Questions tagged [mathematics]

Questions concerning the mathematics of the game of chess, or chess-related mathematical pursuits.

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27
votes
4answers
7k views

Stalemate situation with all pieces on the board

Suppose all the pieces are on the board. Does there exist a position such that A) one of the players can't make any move (a stalemate)? B) neither player can make any move (a double stalemate)?
25
votes
8answers
15k views

Has anyone attempted to characterize chess mathematically?

From years of being student, researcher, and practitioner of chess, I feel like it possesses completeness and elegance that are found in mathematical proofs and theories. Thus, it seems only natural ...
22
votes
5answers
2k views

Least number of moves to achieve a complete switchover of nonpawns in starting setup

[fen "RNBQKBNR/8/8/pppppppp/PPPPPPPP/8/8/rnbqkbnr w - - 0 1"] During a comp, my friend an I had a team kill and we decided to make this position(legally) before drawing. Then we continued to find ...
18
votes
1answer
2k views

How many such positions are there? (chess/math checkmate puzzle)

I'm interested in this kind of position: There are only 4 pieces on the board. If Whites goes first, they can checkmate in one move. If Blacks goes first, they can checkmate in one move. For example: ...
13
votes
7answers
3k views

Arrange 5 non-attacking knights on a 5x5 toroidal board

The celebrated British mathematician and lover of public transport, Simon Norton, was one of those who passed in 2019. He was the subject of a wonderful biography and one of my favourite human beings: ...
13
votes
8answers
15k views

Longest chess game possible (maximum moves)

What is the longest chess game possible in terms of chess moves? I read somewhere that there is a theoretical maximum of 5949 moves. But I don't see any proof and I don't think it is correct. Can it ...
13
votes
3answers
485 views

Fastest King vs King endgame

What is the fastest possible game that ends in a King vs King endgame? Please post a game and tell me the number of half-moves you've achieved. By some simple logic, I can prove that this number is ...
11
votes
3answers
2k views

How to solve this puzzle of Martin Gardner?

Martin Gardner mentioned the following "chess" problem in his column, Mathematical Games, for Scientific American (he got it from Smullyan, who learned about it from a high school math ...
10
votes
2answers
3k views

Maximum number of knights populating a chess board so that no knights are attacking one another

The 8 queen's puzzle is a well-known puzzle involving placing 8 queens on a chessboard so that no queen attacks another queen. It is more or less obvious that 8 is the maximum number of queens that ...
10
votes
3answers
2k views

What is the average number of legal moves per turn?

For example in the first turn there are 20 possible legal moves (16 pawn moves and 4 knights moves). This number increases in the middlegame and then decreases in the Endgame. Are there some ...
10
votes
3answers
515 views

Where can I find Pauls' 1874 article on the n-queens problem?

Where can I find the following article, which concerns the n-queens problem? E. Pauls, Das Maximalproblem der Damen auf dem Schachbrete, II, Deutsche Schachzeitung. Organ f¨ur das Gesammte ...
9
votes
2answers
561 views

Pal Benko's final problem

EDIT: Link added One of my chess heroes, Hungarian GM Pal Benko, is no longer with us. He overcame many difficulties in his early life to live one which was rich and full, and reached the age of 91 ...
9
votes
0answers
295 views

If repetition loses, who wins KvK?

Suppose we're playing a version of chess where you lose if you repeat an earlier position (position = arrangement of pieces + who's move it is). We're down to the wire, and we've just arrived at the ...
8
votes
4answers
2k views

How to calculate the number of combinations for Chess960 positions

Can someone derive how do we get the total number of 960 possible positions for Chess960?
8
votes
2answers
1k views

Longest possible FEN

What is the longest possible FEN? The reason for that question is, that I am designing a PGN viewer (again), and I have noticed that the FEN is dependent on the width of the board (the FEN test area ...
8
votes
3answers
3k views

What is the number of legal positions in a chess game?

There have been several questions regarding the possible number of chess games. for example; Database of every possible move in chess. However,can there be an estimate on the number of "allowable&...
8
votes
4answers
613 views

Does playing tournament chess help one in the field of chess-related math?

