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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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Minimum number of dominoes on an $n \times n$ chessboard to prevent placement of another domino.

OEIS sequence A280984 (based on this Math Stack Exchange question) describes the minimum number of dominoes on an $n \times n$ chessboard to prevent placement of another domino. The sequence ...
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1answer
57 views

Cover a chessboard

Let $2n\times 2n $ board. I cover it with dominoes $1\times 2$ s.t. every cell is adjacent exactly one cell coverd by a domino. I have to find the maximal number of dominoes that can be placed in ...
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22 views

Method for finding all the solutions of the $n$ queens problem for a given $n$?

Is there an easy way of obtaining all the solutions of the $n$ queens problem for a given $n$? An algorithm online? Paper, or something? I only found a code written in c, but I'm not a c programmer, ...
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1answer
45 views

Doubt about $n$ queens problem when $gcd(n,6)=\pm 1$

So I know that the $n$-queens problem only has modular solutions if $gcd(n,6) = \pm 1$ My doubt is: if $gcd(n,6) = \pm 1$ then are all solutions modular? Or can there be normal solutions which are ...
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1answer
32 views

The minimum number to make the whole chessboard black

A 7x7 chessboard that painted in black-white (the corners are black). The operator "inverse - change color" can be run on a single row or column in the panel. the goal is to make the whole panel black....
4
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2answers
117 views

Let consider a square $10$x$10$ and write in the every unit square the numbers from $1$ to $100$

Let consider a square $10\times 10$ and write in the every unit square the numbers from $1$ to $100$ such that every two consecutive numbers are in squares ...
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2answers
44 views

Determine numbers written on a chessboard with the fewest number of questions.

You have a standard chessboard with 64 squares with each square having a different number written on it (say from 1 to 64), but in no particular order. You are allowed to ask questions. A single ...
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20 views

Combinatorial proof of Hamiltonian paths on the rook graph

We can be sure that number of Hamiltonian paths on the rook graph for any single cell on $n\times2$ chessboard equals $$ H(n+1) = \sum_{k=0}^{n} \binom{n}{k} \binom{k}{\lfloor{\...
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29 views

extending 8 queen chess puzzle to n*n chessboard

For any n*n chessboard ( n>8), is it always the case that n queens can be placed on it such that no 2 queens are attacking each other? Also, prove your answer. P.S:I do not know the answer. Just a ...
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22 views

Hamiltonian paths and cycles of rook graph on $n\times2$ chessboard

According to OEIS, there are closed form for directed Hamiltonian paths (A096121) and Hamiltonian cycles (A276356) of rook graph on $n\times2$ chessboard. Are there papers which include proof of those ...
2
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1answer
37 views

Minimum length of ascending sequence on a chessboard

Numbers $1,2,\ldots,n^2$ are placed in some order on an $n\times n$ board. An "ascending sequence" is an increasing sequence of numbers such that any two consecutive numbers in the sequence are in ...
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1answer
117 views

tiles covering a $7\times 7$ square

A $7 \times 7$ board is divided into $49$ unit squares. Tiles, like the one shown below, are placed onto this board. The tiles can be rotated and each tile neatly covers two squares. Note that each ...
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1answer
111 views

How do I solve this tile-covering problem?

Consider an $n\times n$ chessboard whose top-left corner is colored white. But Alice likes darkness, so she wants you to cover those white cells for her. The only tool you have are black L-shaped ...
2
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1answer
49 views

A game-theoretic chess puzzle — Proof verification

I recently came up with the following chess puzzle (which has almost nothing to do with one's Chess skills): Puzzle: Consider a variant of chess where black has to start with $1...e5$ regardless of ...
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0answers
19 views

Number of open and closed rook's tours

Knight's tour is very well known problem, but what about rook's tour? On $n\times1$ chessboard there are obviously $n!$ open and $(n-1)!$ closed tours. Is there a way to easily compute number of open ...
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1answer
42 views

Can two “magical” chess knights meet?

