# Questions tagged [chess]

Mathematical questions in one way or another related to the game of chess.

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### Exact calculation of n-queens solutions [closed]

I'm new to this forum, but I'm hoping this community can help me with some guidance on sharing and improving a mathematical solution that I've developed for the $n$-queens problem and $n$-queens ...
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A couple of days ago, I was thinking about applying the knight (the well-known piece of chess) metric to any cubic lattice $\mathbb{N}^k$, $k \in \mathbb{N}-\{0,1\}$. I suddenly realized that, from $k ... 2 votes 3 answers 1k views ### Strategy-stealing in chess Is it proved that white can guarantee at least draw in chess? A while ago I was told that it was proved using strategy-stealing, but I cannot find a reference. Postscript. Please accept my apology ---... 0 votes 0 answers 142 views ### Number of open tours by a biased rook on a specific$f(n)\times 1$board which end on a$k$-th cell from the right We have a simple structure - biased rook of the two types. Biased rook of the first kind which make open tours on a specific$f(n)\times 1$board where$f(n) = \left\lfloor\log_2{2n}\right\rfloor + 1$... 5 votes 1 answer 1k views ### How many consecutive forced moves are possible in chess? The question concerns chess. I call a move forced if, in a given position, is the unique move consistent with the rules of the game. I wonder what is the largest integer$n$such that there exists a ... 2 votes 1 answer 248 views ### Limited rook moves I have an algebra problem, that could be solved if I could answer the following combinatorial problem. Let$S$and$T$be two nonempty sets. We think of$S\times T$as the index set for the squares ... 11 votes 2 answers 791 views ### Algebraic properties of graph of chess pieces For the purpose of this question, a chess piece is the King, Queen, Rook, Bishop or Knight of the game of chess. To a chess piece is attached a graph which represents the legal moves it can make on an ... 8 votes 1 answer 411 views ### Knight's tour problem It is known that on an infinite board, if all squares of the form$(ki,kj)$are removed,$k$even,$i,j\in\mathbf{Z}$, then there is no knight's tour due to unbalanced black and white squares. My ... 1 vote 1 answer 221 views ### Complexity class of chess when simulated by a Turing machine [closed] Suppose we simulate the game of chess with a Turing machine$M$as follows: The semi-infinite input tape of$M$contains a sequence of symbols beginning in the first cell of the tape. Each symbol ... 46 votes 3 answers 5k views ### Does knight behave like a king in his infinite odyssey? The Knight's Tour is a well-known mathematical chess problem. There is an extensive amount of research concerning this question in two/higher dimensional finite boards. Here, I would like to tackle ... 1 vote 0 answers 180 views ### Search strategy for Babson task in chess I asked this on a computer chess forum (programmers hang out there, etc.) and got no substantive answers, which makes me think it's a research question. Whether it's sufficiently mathematical is ... 67 votes 6 answers 17k views ### What is a chess piece mathematically? Historically, the current "standard" set of chess pieces wasn't the only existing alternative or even the standard one. For instance, the famous Al-Suli's Diamond Problem (which remained ... 8 votes 0 answers 446 views ### Variants of the Angel problem The original Angel problem, first proposed by Conway, Berlekamp, and Guy has two players: the devil and the angel. The angel is placed at the origin of an infinite$2$-D chessboard. The angel's ... 38 votes 5 answers 4k views ### Can one make high-level proofs about chess positions? I realize this question is risky (as the title and the tags indicate), but hopefully I can make it acceptable. If not, and the question cannot be salvaged, I'm sorry and ready to delete it or accept ... 4 votes 1 answer 463 views ### Number of different positions of rooks on chessboard I know that this topic as been mentioned before, but no accurate answer has been provided. Suppose we have to place$n$rooks on$n \times n$chessboard so that no one attacks another. How to count ... 5 votes 1 answer 131 views ### Time to generate a filled-in sub-checkered board Take an$m \times n$checkered board and one-at-a-time add a piece to an empty square. At what point are you guaranteed to have an$s \times t$sub-board where all of its squares are filled? Here I ... 0 votes 1 answer 517 views ### Is it possible to create an infinite sequence in which no subsequence is repeated 3 times in a row? In Chess, there is the Threefold Repetition rule where if a sequence of moves is repeated 3 times in a row, either player can claim a draw. Say two players wanted to play a legal, infinite game of ... 11 votes 1 answer 431 views ### Nonattacking configurations of$k$bishops on an$m$by$n$rectangular board The number of ways to place$k$bishops in a nonattacking configuration on an$n$by$n$square board is a known and can for example be found in http://problem64.beda.cz/silo/... 18 votes 0 answers 935 views ### Are the moves/rules of standard chess delicately balanced? (While the world chess championship is in progress in Sochi...) Is there mathematical evidence that standard chess is somehow ... 17 votes 1 answer 931 views ### Knight's tours in higher dimensions I wonder if Knight's Tours have been explored in higher dimensions, using the following definition of a knight move. In dimension$d=2$, the knight moves left/right and forward/back one step and two ... 1 vote 2 answers 391 views ### Sums Of Independent Random Variables: Pathological Behaviour Background: The result of a chess game between two players is a win ,a loss or a draw which are (usually) scored respectively$1$point,$0$point or$0.5$point for the appropriate player. Team ... 7 votes 3 answers 791 views ### Decidability of the winning-position problem in an infinity chess with a finite number of short-range pieces only Definitions Long-range pieces: queens, rooks, bishops. Short-range pieces: pawns, knights, kings. We can extend the definition of short-range pieces to include also fairy pieces like: Berolina ... 