A question of personal interest. Is "visualization" of various types of boards and graphs helped along by competitive, tournament play? Of course, I'm assuming play beyond simply knowing how the ...
8
votes
1answer
481 views

Which collections of pieces are legal?

Call a collection of (white and black) chess pieces legal if it occurs in the position of a legal chess game. For example, KQRRBBNNPPPPPPPPkqrrbbnnpppppppp is the collection at the beginning of the ...
8
votes
1answer
372 views

Is the number of legal chess positions odd or even?

Is the number of legal chess positions odd or even? Two positions are not the same if they differ in castling rights (i.e. whether K or R have actually moved) or en passant capability (i.e. whether ...
8
votes
1answer
311 views

Mathematical formula to get how broad opening repertoire is

programmers and mathematicians! I'm looking for some opinions how to build meaningfull formula that represents broadness of opponent's repertoire. At the moment I do it like this: I filter all ...
7
votes
3answers
1k views

Square Free Position

[FEN ""] For the purposes of this question, a “square” is a set of four occupied cells on the chessboard whose centres form a geometric square. The cell may have any orientation. For ...
7
votes
3answers
1k views

Longest Sequence For Any Piece To Reach A Certain Square

What is the longest sequence of moves (by both White and Black) wherein White attempts to move one of their pieces from one square to another? Assume Black makes every move optimal towards the purpose ...
7
votes
3answers
1k views

Of all the possible legal chess games, how many end in white win, draw and black win (assuming 50 move rule)?

If both sides make moves completely randomly but legally, how many different games are there? Since the 50 move rule applies, this number should be finite (but HUGE). For example, there are 20 ...
7
votes
3answers
2k views

How many possible combinations for the number of pieces on the board?

My question concerns the total material available on the chess board (black+white pieces). At the very beginning of a game, there are 2 Kings, 2 Queens, 4 Rooks, 4 Bishops, 4 Knights and 16 pawns. ...
7
votes
1answer
254 views

16 roses, 16x16 board; maximum double-attacked squares while all 256 squares are attacked at least once?

The rose is a piece that is best described as a "circular nightrider", able to reach up to 32 other squares from the center of a 13x13 area by making consecutive knight jumps in a rotating fashion in ...
6
votes
2answers
747 views

Biggest Possible Material Advantage But Still Losing

Assuming: Queen=9, Rook=5, Bishop=3, Knight=3, Pawn=1 The below position is evaluated at +411 for White but he still loses by force (mate in 9, pawn promotes to a bishop just for the fun of it) Is ...
6
votes
3answers
506 views

What's the maximum number of moves that can result in a threefold repetition from an arbitrary position?

Assume that the fifty-move rule does not require a player to claim a draw. Instead, the game is automatically declared a draw if no pawns have been moved and no pieces have been captured in the last ...
6
votes
1answer
289 views

Is there a formula that converts a number from 0 to 959 to a corresponding chess960 position?

I am looking into a website with starting positions for chess960. 000 BBQNNRKR 001 BQNBNRKR 002 BQNNRBKR 003 BQNNRKRB 004 QBBNNRKR 005 QNBBNRKR ... 959 RKRNNQBB So there is a table that converts any ...
6
votes
1answer
226 views

A New Year's Math Riddle

Here's a fun and clean riddle for the New Year! I found the material for it on Krabbe's site. I will provide the link when it is solved. How many times can this position be repeated, with both sides ...
6
votes
1answer
603 views

Checkmating with two bishops

Let White have the two bishops. Given any setup, has the maximum number of moves to forced mate been calculated on an nxn board (for large enough n)? If not, are there any bounds on this number? Of ...
5
votes
2answers
2k views

How many moves needed for a knight to go from any square to any other square?

I want to find a sequence of knight moves that has a given origin and destination squares, preferably with the minimum number of moves. I can iterate over knight moves and collect visited squares, ...
5
votes
3answers
7k views

How do you calculate your Tournament Performance Rating?

How do you calculate your Tournament Performance Rating? I can think about two different ways to answer my question: Do you know the formula for calculating your TPR (FIDE or USCF)? Can you link me ...
5
votes
2answers
2k views

Chess480: Why 480?