I have a chess board N*M. Two "magical" knights are standing in random positions (x1,y1) and (x2,y2). They are magical, because they make moves simultaneously. The question is: "how many moves are ...
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1answer
23 views

Coloring and Maximum number of Dominoes

I have joined a course and we are given the following questions, I am still a beginner btw. Find the maximum number of 2×1 dominoes that can be placed on an 8 × 9 chessboard if six of them are placed ...
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A computer screen shows a 98 × 98 chessboard, colored in the usual way. [closed]

This is a question from the 1998 math olympiad I found 2 solutions in the following places: book1 pg.32 book2 pg.163 however, I was having trouble understanding them. Any help would be appreciated ...
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2answers
53 views

Number of possible bishop moves on an $n \times m$ chessboard

For rook we have obviously $$R(n,m)=nm(n+m-2)$$ and for bishop $$B(n,m)=4\left(m\binom{n}{2}-\binom{n+1}{3}+\binom{n-m+1}{3}\right)$$ if we assume $\binom{n}{k}=0$ for $n<0$. Is there a way to ...
0
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1answer
51 views

Using factorials to calculate # of chess combinations

I recently came across a coding problem in which the solution involves writing a program that can take in the starting position and destination square of a chess piece, and then output the number of ...
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68 views

Colorings of a $3\times3$ chessboard

I am having some trouble with the following problem from Brualdi's Introductory Combinatorics (Chapter 14 Exercise 47). The nine squares of a $3\times3$ chessboard are to be colored red and blue. ...
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1answer
57 views

Maximum number of kings on the chessboard subject to some rules

The chess king moves one square in any direction (horizontally, vertically, or diagonally). The goal is to place as many king as possible on an r×c board subject to the following two conditions: ...
2
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1answer
60 views

Tiling for $m\times n$ chessboard problem for both $m, n$ are odd

Consider an $m\times n$ chessboard in which both $m$ and $n$ are odd. The board has one more square of one color, say, black, than of white. Show that, if exactly one black square is forbidden on the ...
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1answer
62 views

Combinatorics. Rook Placing

Suppose that we want to place 8 non-attacking rooks on a chessboard. In how many ways can we do this if the 16 most ‘northwest’ squares must be empty? How about if only the 4 most ‘northwest’ squares ...
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2answers
59 views

A way to calculate all the paths there are through a chessboard with combinatory. [closed]

I am looking to a way to calculate all the paths there are from (1,1) to (8,8) but you can only move the piece one square up, one square to the right and diagonally one square up to the right. I know ...
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0answers
108 views

How many squares can I select from chess board

I am given an $n\times n$ chess board. I have to select $k$ squares of size $1\times1$ so that no two squares share a common side. I have tried for $2\times2 , 3\times3 $ but for big values of $n$ can ...
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1answer
39 views

Put n rooks on a $n \times n$ chessboard so that every empty square is threatened

The answer to this question is $2n^n - n!$, according to the requirement in the question, I think the rooks can attack each other. Then in order to let every square be threatened, we need to put one ...
0
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1answer
35 views

Finding the right grid tessellation, for a chess board with elliptical geometry

I came up with the idea of non-euclidean chess(the chess board I'm working on will have 2D elliptical geometry) but I came across a problem. The question: What kind of grid do I put on it(what shapes ...
2
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1answer
80 views

Matching problem of slightly different chessboard

I try to prove that it is impossible to cover figure 1 with tiles of size $1\times 2$ and $2 \times 1$. When I abstract figure 1 as a graph I realize that it is a bipartite graph. I think this ...
5
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1answer
128 views

Pawn visiting every square on a chessboard exactly once and returning to its right

A pawn moves across $n\times n$ chesssboard so that in one move it can shift one square to the right, one square upward, or along a diagonal down and left. Can the pawn go through all the squares on ...
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1answer
97 views

Chessboard possible ways from A to B [duplicate]

How can I calculate the possible ways from point A to point B on a empty 8x8 chessboard while only being allowed to move up and right. Lets say I have to travel from x1 = 1, y1 = 1 to x2 = 2, y2 = 2 ...
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27 views

Number of knight cycle-free $n$-tours on an $n \times n$ modular chessboard?

Given an $n \times n$ modular (ie: the line resp. column after the last one is identified with the first) chessboard, I'd like to count the number of cycle-free $n$-paths a knight can do, starting ...
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3answers
297 views

Domino Tiling 8 x 8 grid proof

How can I prove that at least 8 dominoes are required to allow a placement to which no further domino can be added without two dominoes sharing an edge Any help will be appreciated. edit : For 9 * 9 ...
2
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1answer
157 views

Solve a recursive equation about chess

If we know that the King stands in the square on the left side at the bottom and has each time 3 possible moves: ...
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1answer
25 views

Reference on properties of the queen graph $Q_n$?

