Questions tagged [chessboard]
Use this tag for questions about the board on which the game of chess is played.
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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 ...
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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 ...
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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 $N$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Jeson Mor chess variant - graph problem equivalent
It is rather a kind of general question, any hints are very pleasant to see :)
There is a chess variant called Jeson Mor: https://en.wikipedia.org/wiki/Jeson_Mor. Briefly speaking, the goal of this ...
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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, ...
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How do I calculate the diagonals of a matrix from given origin?
suppose that I have this kind of cartesian plane within a 8x8 matrix, how can I calculate the amount of squares in the diagonals from any given origin in this matrix?
For instance, the diagonal of Y ...
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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 ...
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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 ...
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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 ...
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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
...
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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 ...
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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 ...
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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....
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The number of placement of $n$ pieces on the chessboard of $n\times n$, such that the piece $k$ cannot be placed in the $k$-th row nor $k$-th column
Considering the chessboard of $n\times n$, we now want to put n pieces $1,2,\cdots, n$ on this chessboard, and satisfy that the piece $k$ cannot be placed in the $k$-th row nor $k$-th column, and each ...
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Calculating the Shannon entropy
Recall that the Shannon entropy of a random variables X taking values in a finite
set S is given by
$H[X] = −\sum_{x∈S}Pr[X = x] \log_2 Pr[X = x].$
(We set $\log_2 0 = 0.)$ For a pair of random ...
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Checking if a configuartion is possible in an infinite chessboard
I have tried this recreational problem for a long time now and I haven't been able to find any approach that would take me to the solution. Any hints or tricks to use would be appreciated.
On a ...
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Can chess moves be identified as a meaningful algebraic structure?
I have been contemplating for some time if one can give the movements of chess pieces on a chess board an algebraic structure. I will give my ideas so far and hopefully someone can help me complete it....
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Placing n non attacking rooks on a value weighted n*n board, such that the value of the sum of the chosen squares is minimal
I have a variation of the n rooks problem: https://en.wikipedia.org/wiki/Rook_polynomial
I have a weighted n*n matrix, say 4 * 4, where each "square" has a value.
I would like to place n non ...
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Grid Colouring: $4\times4$ chessboard case
Every side of every unit square in a $4\times4$ chessboard is coloured either red or blue. What is the maximum number of blue sides such that every unit square has at most one blue side?
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Number of ways to place $n$ rooks on $n$ by $n$ board such that they don't hit each other. But someone has cut out squares of one of diagonals.
I was thinking about asking this is chat because I thought that my concern is not serious enough, but it became too long so here I am.
I hope that the question is clear: you just can't place rooks on ...
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A chess board of $4\times4$ and $4$ different rooks and not allowed to place the rooks on the white main diagonal, each column is at least one rook.
Let's assume we have a chess board of $4\times4$ and $4$ different rooks. We are not allowed to place the rooks on the white main diagonal. So we have $12$ spots to place the rooks. We want to know ...
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Is mate-in-$n$ problem for Trappist-1 undecidable?
Trappist-1 is a variant of infinite chess that has a piece called huygens which leaps any prime number of squares orthogonally. To actually implement this game, it should have decidable mate-in-$0$ (...
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Discrete Mathematics Counting and Combinations/Variations on Chess Board
Knight Movement on Board for Questions
Assume the white Knight starts on square b1 as shown in figure (a). How many paths can
the Knight take to reach the b8 square marked ”X”? Assume that the knight ...
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Placing $n$ rooks peacefully on a board
We got $n\times n$ board colored in different colors. The count of cells with the same color is $\leq \frac{n-1}{16}$. Prove that we can manage to place $n$ rooks on different color cells, so they ...
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Possible positions of the knight after moving $n$ steps in Chessboard.
Problem
There is a knight on an infinite chessboard. After moving one step, there are $8$ possible positions, and after moving two steps, there are $33$ possible positions. The possible position after ...
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Hamilton cycle on chessboard
Suppose we have $8 \times 8$ chessboard such that two squares are adjacent iff they share a common side. In one move pawn can move to adjacent square. Prove that the pawn made a different number of ...
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What's the maximal number of chess pieces under this rule?
Suppose we have a $n\times n$ chessboard, where $n\geq 3$ is a positive integer. We place chess pieces on the board such that any three of them are not standing next to each other and on the same line ...
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Drunken king on chessboard: Why is the probability that the king is on each square proportional to the number of adjacent squares?
On a chessboard there is a (drunken) king. The king moves at the beginning of each minute, in a random direction: up, down, left, right, or the four diagonal directions (unless the king is on an edge ...
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Number of ways to arrange $k$ non-attacking rooks on an $m\times n$ chessboard
I need to calculate the number of ways to place k non-attacking rooks on an $m
\times n$ chessboard where $n\neq m$. I know how to calculate the number of arrangements when the problem is to calculate ...
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Representing graphs by an arrangement of chess rooks
Consider a potentially infinite chessboard on which a number of rooks has been placed, under the restriction that any 2x2 square containing at least 3 rooks must contain a 4th rook.
This way, for ...
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Inverting a $n \times n$ board filled with knights
I am facing the following problem:
Suppose we have a board of size $n \times n$ (in my case, it doesn't go above $n=5$). One diagonal of the board is empty, and the two sides it delimits are filled ...
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How to calculate the number of paths of minimum length possible a knight can take to get from one corner of a chess board to the opposite one?
I've written a small Python script to give me the least number of moves it takes a knight to get from one square to any other on a $n{*}n$ chess board.
But then I've wondered how many paths the knight ...
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Can a chess game be represented by less than 10N bits, where N is the number of moves (ply) in the game?
I started wondering how much information is required to encode a Chess game. Since there are 64 squares on the board, it seemed that 12 bits would be required to encode a move, 6 for the starting ...
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Algorithm to find a stable cycle mean in a bipartite digraph under competing strategies of the two colored nodes.
Given a strongly connected bipartite digraph with a finite number of black and white nodes. Every edge goes either from a white to a black node, or from a black to a white node. Each black to white ...
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