Wikipedia:Reference desk/Archives/Mathematics/2018 March 23

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

March 23

What is the closest math representation of a chess game?

When you represent the chess board and pieces, you could easily use a 8x8 matrix with pieces in each cell. However, you'll miss the whole relation between the pieces. What mathematical structure would better represent a chess position in a meaningful way? --Doroletho (talk) 14:09, 23 March 2018 (UTC)

Sorry, how does a "8x8 matrix with pieces in each cell" "miss the whole relation between the pieces"? A chessboard covers 64 squares - only 50% of them are used at the start of the game and the number of squares in use can theoretically be as low as two. There are two main methods of naming the squares - see Chess notation. 92.31.142.218 (talk) 14:33, 23 March 2018 (UTC)

See also the article at Bitboard, which talks about computer representations of data of this sort. –Deacon Vorbis (carbon • videos) 14:37, 23 March 2018 (UTC)

92.31 misses the point, but Deacon Vorbis in on the money with the bitboard suggestion. You obviously can describe a concrete position unambiguously using any form of chess notation. But you want to understand it, query it, and so on. --Doroletho (talk) 17:03, 23 March 2018 (UTC)

Honestly it's still not clear what your question is. The bitboard article is talking about advantages or disadvantages in terms of memory usage and speed of determining the answer to particular queries, but if you're interested in those things, you seem to have forgotten to mention it in the question. I'm not convinced it's any more "meaningful" than just the 8x8 matrix.

It may also be worth mentioning that a chess "position", in terms of having enough information to determine optimal (or even legal) play, involves more than just what men are in what square. You also need to know a bit about the history. For example, you may need to know whether the king has moved, to know whether castling is allowed. Similarly, you can't always tell whether en passant capture is allowed, just from a snapshot of the board. The fifty-move rule and threefold repetition rule likewise need to know history. --Trovatore (talk) 18:01, 23 March 2018 (UTC)

you need to represent the possible moves, besides just a position, if that is what you are asking. Bubba73 ^{You talkin' to me?} 18:42, 23 March 2018 (UTC)

Not sure whom you mean by "you". I wasn't asking anything; I was attempting to answer. I suppose I was implicitly asking for a clarification of the question.

Representing the possible (next) moves is not enough. For example, there may be a position that has been attained twice, and that the weaker side can force to happen a third time and claim a draw (but cannot force indefinitely). Then you need to know that the position has been attained twice. It's not clear (at least to me) that there's any way of representing this information that is superior to just remembering the whole sequence of moves. --Trovatore (talk) 18:52, 23 March 2018 (UTC)

I guess you can optimize it a little bit by remembering the sequence only back to the last irreversible move (capture or pawn move), together with the having-been-moved status of the king and both rooks. Off the top of my head I think that works. It's possible there's some corner case I'm not thinking of. --Trovatore (talk) 19:06, 23 March 2018 (UTC)

Sorry, by "you" I meant the original poster. I was trying to answer "you'll miss the whole relation between the pieces". Bubba73 ^{You talkin' to me?} 20:46, 23 March 2018 (UTC)