Questions tagged [chess]

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

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The $n$ queens problem with no three on a line

The $n$ queens problem asks if we can place $n$ queens on an $n\times n$ chessboard such that no two queens attack one another. For example, when $n=8$, here are two solutions (images taken from ...
ho boon suan's user avatar
2 votes
2 answers
342 views

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 ...
Dan S's user avatar
  • 21
2 votes
0 answers
204 views

Chess pieces metrics in higher dimensions

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 ...
Marco Ripà's user avatar
  • 1,102
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 ---...
Anton Petrunin's user avatar
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 $f(n) = \left\lfloor\log_2{2n}\right\rfloor + 1$ ...
Notamathematician's user avatar
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 ...
Alessandro Della Corte's user avatar
2 votes
1 answer
251 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 ...
Pace Nielsen's user avatar
15 votes
0 answers
462 views

Does the Angel have to be really smart?

My question is about the computational complexity of the Angel's strategy in the Angels and Devils game, tl;dr does the Angel have a polynomial time strategy. I'm a big Conway fan, so as you can ...
Ville Salo's user avatar
  • 6,337
11 votes
2 answers
930 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 ...
Olivier's user avatar
  • 10.2k
8 votes
1 answer
450 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 ...
Haoran Chen's user avatar
1 vote
1 answer
241 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 ...
user137861's user avatar
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 ...
Morteza Azad's user avatar
1 vote
0 answers
188 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 ...
anonymous's user avatar
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 ...
Morteza Azad's user avatar
8 votes
0 answers
491 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 ...
Paul Burchett's user avatar
39 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 ...
Michał Masny's user avatar
4 votes
1 answer
479 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 ...
maciek's user avatar
  • 173
5 votes
1 answer
133 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 ...
JC1111523's user avatar
0 votes
1 answer
525 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 ...
user2727's user avatar
  • 133
11 votes
1 answer
442 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/...
ruadath's user avatar
  • 321
18 votes
0 answers
952 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 ...
Joseph O'Rourke's user avatar
18 votes
1 answer
1k 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 ...
Joseph O'Rourke's user avatar
1 vote
2 answers
395 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 ...
Ian Calvert's user avatar
7 votes
3 answers
814 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 ...
Waldemar's user avatar
  • 1,107
39 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 ...
Joel David Hamkins's user avatar
0 votes
1 answer
471 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 ...
ARi's user avatar
  • 841
20 votes
1 answer
889 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. ...
Brian Rushton's user avatar
7 votes
5 answers
970 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 $k$ ...
T.R.'s user avatar
  • 133
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 ...
ivan's user avatar
  • 531
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 ...
Roland Bacher's user avatar
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 ...
mmm's user avatar
  • 171
22 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 ...
GM2001's user avatar
  • 223
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 ...
domotorp's user avatar
  • 18.3k
128 votes
13 answers
22k 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 ...
Johan Wästlund's user avatar
1 vote
1 answer
858 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 ...
froogz3301's user avatar
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.
fnasim's user avatar
  • 243
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 ...
Zirui Wang's user avatar
2 votes
1 answer
355 views

A random variable in a game of knights and queens

Suppose that a game is played on an $n \times n$ board as follows. There are two players, Player 1 has (only) $Q$ queens and Player 2 has only $K$ knights. Suppose that $Q, K \leq n/3$. The game is ...
Stanley Yao Xiao's user avatar
4 votes
3 answers
1k views

Probability theory and measuring the true strength of chessplayers

If you wanted to measure the strength of, say, a chess player, the best way would involve knowing the true value of each position: then you could compute the frequency $W$ with which the player finds ...
David Feldman's user avatar
8 votes
1 answer
970 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 ...
Wynand Winterbach's user avatar
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 ...
Barry's user avatar
  • 151
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 ...
David Eppstein's user avatar
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. ...
David Steinberg's user avatar
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 ...
Garabed Gulbenkian's user avatar
67 votes
5 answers
10k 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 ...
Richard Stanley's user avatar
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\...
Timothy Chow's user avatar
4 votes
2 answers
2k views

How to compute the rook polynomial of a Ferrers board?

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 ...
didest's user avatar
  • 1,015
11 votes
1 answer
851 views

Counting colored rook configurations in the cube - when is it even?

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,...
miforbes's user avatar
  • 1,088