I am looking for a reference on the graph approach of the $n$-queens problem, basically something on the various properties of the $Q_n$ graph. I have a pretty good background on regular graph theory ...
0
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1answer
58 views

Chess Problem: What is the minimum number of white pieces required such that the black king is in check, no matter it's position?

I can think of a solution with 7 pieces required. 7 Queens on a diagonal with one missing in the corner. Is there a solution which requires less pieces than this? If so, does it generalise to an nxn ...
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2answers
126 views

$16$ rooks on a chess board.

Place $8$ pairwise non-attacking white rooks and black rooks on a $8\times8$ chess board. If one can swap rows and columns, is it possible for the black rooks to take the initial position of the white ...
3
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3answers
114 views

$5$ rooks on a $5\times 5$ chessboard

$5$ rooks are placed on a $5$ by $5$ chessboard in such a way that no two rooks attack each other. How can I prove that there exists a $2$ by $2$ square on the chessboard which does not contain any of ...
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1answer
75 views

Rooks on a Chessboard

$8$ rooks are randomly placed on different squares of a chessboard. A rook is said to attack all of the squares in its row and its column. Compute the probability that every square is occupied or ...
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2answers
193 views

Square board game with overlapping square sub-parts

Two players, A and B play a game on a board with $NxN$ squares. Player A populates all squares with numbers from 1 to $N^2$ in a completely random way. Here is an example for a 5x5 board: Player B ...
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2answers
118 views

Parity of number of pieces on the black squares of a chessboard [closed]

On a chess board there is an odd number of pieces in each row and in each column. What can we say about the total number of pieces on a black square on the board? The number must be even. The number ...
2
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1answer
281 views

How many ways can I put down two indistinguishable pieces on an ordinary $8 \times 8$ chessboard if they must either be in the same row or column?

I am a student in middle school and I was wondering if anyone could help me with the following problem: How many ways can I put down two indistinguishable pieces on an ordinary $8\times 8$ chessboard,...
2
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1answer
75 views

8×8 chessboard with numbers in them and you are allowed to switch them

Each cell of an 8×8 chessboard has a number written in it. Joonmin is allowed to switch numbers in two adjacent squares and Joonmin is allowed to change numbers in two adjacent squares to the average ...
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1answer
66 views

Cover a 5x5 board with pentominos

In https: this page, there are all the 107 ways to cover a $5\times 5$ board with pentominos. But, it is possible to prove that there is no way to cover the $5\times 5$ board using only L-pentominos ...
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1answer
35 views

12 1x2 dominoes are cut out of a chessboard. Is is true that one will always be able to cut a 1x3 rectangle from the remaning figure?

I have so far been only able to create examples where it is possible, and I am not sure how to create a formal proof using the difference between white and black squares. There are 32 white and 32 ...
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1answer
88 views

How many ways are there to get from one point to another on a (simple) chessboard?

A king is placed on the chessboard starting where the blue circle is (1,1). He can move to the right, up or diagonally. The ending point is marked yellow(8,8). How many ways are there for the king ...
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1answer
94 views

Can Zermelo's theorem be extended to a game which always has a winner?

Zermelo's theorem states that in a finite game, with two players, $a$ and $b$, where each player takes turns, the following is true: Player $a$ has a winning strategy, Player $b$ has a winning ...
2
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1answer
286 views

How many unique solutions are there to the 8 rooks problem?

Say I want to place 8 rooks onto a chess board such that none of them are attacking each other. Trivially, there can only be one rook per row and column, implying that there are $8! = 40320$ solutions ...
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3answers
364 views

Are there any applications of matrices, or linear algebra to chess? If so, are there good books on it?

Chess has never had any appeal to me, but recently my brother bought a chess set, and I realized that the board can be represented as an 8x8 matrix, and each type of of piece as a number from 0 to 6, ...
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1answer
340 views

Graph theory on chessboard

On an 8×8 chessboard, we choose two random squares S and C. We have a peg that is allowed to move always by 1 square horizontally or vertically in each move. What is the expected number of moves, when ...