Questions tagged [chessboard]

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

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Play on a Chessboard

Consider an infinite grid $\mathcal G$ of unit square cells. A chessboard polygon is a simple polygon (i.e. not self-intersecting) whose sides lie along the gridlines of $\mathcal G$. Peter chooses ...
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Tiling a chessboard with triomino (L-shaped tile)

Can someone please help me to prove the following statement? Prove that it is impossible to tile an 8 × 8 chessboard missing two opposite corners using right triominoes. I suppose that the proof ...
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Klein Group in a combinatorics problem on checker board

Came a cross problem 21 here: https://euclid.ucc.ie/mathenr/IMOTraining/2008%20Winter%20Camp%20-%20David%20Arthur%20-%20Invariants.pdf Essentially, for each turn, a checker can jump diagonally over ...
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1answer
21 views

In how many ways you can place 18 chess bishops on a 10x10 chequered Board without hitting each other?

Calculate number of ways 18 chess bishops could be placed on a a 10x10 chequered Board without hitting each other.
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Least number of pieces and squares in chess, given none of them under any attack.

I have seen an interesting chess setup in Puzzling network of Stack Exchange; there white queens attack all the squares except where the black knights are located, on the other hand all black knights ...
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How many ways we can place knights on a $m \times n$ chessboard in such a way that each piece does not attack the others $($modulo $10^9 + 9)$

How many ways we can place knights on a $m \times n$ chessboard in such a way that each piece does not attack the others $($modulo $10^9 + 9) \ ?\ (m \le 4, n \le 10^9)$ For example: $m = 2, n = 2, ...
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2answers
43 views

Chromatic number of a graph on a chess board

For a chess piece Q, the Q-graph is the graph whose vertices are the squares of the chess board and the two squares are adjacent if Q can move from one of them to the other in one move. Find the ...
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1answer
45 views

Probability on a 8x8 chessboard with pieces and kings. What is the chance the piece can be captured?

I recently started playing chess (damn quarantine) and came up with this problem. Supposing we have a blank 8x8 chessboard. We pick a spot at random and place a special black piece which cannot move. ...
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1answer
43 views

Shortest path of a knight on a chessboard

Given a knight on an infinite-size chessboard. Knight starts from $(0,0)$ and the destination is $(x,y)$ with $x\ge 0$ and $y\ge 0$. I want to prove that among all the path with the minimum steps, ...
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1answer
12 views

Expectation of difference between amount of white and black cells on the infinite chessboard that the king will visit

The king is being randomly moved on the infinite chessboard. What is the expectation of modulus of difference between amount of white and black cells, which the king will visit by n steps. Each cell ...
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1answer
75 views

Conditional probability; two queens attack each other

Two queens are randomly placed on a chessboard. What is the probability that they attack each other? A: two queens randomly placed on a chessboard (condition) B: they attack each other I have 2016 ...
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1answer
28 views

Proving T-tetrominoes fit in a chessboard

I'm fairly new to discrete math, and I wasn't sure how to prove the following. Prove that if $n\geq 2$, then every $2^n \times 2^n$ chessboard can be tiled with non overlapping T-tiles. If I draw ...
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1answer
152 views

New Character in Chess game

I having problem to solve the below question: Given $2020\times 2020$ chessboard, what is the maximum number of warriors you can put on its cells such that no two warriors attack each other. ...
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Solutions to N-queens when each queen is threathened by two queens.

This is the problem of placing N chess queens on an N × N chessboard so that queens threaten each other sequentially. That is queen[x] threatens queen[x+1] for x = 0 to n-1, and queen[n] threatens ...
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Colorings versus geometric group theory method

Consider the following famous problem: Two opposite corner cells of a chessboard are removed. Prove that it is impossible to cover remaining cells with dominoes $2\times1$. The standard proof uses ...
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1answer
150 views

A problem on the $n\times n$ square [closed]

This problem is stated directly as following Problem A $n\times n$ square that satisfies the following conditions: $n\geq 4$ In each unit square we write a number $x$, with $0\le x \le 1 $. No ...
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1answer
361 views

The spreading game and its expansion

For all those who lost their lives and due to this tragic disease CONTEXT This question is inspired by the following question that was proposed by my math teacher Lam Nguyen, I shall cite this ...
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1answer
19 views

Largest number of moves while avoiding threefold repetition

I have a rook confined to a $1$ by $8$ chessboard, with the squares numbered 1 through 8. As my rook moves to different spots on the board, I create a sequence of positions $a_n$. For example, we ...
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38 views

Are chess PSPACE-complete or PSPACE-hard only?

