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Questions tagged [chessboard]

Use this tag for questions about the board on which the game of chess is played.

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Number of ways to place $4$ kings on an $n \times n$ chessboard

I have an $n \times n$ chessboard and $4$ kings on it. My goal is to count the number of arrangements where some of them are non-attacking or mutually attacking, for example: In the case where the $4$...
Cardstdani's user avatar
7 votes
1 answer
90 views

How many ways are there to place 6 salamanders on a 6-by-6 grid such that no 2 salamanders share the same row, column, or diagonal?

I saw this problem on a 4th grader's math homework. "You have a 6-by-6 grid. You must place 6 salamanders on the board such that none of them share the same row, column, or diagonal. How many ...
Lauren S's user avatar
  • 409
3 votes
1 answer
34 views

Hamiltonian graph on a $8\times 8$ chessboard with upper left corner and bottom right corner square removed

Suppose we are given the setup in the title. Two squares are adjacent if and only if they share a common edge. I want to find out whether the obtained graph considering squares as nodes would be ...
Sj2704's user avatar
  • 79
8 votes
2 answers
989 views

Number of ways a chess king can move from a1 to h8

Assume a king sits on a1 on an 8x8 chessboard. The king is restricted so he can only move up, right, or diagonally towards top-right. How many paths are there to h8. I know this is a duplicate ...
Jack Hueson's user avatar
10 votes
3 answers
224 views

What is the minimal number of pieces to surround n pieces in a Go game?

Go is a game of black and white pieces on a lattice of $19\times 19$. Pieces have liberty by having empty spaces next to them and are killed if the liberty are occupied by the opponent. Pieces are ...
ZhenRanZR's user avatar
  • 399
0 votes
1 answer
25 views

Understanding Perimeter of Infected Area in Chessboard Virus Puzzle

In the Chess Board Virus Simulation found here: https://ves.ac.in/wp-content/uploads/sites/11/2015/12/Mangala_Gurjar_-_Use_of_invariants.pdf, I'm trying to understand the following statement: "...
Robin Andrews's user avatar
1 vote
0 answers
55 views

Ball's Diagonal Perfect Path in a board

This question sounds ridiculous as it seems. Setup Assume there is an m x n checker board (m, n is a natural number). Place a ball in any square on the board (assumes this square has the color ...
Oscar Nguyen's user avatar
1 vote
1 answer
67 views

How many unique patterns can be made by placing $k$ counters on an $n\times m$ chess board?

How many unique patterns can be made by placing $k$ counters on an $n\times m$ chess board? Obviously we take $k\leq n\times m$. Rotations and reflections should all be counted as distinct patternns (...
J.Smith's user avatar
  • 83
0 votes
0 answers
66 views

Depth to Mate in a random $N$-piece chess position

What is the formula or piecewise function that gives how many moves [or half moves (plies)] are the DTMs if there are multiple checkmate variations (Depth to Mate ignoring 50-move rule) for a random $...
nirates biadenroc's user avatar
3 votes
1 answer
117 views

Covering dark squares on a large chess board

Imagine we have a $8 \times 8$ chessboard and a person situated on one of the dark squares. The person is allowed to jump diagonally, but only by $1$ square and the person cannot revisit the squares ...
SStormM's user avatar
  • 92
2 votes
0 answers
64 views

Find minimum number of moves between two squares for a chess knight

I have asked this question once before, but I forgot about it and it's been a long time since then. Also, I have learned new things since then so my approach has changed. Therefore I am creating a new ...
Alice's user avatar
  • 508
-1 votes
1 answer
50 views

Chess and math. How many possible combinations?