From wiki: Lewis has named this chess variation "Chess480"; it follows the rules of Chess960 with the exception of the castling rules. Although a Chess480 game can start with any of 960 ...
5
votes
6answers
663 views

How to determine if a position is open or closed in a "rigorous" fashion?

I am creating a set of chess positions to be used [as a data set] for a project. It's based on supervised machine learning, but I need to have labeled data (see more italicized text below) before any ...
5
votes
2answers
1k views

How to calculate the number of positions in chess?

For layman, can someone explain how you calculate the number of available positions in a chess game, and how do you get the huge number of 10^120 or 10^46?
5
votes
1answer
232 views

Is it possible to reach King vs King by moving each piece at most twice?

I'm interested in short games that reach a King vs King endgame. Sam Lloyd reached King vs King after only 17 moves. This has since been improved to 16.5 moves. (See this and that.) One can show that ...
5
votes
2answers
302 views

What is the maximum number of squares 8 queens and 8 nightriders can attack on a 16x16 board?

Including but still obstructed by occupied squares, what is the maximum number of squares on a 16x16 board that 8 queens and 8 nightriders can collectively attack at least once each? Even if attacking ...
5
votes
2answers
120 views

Endgames: How was "maximum moves required" determined before tablebases?

To use a simple example, tablebases show that KQ vs. K is won in at most ten moves. But chess books going back many decades, well before tablebases, say the same thing. How was this calculated?
5
votes
1answer
507 views

Longest possible game of Horde Chess

The longest possible normal game of chess has been calculated, but I want to know what the longest possible game of Horde Chess is. What makes this an interesting case is the crazy amount of pawn ...
4
votes
2answers
2k views

How many times can the same pawn move be made in a game?

How many times in a game can "the same pawn move" be made? By "the same pawn move", I mean, a move made by a pawn that would be notated the same way. For instance, after 1.e4 e5 2....
4
votes
3answers
1k views

How does Elo rating gain scale with games played?

I only take two players starting out at 1500 Elo and have them play against each other with one player winning all games. Also I leave out any artificially introduced cut-offs. So the winner is ...
4
votes
4answers
1k views

Longest path for knight

Suppose the knight is on square X and he wants to get to square Y, and in order to do that he has to move N moves. how big could N be? Could that be solved using a mathematical equation? Say Nf3 ...
4
votes
3answers
460 views

Puzzles that highlight the finiteness of the chessboard

I'm curious as to whether there are compositions of chess puzzles/problems where the solution really emphasizes the boundedness of the board. Put differently, cases where the solution would be ...
4
votes
1answer
483 views

Minimal number of moves before castling in Chess960

As you know, in the initial position Chess960 the king lies between the two rooks. Here are the rules for castling. For a starting position P, let n(P) be the minimal number of moves that White needs ...
4
votes
1answer
175 views

Leela Chess PUCT Mechanism

How do we know w_i which is not possible to calculate using the tree search only? From the lc0 slide, w_i is equal to summation of subtree of V ? How is this equivalent to winning ?
4
votes
1answer
126 views

Longest sequence before a certain capture

Inspired by a recent question, what is the longest possible sequence before White can make a certain capture? Basically, White is trying to capture a chosen piece, and Black optimally defends that ...
4
votes
0answers
117 views

Relation between ratings for Chess960 and standard chess

Is there any relation or any attempt at discovering a relation between Chess960 ratings and standard chess ratings? This can be for GMs, for (any)M's, for pro players, or for all players on lichess/...
3
votes
2answers
370 views

How many different knight's tours are there?

How many solutions does the Knight’s Tour puzzle have when the knight first moves from square a1, through all squares (on each square once), and return back to a1? Please give some solutions. Also, ...
3
votes
2answers
270 views

How many games of optimal chess are there?

Assume that we have weakly solved chess, in the sense that we have an algorithm that when used, allows us to force a win if possible, and if that is not possible, force a draw. Define a ply as optimal ...
3
votes
1answer
1k views

What is the minimum number of pieces needed in a chess set to account for all possible promoted pieces?

How many pieces would be needed in a chess set where all possibilities for promoted pieces are included if pawns are no longer used after they promote?