thought they knew. I also don't always need a great deal of precision to spot numbers that are mathematically hopeless and clearly wrong, and you have plenty of history to suggest that in any case.

Back to my questions:
1) What exactly do we mean when we count the number of legal positions with three pieces on the board? Are we ignoring en passant and castling when we count? What about symmetry?
2) With an exact definition in hand, what is the number of positions with three pieces on the board? How do you prove it?

I suggest we finish these before we get on to the more complicated problems of counting the number of legal positions with four pieces on the board through 31 pieces on the board, which we will need as part of counting all the legal positions.

I take legal to mean "achievable without violating the accepted rules of chess", by players who have reaching the specified position as a goal.[/QUOTE]

my original idea was a chess position count. symmetry I would think not because each place on the board has a specific name. Also what would we do with symmetrical variants of the openings ( which usually come with a position attached)? as for the en passant and castling I don't see why not (though I doubt a castle happening with 3 pieces left is likely from the odds of check and having to move out of it, capture the piece, or block).

[QUOTE=science_man_88;257732]if I could remember my Wikipedia account information I'd suggest we make a page walking through them they have a setup of how to display them.[/QUOTE]

Since I actually rethought I didn't have a wikipedia account ( I think it was a rosettacode.org account I was thinking of) I've made an account on wikipedia and can now use my user page for explanation.

[QUOTE=science_man_88;257734]my original idea was a chess position count. symmetry I would think not because each place on the board has a specific name. Also what would we do with symmetrical variants of the openings ( which usually come with a position attached)? as for the en passant and castling I don't see why not (though I doubt a castle happening with 3 pieces left is likely from the odds of check and having to move out of it, capture the piece, or block).[/QUOTE]

I still don't know what a chess position count means to you. Does it mean
a) A particular arrangment of chess men on a board that might be reached in a game? (Mr P-1's "configuration")?
b) The arrangement of chess men AND the additional information sufficient to determine all possible games that might follow?(Mr P-1's "position"), including which side is to move next.

Concretely, I'm trying to understand your answers to the following question:
How many distinct positions do we have if the white king is on a2 and the black king on h1? If we keep track of whether white or black is to move next, we have two positions. If we don't, we have one position.

I think you are telling me that in the three piece case, we count it as two distinct positions if a king and rook are in castling position. The one position still has castling allowed and the other doesn't.

I think you are telling me that if two pawns are adjacent, it is three distinct positions depending on whether one of the pawns can be taken en passant.

I think you are telling me that the following two positions are distinct:
a) white king on a1, black king on h8
b) white king on a8, black king on h1
That is, we are counting symmetric, tactically identical positions as distinct.

Let me know the answer to the first question and if the answers to the remaining questions are correct; I'm pretty sure the answers are sufficient to let us prove the number of positions with three pieces on the board.

Mr Greathouse, have I missed anything?