A chessboard has 64 squares 16 pawns 4 rooks , bishops and knights each 2 queen and king each Total pieces are 32. Total squares are 64. (No pieces are captured) Let's now focus on one piece One ...
Krishang Rana's user avatar
2 votes
1 answer
106 views

The stationary distribution of a rook's random walk

Let us consider a classic $8\times 8$ chessboard (grid), where a lone rook is doing a random walk. We assume that the rook starts its journey from the $(7, 0)$ square ($A1$), and that there are no ...
o.spectrum's user avatar
  • 1,170
12 votes
2 answers
653 views

Placing kings on a 6x6 board - who wins?

Two players alternate placing kings on a $6\times6$ chessboard, such that no two kings are allowed to attack each other (not even two kings placed by the same player). The last person who can place a ...
Akiva Weinberger's user avatar
-2 votes
2 answers
75 views

Chess board squares counting [closed]

Let a chess board is a $(8*8)$ grid. How many squares are there? I have tried it manually. After some time I lost count.
Math ROCKER's user avatar
0 votes
2 answers
142 views

A beetle on each square of a $9 \times 9$ chessboard. Each beetle crawls one square diagonally, find minimal possible no. of free squares.

Question: A beetle sits on each square of a $9 \times 9$ board. At a signal each beetle crawls diagonally onto a neighboring square. Then it may happen that several beetles will sit on the same ...
D S's user avatar
  • 5,315
4 votes
0 answers
156 views

Books on chess mathematics [closed]

My question is whether there are some books on chess mathematics with theorems and their proofs concerning winning strategies for classes of initial positions. Despite there being a lot of books on ...
Bertrand Haskell's user avatar
2 votes
1 answer
126 views

$E($bishop moves$) = E($knight moves$)$ when each square on $\mathbb{Z}^2$ is blocked with probability $p$

I came across this probability problem in a placement exam a few days ago , and i was not able to solve it. so, i am trying to solve it since then but i couldn't come up with a solution so please ...
nitish's user avatar
  • 31
2 votes
1 answer
116 views

If an ant starts from A on a chessboard and can only move in the forward and upward direction, what is the number of ways in which it can reach B?

This is the chessboard, where A and B are the 2 ends of the board. An ant starts from end A, and can only move forwards (horizontally) and upwards (vertically). The ant cannot move diagonally and is ...
Bongo Man's user avatar
  • 331
5 votes
1 answer
1k views

Can a {p,q}-powered knight return to the origin?

I am working on the following problem: Suppose we have an arbitarily large chessboard. For two positive integers $p$, $q$, define a $[p,q]$-knight as an object that on each turn, must move p squares ...
Yonas Oberlin's user avatar
0 votes
1 answer
244 views

Minimize number of rooks in the final configuration in a chess board with $K$ rooks $(1 \leq K \leq N \times N )$

Given $N \times N$ chessboard with $K$ rooks placed on it initially. The initial position of the $K$ rooks is given as the input. Any rook can move to any other position only if there is an uncaptured ...
Tushar Sinha's user avatar
0 votes
1 answer
157 views

How can I calculate the limit of a transition matrix?

I've been wondering about how can I calculate the limit of this matrix: These states the different movements of the knight within a $4\times3$ chessboard, and what I'm trying to do with the limit ...
Daniel Alonso Paz's user avatar
1 vote
0 answers
183 views

Probability and Markov chains in a knight's tour

I've been working on my math IA for a month now, but I cannot happen to stumble across anything important to calculate within my chosen topic. I've tried optimization of algorithms but it went wrong, ...
Daniel Alonso Paz's user avatar
1 vote
0 answers
64 views

Tied down chess pieces forming knots

I want to make a public chess table, with every piece tied down securely so it cannot be stolen. Each of the pieces have attached one arbitrarily long thin steel cable, with one end in the side of the ...
Quebec's user avatar
  • 11
1 vote
0 answers
249 views

Shortest path between arbitrary squares for a chess knight

I have been working a bit on the following math problem: Assume a knight is placed on a random square on a chess board with the coordinates $(x, y)$, where the x-coordinate is the letter of the given ...
Alice's user avatar
  • 508
0 votes
0 answers
42 views