There are links that claim that chess are PSPACE-complete but the snippet below from the Handbook of theoretical computer science says that they are harder. Which statement is true ?? EDIT: included ...
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5answers
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King on a $15 \times 15$ chessboard

A king is placed in the center tile of a $15 \times 15$ chessboard. He can move in the usual ways, 1 move in any direction. In how many ways can he return to his original position if he can make a ...
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1answer
32 views

Maximum Number of Attacking Rooks placed on a chessboard

First, we put a rook on the 8*8 board. ● Then we put a second one on the board, so any of the two rooks can take the other. ● This continues with a third one, so any of the three rooks can take any ...
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1answer
104 views

Non-attacking knights and rooks on a chessboard

This question from math contest olympiad phystech I tried to find the solution but I can't find it. Please help me to find solution of this problem. Given a board with a size of $11 × 11$ cells. Jack ...
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1answer
39 views

Chess parity problem [duplicate]

On an 8x8 chess board we place rooks so that the number of them is odd on each line or collumn. Show that the number of black squares that have rooks is even. What I found: Obviously, since the sum ...
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Finding Diagonals in 1D Representation of a Square Matrix

The context here is detecting illegal moves in the N-Queens problem. If I have a zero-index based 1D list storing say 0s for empty squares and ...
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1answer
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Chess Knight Distance

On an $a×a$ chess-board knight takes $n$ jumps from the bottom left to the bottom right and $m$ jumps from the bottom left to the upper right. For $a=7$ are 4 steps necessary. Is $a=7$ the only ...
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1answer
165 views

5 indistinguishable rooks on 8x8 chessboard

In how many ways can five indistinguishable rooks be placed on an 8-by-8 chessboard so that no rook can attack another and neither the first row nor the first column is empty? I started out by ...
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1answer
137 views

Swap the place of two knights on a standard chess board?

given the place of two knights on a chess board what is the minimum steps to swap their places? note that they are moving in turns i.e. the white moves first then the black etc. they can't -at the ...
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103 views

Minimal covering of a $5 \times 5$ square with $T$ shaped tetrominoes.

The title almost completely describes the problem. So you have a $5 \times5$ square. You have to fit the minimum number of $T$ shaped tetrominoes such that no more $T$ shaped tetrominoes can be fitted ...
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2answers
26 views

In how many ways we can choose $k$ squares on $n \times n$ chessboard, given certain restrictions?

In how many ways we can choose $k$ squares on $n \times n$ chessboard, so that none of these squares are on the same column nor the same row? Sorry, but the question is the same as in the title. Here ...
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Ways to cross out $n^2/2$ squares on a $n\times n$ chessboard

How many patterns $P_n$ are there to cross out $n^2/2$ squares on a $n\times n$ chessboard, so that the number of crossed out squares in each row and each column are all even? Is there a way to get a ...
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1answer
92 views

Distance formula for generalized knight movement on infinite chessboard from a corner

Consider a chessboard infinite in positive x and y directions, all square has non-negative integer coordinates, and the only corner is at $(0,0)$. A $(p,q)$-knight is a piece that can move so that ...
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3answers
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How to explain expected (lone) Knight moves on a chess board

(* edited as per comments below *) I've been working on the expected number of moves a lone knight on a chess board needs to make to get from position (i,j) on a chess board to position (8,8) at the ...
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1answer
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Path on a chessboard

Consider a chess board and a pawn. The game is simple (and I'm sure you have all heard it before), pawn wants to move from position (1,1), or the bottom left corner, all the way to (n,n), or the upper ...
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1answer
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N Queens: possible arrangements on 8x8 board

