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

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### How many edges are in the kth-iterated common neighborhood knight's graph?

For terminology see https://www.academia.edu/2180382/The_Common_Neighborhood_Graph_and_Its_Energy)
It has been known for quite some time by players that a vertex whose corresponding square is in the ...

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**0**answers

264 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 ...

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**1**answer

131 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 ...

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**3**answers

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### 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 ...

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**9**answers

3k 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 ...

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**1**answer

188 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 ...

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**1**answer

320 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. ...

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**5**answers

606 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 $k$ ...

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**2**answers

510 views

### A Chess Question Of The Late Great W.T.Tutte

In "Graph Theory As I Have Known It", p.12, Knights Errant, Tutte mentions as an aside the chess question " does either Black or White have a certain win from the initial position, given perfect ...

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**1**answer

265 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 ...

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**3**answers

1k 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 ...

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**5**answers

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### 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 ...

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**0**answers

1k 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 ...

**71**

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**11**answers

7k 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 ...

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**1**answer

571 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 ...

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**1**answer

1k 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.

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**4**answers

1k 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 564. If the true value is intractable, we may give up solving chess. But if it's small, there still could be ...

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**1**answer

600 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 ...

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**2**answers

1k 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 ...

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**2**answers

1k 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 ...

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**1**answer

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### 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. ...

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**3**answers

3k 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 ...

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**5**answers

4k 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 ...

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**3**answers

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### 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 ...

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**2**answers

990 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 ...

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**1**answer

641 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 ...