Way to move the chess piece

Donald loves to play chess and often invents new pieces, making the game more interesting. Donald's chessboard is seen as a grid of $N$ lines and $M$ columns. The rows are numbered from $1$ to $N$ ...
trungbk's user avatar
  • 165
1 vote
1 answer
50 views

the number of ways in which you can place $m*n$ kings on only the white coloured squares

The number of ways in which you can place $m*n$ kings on only the white coloured squares on a $2m*2n$ chessboard such that no two kings are diagonally adjacent to each other? My approach: I noticed ...
hmm191's user avatar
  • 33
0 votes
2 answers
94 views

The number of ways to arrange 8 rooks on the chessboard satisfying the condition

I have an interesting combinatorial math problem as follows How many ways are there to arrange 8 rooks on the chessboard such that no rook is on the main diagonal (the diagonal connecting the top left ...
Question 's user avatar
3 votes
1 answer
249 views

Number of ways of placing $4$ knights on a $4 \times 4$ board such that for every knight, there is an unique knight attacking it

I extracted this problem from an online contest. On the day of the contest (while it was still accepting responses), someone posted this very question on this site. It was closed because of lack of ...
Nothing special's user avatar
3 votes
1 answer
102 views

When can we flip the entire grid if we contunue flipping the cells of a subgrid?

Consider a 50-by-50 square grid. Initially all cells have the number $0$. For each operation, one selects a 1-by-7 or 7-by-1 subgrid and changes the number in the cells of the subgrid from 1 to 0 and ...
maomao's user avatar
  • 1,219
0 votes
1 answer
137 views

Placing chess pieces so that n such pieces are always non-attacking

I have a $12 \times 12$ chess board. I need to find: the minimum number of rooks I can place on the chess board so that there always exist 7 rooks no two of which are attacking. the minimum number of ...
Sahaj's user avatar
  • 3,290
0 votes
0 answers
29 views

Tifr gs 2021: symmetry of coloring chessboard under rotation [duplicate]

Let $\mathcal{C}$ denote the set of colorings of an $8\times 8$ chessboard, where each square is coloredeither black or white. Let $\sim$ denote the equivalence relation on $\mathcal{C}$ defined as ...
user avatar
1 vote
1 answer
175 views

Number of ways for a king to move from the lower left corner to the upper right corner by moving one step right, one step up, or one step up-right

There is a king in the lower left corner of the $n×n$ chessboard. The king can move one step right, one step up or one step up-right. How many ways are there for him to reach the upper right corner of ...
Isam's user avatar
  • 67
0 votes
0 answers
29 views

Counting "isomorphism-classes" of configurations of pieces which 'maximize' the possible movesets.

Similar question: How can one determine the chess configuration that maximizes the number of possible moves? Say you have an $8 \times 8$ chessboard with only 8 white rooks (don't want to worry about ...
John's user avatar
  • 1,950
21 votes
0 answers
566 views
+500

Number of Chess Games Possible: Parity Discussion

Chess is an incredibly intricate game, offering an immense number of moves and combinations. Due to this complexity, determining the precise count of legal chess games poses a significant challenge. ...
zero2infinity's user avatar
-3 votes
1 answer
110 views

Texts on the logic of chess

I am looking for texts that discuss the logic of the game of chess. I am sure there are a few such texts out there. Such a text might formalize chess in first-order logic. I would be very grateful if ...
user107952's user avatar
  • 21.4k
1 vote
0 answers
77 views

Generalizing standard chess pieces in $3$ and more dimensions

Writing a graph theory preprint about metric spaces in chess, I am struggling with the formalization of how every chess piece would move on a $k$-dimensional ($n \times n \times \dots \times n \subset ...
Marco Ripà's user avatar
  • 1,162
1 vote
1 answer
207 views

Finding distinct paths for an NxN chessboard. Is there a pattern? [duplicate]

I decided to come up with a puzzle for my friends at lunch today, and it seems to be much more complicated then I would have ever imagined. Here's the puzzle: A rook is located at the top left corner ...
waffles2124's user avatar
0 votes
1 answer
1k views

Chessboard Prisoner Puzzle - Uniqueness and Quality of Solution?