I am studying the $N$-Queens problem and have found the following statement Now number of possible arrangements of $N$ Queens on $N \times N$ chessboard is $N!$, given you are skipping row or ...
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Number of ways to put two different knights on a chessboard so they attack each other

This question is in my textbook: What is the number of ways to put two non-identical knights on a chessboard so they attack each other? My solution: If two knights attack each other, they can be ...
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1answer
267 views

Please explain the proof of the Mutilated Chessboard Problem

Can someone explain the proof behind why the mutilated chessboard problem is unsolveable? The problem asks, given an 8x8 chessboard with two diagonally opposite corners removed, is it possible to fill ...
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2answers
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How many rectangles do you have on a cylindrical chessboard?

Say I roll up a square chessboard into a cylinder. How many rectangles, if finite, will it have? Provided that no lateral sides of the rectangle must overlap (so you can't basically go round and round ...
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1answer
61 views

Knightwise “Nearness” By Number Of Moves Required

Given an otherwise empty $n\times n$ chessboard with a knight on one of the squares, define the “knight-closedness” of this board as the maximum possible length of a minimal knight route from one ...
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Help me with this interesting and challenging to prove “fill the chessboard” problem

A $15×15$ chessboard was covered by $3×3$ and $2×2$ plates in such a way that the plates don't stick out of the chessboard, they don't overlap each other and every field of the chessboard is covered. ...
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1answer
60 views

Given an $n\times m$ grid and $x$ superqueens, what's the minimum number of moves for the superqueens to be symmetric?

Given an $n\times m$ grid and $x$ elements on the grid, what's the maximum number of moves required to arrange the elements, for all possible starting permutations of the elements, into a symmetric ...
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1answer
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A Hamiltonian cycle generated by lame rooks moves

I have got this problem at high-school math-contest seminar on Graph Theory Let us have a chessboard, where one black and one white lame rooks stand. Lame rook can move to edge-adjacent field only. ...
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Expected size of colored block on chessboard?

Randomly color the squares of an $m\times n$ chessboard red or black (each square has a fifty-fifty chance of being red or black). A monochromatic region is a set of squares that are connected along ...
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Tour a $M\times N$ chess board with a piece that moves in each turn along the diagonal of a $m\times n$ rectangle

Here a "tour" is a sequence of moves that visit each square of the board. A square can be visited more than once if necessary. For example, in a $8\times 8$ chess board, a knight which makes $2\...
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What is the fewest vertices a non-planar subgraph of the Knight's graph can have?

I was wondering about the title question: What is the fewest vertices a non-planar subgraph of the Knight's graph can have? I have found a subgraph with 14 10 vertices that is non-planar And it ...
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1answer
60 views

Minimum number of steps for a knight on chess board (5 x 5 or under)

Given two squares on a chess board of 5x5 or under, how can we determine the minimum number of moves required by a knight to reach one square starting from the other, including a way to determine if ...
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2answers
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Chessboard problem [closed]

The 64 squares of an 8×8 chessboard are filled with positive integers in such a way that each integer is the average of the integers on the neighbouring squares. Show that in fact all the 64 entries ...
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One knight and one queen on an infinite chess-board (a simple game)

On an infinite chessboard, player A places the queen wherever they want. Then, player B places the knight wherever they want. Finally, the game starts. The main rule is that any square where player ...
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1answer
329 views

Is it possible to start with a knight at some corner of a chess board and reach the opposite corner passing once through all the squares?

Is it possible to start with a knight at some corner of a chessboard and reach the opposite corner passing once through all the squares? The knight can reach the other corner or any square for that ...
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5answers
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Subset of knight's move in chess.

A particle is allowed to move in the $\mathbb{Z}\times \mathbb{Z}$ grid by choosing any of the two jumps: 1) Move two units to right and one unit up 2) Move two units up and one unit to ...
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2answers
284 views

Two Knight on chessboard

We have given an integer n, we need to find the number of ways two knights can be placed on an n×n chessboard so that they do not attack each other. I tried the simulation strategy but it is too ...