40 votes 9 answers 8k views ### What proportion of chess positions that one can set up on the board, using a legal collection of pieces, can actually arise in a legal chess game? 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 ... 0 votes 1 answer 456 views ### Infinite board games: sentences about As a unified approach if we have an ( read any) infinite board game described as$\mathcal{G}$using a particular axiom set A.. can a sentence be devised in A which automatically answers the basic ... 20 votes 1 answer 860 views ### Discrete Morse theory and chess There are many mathematical objects that are similar to groups and Cayley graphs of groups but lack homogeneity in some sense. Graphs of webpages with edges corresponding to links are one example. ... 7 votes 5 answers 967 views ### Collisions between rooks taking random flights on an N by M chessboard I randomly place$k$rooks on an (arbitrarily sized)$N$by$M$chessboard. Until only one rook remains, for each of$P$time intervals we move the pieces as follows: (1) We choose one of the$k$... 36 votes 2 answers 3k views ### Rooks in three dimensions Given is an infinite 3-dim chess board and a black king. What is the minimum number of white rooks necessary that can guarantee a checkmate in a finite number of moves? (In 3-dimensional chess rooks ... 11 votes 1 answer 426 views ### Exceptional points for generalized north-eastern knight walks in a quarter plane Given two coprime integers$a < b$of different parities, only a finite number of points in$\mathbb N^2$cannot be reached by a walk in$\mathbb N^2$, starting at the origin and using only steps ... 17 votes 3 answers 2k views ### Traversing the infinite square grid Starting somewhere on an infinite square grid, is it possible to visit every square exactly once, if at move$n$, one must jump$a_n$steps in one of the directions north,south,east or west, and mark ... 21 votes 5 answers 3k views ### Irreversible chess Suppose we play a chess-variant, where any finite number of pieces are allowed, and the board is as large as we wish, but only two kings in total. And there is no 50 move-rule, no castling and no ... 7 votes 0 answers 2k views ### Is there a chess position equivalent to the Collatz conjecture? Suppose we have an infinite board with a finite number of chess pieces. The question is whether white can checkmate black (without the after 50 moves it is a draw rule). Can you give an explicit ... 124 votes 13 answers 20k views ### Checkmate in$\omega$moves? Is there a chess position with a finite number of pieces on the infinite chess board$\mathbb{Z}^2$such that White to move has a forced win, but Black can stave off mate for at least$n$moves for ... 1 vote 1 answer 849 views ### Generating fixtures for a chess league, with a twist Hello, I am in the process of building some software to generate fixtures for a chess league. Which has a little twist which complicates matters. I would like to introduce a constraint. Where by a ... 14 votes 2 answers 3k views ### How to place k bishops on an nxn chessboard In how many different ways can k bishops be placed on an nxn chessboard such that no two bishops attack each other? Please try to respond with a formula and explanation. 5 votes 5 answers 2k views ### How long is the longest path in the game tree of chess? I can only think of an upper bound, which consists of all configurations and so has length$5^{64}$. If the true value is intractable, we may give up solving chess. But if it's small, there still ... 7 votes 1 answer 956 views ### Rooks on a lifeline The short version of this question is: If$G$is a graph whose nodes are associated with squares of a chessboard, such that no two nodes in the same row or column of the board are adjacent, we want ... 4 votes 2 answers 2k views ### Elo Rating System Help with the Maths around number of matches I'm creating a system that will allow people to rate images. My idea is to use an Elo Rating system (http://en.wikipedia.org/wiki/Elo_rating_system) for each image and then use crowdsourcing to have ... 18 votes 2 answers 2k views ### Is the 4x5 chessboard complex a link complement? The 2x3 and 3x4 chessboard complexes (form a square grid of vertices and make a simplex for any set of vertices no two of which are in the same row or column) are a 6-cycle and a triangulated torus ... 11 votes 1 answer 2k views ### Is the space of solutions to the Queens Domination Problem connected? A configuration of queens on an 8 by 8 chessboard (or n by n if you like) is a queen domination if every square on the board lies in the same row, column, or diagonal as at least one of the queens. ... 31 votes 3 answers 6k views ### Is there a good argument for why you can't place 4 queens which cover a chessboard? It has been known since the 1850's (or even much earlier) that 5 queens could be placed on an 8*8 chessboard so that every square on the board lies in the same row, column, or diagonal as at least ... 64 votes 5 answers 9k views ### Decidability of chess on an infinite board The recent question Do there exist chess positions that require exponentially many moves to reach? of Tim Chow reminds me of a problem I have been interested in. Is chess with finitely many men on an ... 52 votes 4 answers 10k views ### Do there exist chess positions that require exponentially many moves to reach? By "chess" here I mean chess played on an$n\times n$board with an unbounded number of (non-king) pieces. Some care is needed if you want to generalize some of the subtler rules of chess to an$n\...
Given a Ferrers board of shape $(b_1,\ldots,b_m)$, we define $r_k$ as number of ways to place $k$ non-attacking rooks (as in Chess). In section 2.4 of Stanley's Enumerative Combinatorics (vol. 1) it's ...
Informal Statement In the $n\times n \times n$ grid, we can places rooks (those from chess) such that no two rooks can attack each other. One way to achieve this is to place a rook in position \$(i,j,...