Questions tagged [chessboard]

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

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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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  • 747
-1 votes
1 answer
52 views

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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5 votes
1 answer
122 views

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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2 votes
1 answer
100 views

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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  • 1,192
1 vote
2 answers
67 views

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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  • 29
12 votes
1 answer
271 views

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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3 votes
1 answer
88 views

Inverting a n x n board filled with knights

I am facing the following problem: Suppose we have a board of size $n x n$ (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 with ...
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9 votes
5 answers
624 views

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 a knight takes on a $n{*}n$ chess board to get from one square to any other. But then I've wondered how many paths the knight ...
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  • 105
5 votes
2 answers
115 views

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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1 vote
0 answers
49 views

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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4 votes
0 answers
76 views

Chessboard 5x5 with twentyfour"+1" and one "-1" [duplicate]

A number $+1$ has been entered in 24 fields of the $5 \times 5$ array, and the number $-1$ in one. In one move, you can multiply by $ -1 $ all numbers in the subarray $ k \times k $ where $ k> 1 $...
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1 vote
0 answers
54 views

Number and Density of Possible Checkmates

Is there a way to estimate the number of possible checkmates on a chess board? By possible, I mean that it corresponds to some board state. If it's easier to estimate, I don't mind whether the number ...
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4 votes
1 answer
56 views

Number of ways from top of the black square on chess board to any bottom black square when moving downwards diagonally to black squares only.

I have attempted this problem by drawing out grid like this: \begin{array}{|c|c|c|c|c|c|c|c|} \hline1&&1&&1&&1&\\\hline&2&&2&&2&&1\\\hline2&&...
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2 votes
1 answer
55 views

How to select some black color cells in a chess board such that each white cell has exactly 1 adjacent selected black cell?

Given a chess board of size $n$ x $n$, where $n$ is an even number. How do i prove that for any given even number n, there exists a way to select some of the black cells of the chess board such that ...
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  • 111
7 votes
1 answer
128 views

Tour of chess king

Consider lame chess king that can move only one cell left, down and diagonal upright. Consider square chess board. Question: Can such a king visit all cells of a board (each cell only once) and end up ...
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1 vote
1 answer
112 views

Counting the number of dominating rook placements in a chessboard

Given a square $n \times n$ chessboard and $m$ rooks (with $m \geq \lceil{n/2}\rceil$ and $m \leq n^2$) I would like to count how many of the total $\tbinom{n^2}{m}$ possible combinations cover each &...
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  • 133
2 votes
0 answers
81 views

single king on a chessboard game

Consider the following two players game: Giving chessboard NxN, and single king placed on some square on the board. Each player in his turn, can move the king in each direction in the same way player ...
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1 vote
1 answer
119 views

How many bishops can be placed on a $m \times n$ chessboard?

I came across this question and since I did not find it answered anywhere I do not know if my thoughts are correct, but I went like this: We know, that if we want to place bishops so they cant attack ...
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0 votes
0 answers
138 views

General Solution to Kings on Chessboard

This question was inspired by the numerous other king on chessboard questions asked on this site. Specifically this one: Kings on a chessboard My question is "just more general: "What is the ...
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3 votes
1 answer
141 views

Why pawns starting at chessboard squares $(1,1)$ and $(8,8)$ that move orthogonally at each step will never swap positions?

Let's say we have a chessboard (i.e an $8×8$ grid). Let's assume each cell is identified by two coordinates (integer numbers) ranging from $1$ to $8$. Assume to have a red pawn in position $(1, 1)$ ...
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0 votes
0 answers
58 views

How many rectangles of different sizes are there on the standard chessboard that have equal area black and white?

I know that the number of rectangles on a standard chessboard are ${9 \choose 2}\cdot {9  \choose 2}$. How to compute the ones having equal areas of black and white?
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0 votes
0 answers
39 views

Chessboard combinatorics question

A white square and a black square have to be chosen on a chessboard such that they do not lie in the same row or column or diagonal. In how many ways can they be chosen. My approach:- Ways of ...
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  • 633
0 votes
1 answer
136 views

What is the maximum number of knights we can put on a chessboard such that no knights of different colors attack each other?

