Questions tagged [chessboard]

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

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56 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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Can someone help me with this problem. [closed]

Consider an n × n board divided into unit squares, with 1s in all of them. At each step, we take a 2 × 2 square and change the sign of all the numbers in it. Is it possible to reach a state in which ...
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1answer
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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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1answer
52 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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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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1answer
109 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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1answer
37 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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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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1answer
41 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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38 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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2answers
63 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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1answer
163 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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1answer
75 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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85 views

Different legal moves for a king in chess

Suppose you have an $n\times n$ chess board and a king can attack all adjacent squares, but not the diagonals (i.e. attacks $5$ squares including itself). What is the least number of kings necessary ...
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62 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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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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1answer
66 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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0answers
52 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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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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1answer
55 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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3answers
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How many ways can 2 rooks and a bishop be placed on a 4x4 board such that no piece attacks another piece?

How many ways can $2$ rooks and a bishop be placed on a $4\times 4$ board such that no piece attacks another piece? I stumbled upon this question and I don't know how to find the answer ...
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63 views

How I can shortly prove that you can have a closed knight's tour on the 6x6 chessboard?

On the website, the explanation that a knight's tour on a $6\times6$ board is possible is the continued proof of around $1\frac{1}{2}$ pages! It will be great if one of you could provide a simple, ...
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1answer
47 views

Question about closed knight's tours for n x m chessboard

Is there a simple mathematical algorithm where you can get a CLOSED knight' tour on an n x m chessboard? I need a way to prove that it is mathematically possible or impossible to have a closed knight'...
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1answer
75 views

Invariant problem-Chess board

The Question There is an integer in each square of an $8\times8$ chessboard. In one move, you may choose any $4\times4$ or $3\times3$ square and add $1$ to each integer of the chosen square. Can you ...
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White orthogonal (4-connected) path through a random chessboard.

Problem Let there be a chessboard of side length $n \geq 2$. The color of a single square (white or black) is a random variable $C \sim Be(p)$ (white with probability $p$), independent of all other ...
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Nonattacking “queens” on tiled triangles

I'll start with what my question is not. My question is not Nonattacking rooks on a triangular chessboard or The number of spacing $k$ non-attacking towers on the board $\left\{(i,j):1 \le i \le j \le ...
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1answer
47 views

Counting chessboard rectangles

I am looking into the puzzle count the number of rectangles in a regular $8*8$ chessboard. For a 1 by 1 chessboard there are 0 rectangles For a 2 by 2 chessboard there are 4 rectangles (2 by 1) For a ...
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1answer
104 views

Combinatorics - prove white can always force a win or draw in double chess

The game of double chess is played like regular chess, except each player makes two moves in their turn (white plays twice, then black plays twice, and so on). Show that white can always win or draw. ...
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Zobrist hashing collision probability 64-bit vs twice 32-bit

I've been thinking of the right site to post this, finally decided against SO and chess.stackexchange.com, since it's really more a mathematical question. I'm looking into how position databases in ...
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1answer
136 views

No. of ways of selecting 3 squares when they do not lie in same row, column, or diagonal

Total number of ways of selecting 3 small squares on a normal chess board so that they don’t belong to the same row, column or diagonal line, is equal to: No. of ways of selecting 3 squares when they ...
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1answer
63 views

Neighbours of white squares on a chess board and Hall's theorem.

Consider an $m \times n$ chess board with $mn$ even, $m,n \geq 2$, and one black and one white square removed. Label the white squares $1,\dots, \ell$ where $l = mn/2 - 1$, and for each $i \in \{1,\...
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1answer
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System of distinct representatives and chessboards

I encountered the following problem, which was presented in the context of the topic of SDRs (system of distinct representatives) - I am able to solve the problem, but I make no use of a SDR, and I am ...
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1answer
223 views

How many regions can be selected in a chess board?

