[QUOTE=Christenson;257743]Mr Greathouse, have I missed anything?[/QUOTE]

Looks like you have everything I would have asked covered.

[QUOTE=Christenson;257743]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?[/QUOTE]

I'd more agree with a) in the first group but all we need is the way things move ( regardless if they can) and it's his definition of position. I do however see the reason that a position can happen to either player and it's mostly in my opinion based on the 2 space pawn move. as to the number with the specific king positions a2 is a pawn position lost so there are only 47 pawn places for either persons last pawn, if we allow check ( which we'd have to even for the pawn calculation to be that simple) then most pieces( like both bishops combined being treated as a bishop that can go on any diagonal) can go on the remaining squares allowing for a very maximum of ( 62*4 = 248) possible positions between the rest for each team 2*(248+47) = 590 positions in total. as to the pawns and castling I'm unsure, but the tactically identical positions I'd agree they're different.

[QUOTE=science_man_88;257758]I'd more agree with a) in the first group but all we need is the way things move ( regardless if they can) and it's his definition of position. I do however see the reason that a position can happen to either player and it's mostly in my opinion based on the 2 space pawn move. as to the number with the specific king positions a2 is a pawn position lost so there are only 47 pawn places for either persons last pawn, if we allow check ( which we'd have to even for the pawn calculation to be that simple) then most pieces( like both bishops combined being treated as a bishop that can go on any diagonal) can go on the remaining squares allowing for a very maximum of ( 62*4 = 248) possible positions between the rest for each team 2*(248+47) = 590 positions in total. as to the pawns and castling I'm unsure, but the tactically identical positions I'd agree they're different.[/QUOTE]FWIW, I've been assuming these definitions:

1) A position is an arrangement of zero or more chessmen on a chessboard, each square of which contains zero or one chessmen.

2) A legal position is a position which can be achieved by a sequence of zero or more legal moves from the standard starting position.

AFAICT, that's equivalent to option (a).


Paul

[QUOTE=xilman;257759]FWIW, I've been assuming these definitions:

1) A position is an arrangement of zero or more chessmen on a chessboard, each square of which contains zero or one chessmen.

2) A legal position is a position which can be achieved by a sequence of zero or more legal moves from the standard starting position.

AFAICT, that's equivalent to option (a).


Paul[/QUOTE]

I had to look up one of the abbreviations but yes I basically thought that this would be logical.

[QUOTE=science_man_88;257758]I'd more agree with a) in the first group but all we need is the way things move ( regardless if they can) and it's his definition of position. I do however see the reason that a position can happen to either player and it's mostly in my opinion based on the 2 space pawn move. as to the number with the specific king positions a2 is a pawn position lost so there are only 47 pawn places for either persons last pawn, if we allow check ( which we'd have to even for the pawn calculation to be that simple) then most pieces( like both bishops combined being treated as a bishop that can go on any diagonal) can go on the remaining squares allowing for a very maximum of ( 62*4 = 248) possible positions between the rest for each team 2*(248+47) = 590 positions in total. as to the pawns and castling I'm unsure, but the tactically identical positions I'd agree they're different.[/QUOTE]

I found a few flaws in my calculation that I'd like to forward:

1) 47 positions of the pawn might be possible but depending on where it is shows what is legal for the other king (for example if the pawn is in it's starting square the opponent king can't be in the capture square(s) for that pawn because it would have to illegally move itself into check for that to happen ( this doesn't happen with any other positions that I'm aware). With that in mind, I can say at least 2*7 positions for the opponent king in situations like this overall are definitely illegal.

I made this [url]http://en.wikipedia.org/wiki/User:Science_man_88[/url] for you who want to look.

[QUOTE=xilman;257759]FWIW, I've been assuming these definitions:

1) A position is an arrangement of zero or more chessmen on a chessboard, each square of which contains zero or one chessmen.

2) A legal position is a position which can be achieved by a sequence of zero or more legal moves from the standard starting position.

AFAICT, that's equivalent to option (a).


Paul[/QUOTE]
Paul, until Mr P-1 came along and made it clear there was an alternative, and called the above a "configuration", I certainly had thought that yours was the operational definition.