Let's suppose we have three different colors of knights: red, yellow, and green. What is the maximum number of knights that we can put on a chessboard, such that no knights of different colors attack ...
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1 vote
1 answer
93 views

A $7\times 14$ chessboard is cut along its gridlines into squares and three-square corners. Show that there are at least as many corners as squares...

The problem: A $7$ by $14$ chessboard is cut along the grid lines into some squares and some three-square corners. Show that there are at least as many corners as squares. Can equality hold? I do not ...
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  • 187
6 votes
2 answers
334 views

What is the maximum number of warriors one can put on a chess board so that no two warriors attack each other?

In chess, a normal knight goes two steps forward and one step to the side, in some orientation. Thanic thought that he should spice the game up a bit, so he introduced a new kind of piece called a ...
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  • 3,823
3 votes
2 answers
89 views

A book claims there are $10^{120}$ board positions on a chess board. How would one prove this?

I have recently bought an exercise copy and there, in the cover page I got an amazing fact about chess board, "There are total $10^{120}$ board positions in a chess board." But I was just ...
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  • 103
8 votes
2 answers
386 views

Is it possible to place one queen and at least 29 knights in a 8x8 chess board such that no 2 pieces attack each other?

Is it possible to place one queen and at least 29 knights in a 8$\times$8 chess board such that no 2 pieces attack each other? I thought to try to use the bound $\lceil \frac{mn}{2} \rceil $ for the ...
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2 votes
1 answer
170 views

n-"Kings" Problem

I was thinking on some variations of the n-Queen problem until I reached the following problem I couldn't solve: How many ways are there to put $n$ kings (as in a chess game) on a $n\times n$ ...
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  • 1,411
0 votes
1 answer
43 views

How many distinct patterns can be formed using 3 chesspieces on a 8x8 standard board?

Scenario 1: You have 3 same colored pawns, and there are no restrictions as to their placement. How many unique patterns can be formed? By unique, I mean that the patterns can't coincide by mere ...
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1 vote
1 answer
58 views

Discrete Geometry Violating The Triangle Inequality

My question today is spurred by something I came across in chess called The Crooked Path. The idea is simple enough and essentially comes down to this: if the king wants to move say 6 squares up the ...
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5 votes
1 answer
136 views

A chessboard Combinatorics Problem

How many ways are there to put numbers $1,2,3,\cdots, n^2$ into a $n\times n$ chessboard s.t. the sum of the numbers on every row and every column is even. My approach to this problem (Ideas) I want ...
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  • 549
0 votes
1 answer
62 views

Painting the chessboard symmetrically

We are asked to find all natural numbers $n$ for which we can paint the cells of a $2n \times 2n$ chessboard black and white such that: each pair of cells symmetric about the center of the board ...
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2 votes
2 answers
186 views

Recommendations for chess themed math exercises

I am trying to organize a recreational math class for a group of high school students (mixed years), themed around the game of chess. Ideally, I would like to prepare exercises that simply require a ...
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  • 1,354
0 votes
1 answer
51 views

How many short rooks can be placed on a chessboard

Placing rooks on the table is a commonly known problem. What about "short rooks" (they behave in the same manner, but on distance less or equal to 2. I've noticed that splitting the board ...
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0 votes
1 answer
68 views

Pigeonhole principle for n queens

Suppose we have a board $100 \times 100$ and place $100$ queens such that none attack another. Prove that each of the four $50 \times 50$ sub-boards (gotten by dividing the board in $4$) contains at ...
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0 votes
2 answers
73 views

Adjacent houses

We propose a game that is quite simple: You choose a house to start (the starting house). From this square you must move to the adjacent square (left, right, above or below) with the lowest value yet ...
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  • 3,816
4 votes
1 answer
156 views

Covering $6 \times 6$ chessboard with $1 \times 2$ tiles [duplicate]