How many regions can be selected in a 8 by 8 chess board? Definition of region: A region is a set of cells that are all connected together(by edge). i.e. a possible region: I want to run(or do ...
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17 views

number of paths passing by each tile of a chessboard by adjacent tiles

I have a $N\times N$ chessboard, and I need to compute all the paths from a corner to the opposite, walking once on each tile, and only walking from a tile to one of the nearest tiles (i.e., no jumps, ...
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Clarification On A Pigeonhole Principle Problem

I have encountered the following pigeonhole principle problem. I'm not sure what the question means, so I would like to clarify what it means: 17 rooks are placed on an 8×8 chessboard. Prove that ...
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1answer
66 views

Using dominoes to cover a chessboard

The question is You have a chessboard (8 × 8) plus a big box of dominoes (each 2 × 1). I use a marker pen to put an “X” in the squares at two locations. These two locations correspond to a black and ...
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0answers
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Coloring of small chessboards on the torus

Take $\mathbb{Z}[i]\pmod{4-i,1+4i}$. It has seventeen elements. Each element $m+ni$ has eight neighbours, that differ from it by $1,-1,i,-i,1+i,1-i,-1+i,-1-i$. Two copies of this torus can be ...
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1answer
92 views

Ways to place 3 chess pieces so that none are in the same column or row

How many ways can we place $3$ chess pieces so that none are in the same column or row? Chess pieces are distinguishable, so we can imagine them as one pawn, one knight, one rook. A chess board is a $...
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1answer
60 views

Place as few knights as possible on an 8 by 8 chess board so that every square is controlled by at least one knight.

Squares containing knights should be controlled as well. I think you need at least 3 knights to cover a corner, i.e. have the outside squares covered by knights, so that would be at least 12 knights ...
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2answers
577 views

Arranging Eight Queens on a Chess Board

I am tasked with finding the answers to the following questions: Part $1$: Consider the classic puzzle of placing eight queens on an $8$ × $8$ chessboard so that no two queens are in the same row or ...
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0answers
62 views

Combinatorics chess pawns placement

How many ways can one place $8$ pawns on a $4$x$4$ chess board such that each row and each column contains exactly $2$ pawns? I figured a brute-force way where I go row by row placing $2$ pawns each ...
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1answer
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Minimum number of moves a chess piece has to do in order to pass through all squares of a 8x8 board from an X position.

I need to know what is the minimum number of moves a chess piece has to do from an X position to pass through all squares of a board. Mathmatic and programming solutions are welcome. I need because I ...
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2answers
53 views

Minimize the greatest value labeled in a board in such a way that the sum of the numbers labeled is different for each possible path.

Consider a $n\times n$ board. A pawn is placed on the top left square of the board. The pawn must go to the bottom right square of the board by moving right and down on the board. We must label each ...
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Combinations of tiles on a chessboard —— how to root out symmetrical sets?

Let's say we're trying to find the number of combinations 3 tiles could have on a 3*3 chessboard, but excluding all symmetrical cases. For example: The two combinations above would be symmetrically ...
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30 views

Is it possible to represent 5-d chess spatially?

Recently a five-dimensional variant of chess came out, using time and multiple universes to represent 5-d chess on a screen. I know 4-d chess can be represented by splitting it into multiple 3-d ...
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0answers
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Let $n \geq 3$. Take an $2n \times 2n$ chessboard, and remove $2$ white pieces and $2$ black pieces, can you always cover it with dominoes?

I am reading "Kombinatorika" by Laszlo Lovasz, Katalin Vesztergombi and Jozsef Pelikan(in Japanese, translated and arranged by Jin Akiyama and Peter Frankl). There is the following problem ...
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1answer
42 views

modified queens

For the formulation of a modified N queens. Unlike the original Queens problem, there is just one rule-all N queens must be placed row-wise first. The goal is to select the smallest integer $p$ such ...
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2answers
78 views

A combinatorial chessboard strategy-game 2 players

We have a regular chessboard $ 8 * 8 $ and a coin. There are two players: A, B. Firstly, A places the coin somewhere on the chessboard Then, B can move the coin in a cell in the same line or the same ...
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1answer
21 views

What arrangement of 8 each of these 4 different compound leapers has the “strongest coverage properties” on a $16\times 16$ board?

The 4 compound leapers I'd like to look at are: (1,2), (0,3), (2,2); knight+threeleaper+alfil (1,2), (0,2), (1,1); knight+kirin (1,2), (1,3); knight+camel (1,2), (2,3); knight+zebra What ...
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80 views

Which theorems and definitions do I need to know to prove the impossible chessboard puzzle has a solution for every number of squares?

I saw this problem on the youtube channel called 3blue1brown. The problem is the following: There is a 8 x 8 chessboard, two prisoners (Prisoner 1 and prisoner 2), a key and a warden. Each of the ...

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