# Questions tagged [chessboard]

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

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2answers
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### 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 ...
0answers
32 views

### 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 ...
1answer
45 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 ...
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 ...
2answers
58 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 ...
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: ...
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 (...
0answers
54 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 (...
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 ...
0answers
38 views

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 ...
0answers
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, ...
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'...
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 ...
0answers
22 views

### 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 ...
2answers
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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 ...
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 ...
0answers
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, ...
0answers
59 views

### 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 ...
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 ...
0answers
40 views

### 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 ...
1answer
92 views

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 $... 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 ... 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 ... 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 ... 1answer 48 views ### 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 ... 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 ... 0answers 31 views ### 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 ... 0answers 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 ... 0answers 61 views ### 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 ... 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 ... 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 ... 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 ...
0answers
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 ...