Problem: prove that after putting in 11 1 by 2 tiles in the 6 by 6 chessboard, there is definitely room to put in another tile. Tiles cannot overlap with each other. So I found this similar problem: ...
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  • 335
2 votes
1 answer
46 views

How is this function surjective? (Double Counting)

We call a positive integer $n$ "good" if and only if , in a $6 \cdot 6$ checkerboard , no matter how we put $n$ $1 \cdot 2$ dominoes in the board, there will be a space for another domino (...
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3 votes
0 answers
60 views

Can the Knight go through all spaces on a $7\times7$ chessboard?

On a $8\times8$ chessboard if the Knight starts at one of the corners it can move through all $64$ squares only once. But on a $7\times7$ board can the Knight go through all $49$ squares only once (...
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1 vote
1 answer
43 views

Chess puzzle with king [closed]

There are 2 players who share the same king on a chessboard. The starting point is A1 and they want to finish at H8.The King can move up (A1-A2) right (A1-B1) and diagonaly (A1-B2) but back tracking ...
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4 votes
0 answers
66 views

Generalized Hertzsprung Problem

The Hertzsprung Problem goes as follows: In how many can we place exactly $n$ non-attacking kings on a $n \times n$ chessboard such that there is exactly $1$ king in each row and column where $n \in \...
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2 votes
2 answers
108 views

Tiling a $(2n - 1) \times (2n - 1)$ chessboard with one corner cut out

One corner of a $(2n - 1) \times (2n - 1)$ chessboard is cut off. For which $n$ can you cover the remaining squares by $2\times 1$ dominoes, so that half of the dominoes are horizontal? My half-...
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  • 142
1 vote
1 answer
275 views

Tiling a rectangle with tetris pieces of T-shape

I know that Walkup published a paper stating that $m\times n$ table can be tiled with T-tetrominoes iff $4\mid m$ and $4\mid n$. The converse is clearly true because we can tile a $4\times 4$ table ...
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  • 164
0 votes
1 answer
469 views

Whys is the maximum number of non-attacking pairs of queens in the 8-queen problem?

In the 8-queen’s problem we want the number of attacking pairs of queens to be zero in the solution assignment of the queens. A possible fitness function is the number of non-attacking pairs of queens ...
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1 vote
0 answers
115 views

How many ways can a rook move from the top left to the bottom right corner. How do you generalize a formula for this.

A game of chess is played on a board with 64 squares. generalize a formula for how many ways the rook could move from the top left to the bottom right corner. I know how to write the formula for ...
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3 votes
0 answers
51 views

Forming circuit with straight moves

I am reading about a problem where we have a rectangular board, and we have to show that it is impossible to complete a circuit of the board if both sides have odd length. Circuit is a sequence if ...
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  • 1,507
1 vote
1 answer
109 views

What is the minimum number of broken queens required to cover an $n\times n$ board?

Consider an $n\times n$ board. Assume that the sides of the board are parallel to the north-south and the east-west directions. If a piece of "broken queen" is placed on this board, it &...
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  • 164
4 votes
0 answers
96 views

Number of sudoku puzzles vs valid chess positions

The basic question is - which one is bigger? This possibly needs some clarification: By a sudoku puzzle I mean a grid with some cells filled with numbers and others empty so that they can be filled ...
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  • 2,274
3 votes
0 answers
252 views

Non-attacking kings on an $n\times n$ board with the same number in each row and column

Building on my previous question: Kings on a chess board. What is the smallest square board that could have three non-attacking kings in every row and column. (Note: kings attack each other by being ...
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1 vote
1 answer
107 views

Check if polynomial can be rook polynomial

How to verify if a given polynomial is a rook polynomial? Let’s assume I have a chessboard 5x5 and those 5 polynomials: $r(x) = 2 + 11x + 35x^2 + 50x^3 + 26x^4 + 5x^5$ $r(x) = 1 + 15x + 35x^2 + 50x^...
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