I was fascinated yesterday to discover a solution to the chessboard prisoner puzzle (Can anyone help me understand this particular solution of the famous two prisoners and a chessboard problem?). My ...
Barri's user avatar
  • 541
3 votes
1 answer
202 views

Compact Way of Storing Chess State Information

I'm looking for a method or a solution to store the chess state information in the most compact way possible without losing information and without too much surplus in computers. According to Deedlit ...
Karel Matthieu L. Logro's user avatar
1 vote
0 answers
67 views

Random Movements in Chess

This is a problem that I just thought of and could not find online- nor any similar problem. Suppose we have two different chess piece stationed on opposite corners of a chess board. For this example, ...
Evan Semet's user avatar
-1 votes
3 answers
86 views

Possible combinations that can arise in arranging half a Chess set when we get to pick each piece from either colors

Let's say we have a full Chess set - 2 Kings (1W, 1B), 4 Knights (2W, 2B), 16 Pawns and so on. How many ways can we arrange just half a set by picking pieces of either color. One example arrangement ...
abhi's user avatar
  • 99
6 votes
1 answer
107 views

Coloring game on a chessboard

On an $8×8$ board, you and your friend play the following game. Initially, all the unit squares are white. First, you color any $n$ squares blue. Second, your friend chooses $4$ rows and $4$ columns, ...
11235's user avatar
  • 113
1 vote
0 answers
59 views

Fomin et al., Mathematical Circles Chapter 4- Pigeon Hole Principle Problem 12. Max. no. of kings that can be placed so no two put each other in check

I found this problem in Mathematical Circles in the Pigeon Hole Principle chapter: What is the largest number of kings which can be placed on a chessboard so that no two of them put each other in ...
D S's user avatar
  • 5,315
3 votes
1 answer
57 views

To prove that no perfect covers exist for the staircase board.

Let $S_n$ denote the staircase board with $1 + 2 + ... + n = $$n( n + 1)\over2$ squares. Prove that $S_n$ does not have a perfect cover with dominoes for any $n \ge 1$. This problem is provided in ...
Camelot823's user avatar
  • 1,467
0 votes
0 answers
70 views

Need Help Proving The Impossibility of A Prisoner Problem

Imagine a prison consisting of 64 cells arranged like the squares of an 8-by-8 chessboard. There are doors between all adjoining cells. A prisoner in one of the corner cells is told that he will be ...
Camelot823's user avatar
  • 1,467
3 votes
0 answers
145 views

Are there large $n$-queens solutions without 3-in-a-line?

Wikipedia references: N-queens & No-3-in-line The 4-queens board is few enough pieces that it never has three queens on the same line ...
me22's user avatar
  • 181
1 vote
1 answer
125 views

Queen's graph diameter and knight's graph diameter for a $n \times n \times \dots \times n$ chessboard

Let a $n \times n \times n$ chessboard be given. I have just proven (by brute force) that, the queen's graph diameter, ${d_{n}^k}(Q)$ (i.e., the number of moves needed to move a queen from any 3D cell ...
Marco Ripà's user avatar
  • 1,162
0 votes
1 answer
115 views

Storing the board state of Hexagonal chess in a multi-dimensional array

Chess is normally played on a square board. This means, the board state can be easily represented in a square 8x8 2-dimensional array. On the other hand, Gliński's hexagonal chess is played on a ...
Richie Bendall's user avatar
1 vote
0 answers
72 views

Number of moves required to create a 5×5 chessboard under the given conditions

Consider a 5×5 square divided into 25 cells. Initially all cells are white. In each move it is allowed to change the colour of any three consecutive cells in a row or column to the opposite colour (i....
Toba's user avatar
  • 309

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