Sm88, do I correctly have you that distinct positions we are counting are in fact as defined above, and we are ignoring which side is to move next?

Next, with three pieces on the board, and 3608 distinct ways to place the two kings, what are the distinct choices for the third piece on the board? I count 10 if we let the color of a bishop be determined by the square we place it on.

[QUOTE=Christenson;257811]Paul, until Mr P-1 came along and made it clear there was an alternative, and called the above a "configuration", I certainly had thought that yours was the operational definition.

Sm88, do I correctly have you that distinct positions we are counting are in fact as defined above, and we are ignoring which side is to move next?

Next, with three pieces on the board, and 3608 distinct ways to place the two kings, what are the distinct choices for the third piece on the board? I count 10 if we let the color of a bishop be determined by the square we place it on.[/QUOTE]


first it's 3612 king on king. A pawn,knight,rook,white squared bishop,black squared bishop ( though we could count them both the same), or queen last I checked. really because there's 47 places for a pawn to go and that should lead to many more positions.

[QUOTE=science_man_88;257838]first it's 3612 king on king. A pawn,knight,rook,white squared bishop,black squared bishop ( though we could count them both the same), or queen last I checked. really because there's 47 places for a pawn to go and that should lead to many more positions.[/QUOTE]

I realized after looking that the person moving determines what's next and it looks like it's the two square pawn move that messes it up by allowing either 2*(1 square moves) or 1*(2 square move) you can change the person to move at least 8 times in a game.

SM: 3612 is indeed the correct number; 3608 is a typo.

I still maintain there are 10 possible choices for the third piece on the board -- remember that a black rook is distinct from a white rook, according to the answers above.

Each of these 10 pieces can be placed in *at most* 62 places (there are restrictions on the pawns), so we get an upper bound of 3612*10*62 (2,239,440) positions with three pieces on the board.

We can also get a lower bound. Since a legal position can include checkmate, the only real restrictions are that a pawn cannot be on the first or last row(Assumption: pawn on 8th rank *has* to be promoted -- missed in my questions about the rules).
Calculating, I get 3612 positions for two * ( (8 choices for non-pawns * 62 spots)+(2 choices for a pawn * At least 46 spots where the pawn can be placed) ) = 3612*588 = AT LEAST 2,123,856 positions with three pieces on the board.

SM, can we make these bounds tighter? There's an obvious easy trick here, and I'll entertain ideas on how to make the bounds exact.

[QUOTE=Christenson;257893]SM: 3612 is indeed the correct number; 3608 is a typo.

I still maintain there are 10 possible choices for the third piece on the board -- remember that a black rook is distinct from a white rook, according to the answers above.

Each of these 10 pieces can be placed in *at most* 62 places (there are restrictions on the pawns), so we get an upper bound of 3612*10*62 (2,239,440) positions with three pieces on the board.

We can also get a lower bound. Since a legal position can include checkmate, the only real restrictions are that a pawn cannot be on the first or last row(Assumption: pawn on 8th rank *has* to be promoted -- missed in my questions about the rules).
Calculating, I get 3612 positions for two * ( (8 choices for non-pawns * 62 spots)+(2 choices for a pawn * At least 46 spots where the pawn can be placed) ) = 3612*588 = AT LEAST 2,123,856 positions with three pieces on the board.

SM, can we make these bounds tighter? There's an obvious easy trick here, and I'll entertain ideas on how to make the bounds exact.[/QUOTE]

well for pawns on the initial position ( and possibly others) I see a way to get a lower bound on the amount of places the other king can be depending on a few thing ( I've made diagrams for that on my Wikipedia page I posted in this thread). if a pawn is within the edge of the board and the kings protection such that it doesn't have squares outside of the protection it can attack it acts just like 2 kings but only on certain places on the board ( so those positions are strictly countable). Also you can have positions with the pawn with 1 square under attack ( if the pawn is on the initial position that square is illegal for the opponent king ( otherwise it could be the pawn creating a check)), and ones with 2 places under attack( and at least this position
type takes another away due to the pawn being on it).