# 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 $4$...
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### 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 ...
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### 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 ...
• 79
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### 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 ...
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### 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 ...
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### 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: "...
1 vote
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### 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 ...
1 vote
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 (...
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1 vote
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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 ...
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 ...
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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 ...
1 vote
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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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1 vote
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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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1 vote
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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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