A chessboard is covered with $32$ dominos. Each domino covers two adjacent squares. Show that the number of horizontal dominos with a white square on the left equals the number with a white square on the right.

Can the squares of a $1990 \times 1990$ chessboard be colored black or white so that half the squares in each row and column are black and cells symmetric with respect to the center are of opposite color?

Given natural number n. Suppose that $N$ is the maximum number of elephants that can be placed on a chessboard measuring $2 \times n$ so that no two elephants are mutually under attack. Determine the number of ways to put $N$ elephants on a chessboard sized $2 \times n$ so that no two elephants attack each other. Alternative Formulation: Determine the number of ways to put $2015$ elephants on a chessboard measuring $2 \times 2015$ so there are no two elephants attacking each othe PS. Elephant = Bishop

Let $n\geq 3$ be an integer. Find the number of ways in which one can place the numbers $1, 2, 3, \ldots, n^2$ in the $n^2$ squares of a $n \times n$ chesboard, one on each, such that the numbers in each row and in each column are in arithmetic progression.

We put a figure of a king on some $6 \times 6$ chessboard. It can in one thrust jump either vertically or horizontally. The length of this jump is alternately one and two squares, whereby a jump of one (i.e. to the adjacent square) of the piece begins. Decide whether you can choose the starting position of the pieces so that after a suitable sequence $35$ jumps visited each box of the chessboard just once.

What is the number of ways in which one can choose $60$ unit squares from a $11 \times 11$ chessboard such that no two chosen squares have a side in common?

Some cells of a $10 \times 10$ are colored blue. A set of six cells is called [i]gremista[/i] when the cells are the intersection of three rows and two columns, or two rows and three columns, and are painted blue. Determine the greatest value of $n$ for which it is possible to color $n$ chessboard cells blue such that there is not a [i]gremista[/i] set.

Hugo places a chess piece on the top left square of a $20 \times 20$ chessboard and makes $10$ moves with it. On each of these $10$ moves, he moves the piece either one square horizontally (left or right) or one square vertically (up or down). After the last move, he draws an $X$ on the square that the piece occupies. When Hugo plays the game over and over again, what is the largest possible number of squares that could eventually be marked with an $X$? Prove that your answer is correct.

Player $A$ and player $B$ play the next game on an $8$ by $8$ square chessboard. They in turn color a field that is not yet colored. One player uses red and the other blue. Player $A$ starts. The winner is the first person to color the four squares of a square of $2$ by $2$ squares with his color somewhere on the board. Prove that player $B$ can always prevent player $A$ from winning.

Let $n$ be a positive integer. There is a pawn in one of the cells of an $n\times n$ table. The pawn moves from an arbitrary cell of the $k$th column, $k \in \{1,2, \cdots, n \}$, to an arbitrary cell in the $k$th row. Prove that there exists a sequence of $n^{2}$ moves such that the pawn goes through every cell of the table and finishes in the starting cell.

What is the maximum number of rooks one can place on a chessboard such that any rook attacks exactly two other rooks? (We say that two rooks attack each other if they are on the same line or on the same column and between them there are no other rooks.) Alexandru Mihalcu

Determine the smallest positive integer $k{}$ satisfying the following condition: For any configuration of chess queens on a $100 \times 100$ chequered board, the queens can be coloured one of $k$ colours so that no two queens of the same colour attack each other. [i]Russia, Sergei Avgustinovich and Dmitry Khramtsov[/i]

What maximal number of the men in checkers game can be put on the chess-board $8\times 8$ so, that every man can be taken by at least one other man ?

Place eight rooks on a standard $8 \times 8$ chessboard so that no two are in the same row or column. With the standard rules of chess, this means that no two rooks are attacking each other. Now paint $27$ of the remaining squares (not currently occupied by rooks) red. Prove that no matter how the rooks are arranged and which set of $27$ squares are painted, it is always possible to move some or all of the rooks so that: • All the rooks are still on unpainted squares. • The rooks are still not attacking each other (no two are in the same row or same column). • At least one formerly empty square now has a rook on it; that is, the rooks are not on the same $8$ squares as before.

A closed, non-self-intersecting broken line $L$ is drawn over a $(2n+1) \times (2n+1)$ chessboard in such a way that the set of L's vertices coincides with the set of the vertices of the board’s squares and every edge in $L$ is a side of some board square. All board squares lying in the interior of $L$ are coloured in red. Prove that the number of neighbouring pairs of red squares in every row of the board is even.

Originally, every square of $8 \times 8$ chessboard contains a rook. One by one, rooks which attack an odd number of others are removed. Find the maximal number of rooks that can be removed. (A rook attacks another rook if they are on the same row or column and there are no other rooks between them.) [i](6 points)[/i]

Find the number of ways one can put numbers $1$ or $2$ in each cell of an $8\times 8$ chessboard in such a way that the sum of the numbers in each column and in each row is an odd number. (Two ways are considered different if the number in some cell in the first way is different from the number in the cell situated in the corresponding position in the second way)

On an $8 \times 8$ chessboard, a rook stands on the bottom left corner square. We want to move it to the upper right corner, subject to the following rules: we have to move the rook exactly $9$ times, such that the length of each move is either $3$ or $4$. (It is allowed to mix the two lengths throughout the "journey".) How many ways are there to do this? In each move, the rook moves horizontally or vertically.

Investigate whether a chess knight can traverse a $4 \times 4$ mini-chessboard so that it reaches each of the $16$ squares only once. Note: the drawing below shows the endpoints of the eight possible moves of the knight $(C)$ on a chessboard of size $8 \times 8$. [asy] unitsize(0.4 cm); int i; fill(shift((2,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((4,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((1,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((5,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((1,5))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((5,5))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((2,6))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((4,6))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); for (i = 0; i <= 8; ++i) { draw((i,0)--(i,8)); draw((0,i)--(8,i)); } label("C", (3.5,4.5), fontsize(8)); [/asy]

The numbers from $1$ to $64$ are written on the squares of a chessboard (from $1$ to $8$ from left to right on the first row , from $9$ to $16$ from left to right on the second row , and so on). Pluses are written before some of the numbers, and minuses are written before the remaining numbers in such a way that there are $4$ pluses and $4$ minuses in each row and in each column . Prove that the sum of the written numbers is equal to zero.

Given a board with size $25\times 25$. Some $1\times 1$ squares are marked, so that for each $13\times 13$ and $4\times 4$ sub-boards, there are atleast $\frac{1}{2}$ marked parts of the sub-board. Find the least possible amount of marked squares in the entire board.

Some bishops and knights are placed on an infinite chessboard, where each square has side length $1$ unit. Suppose that the following conditions hold: [list] [*] For each bishop, there exists a knight on the same diagonal as that bishop (there may be another piece between the bishop and the knight). [*] For each knight, there exists a bishop that is exactly $\sqrt{5}$ units away from it. [*] If any piece is removed from the board, then at least one of the above conditions is no longer satisfied. [/list] If $n$ is the total number of pieces on the board, find all possible values of $n$. [i]Sheldon Kieren Tan[/i]

Given a chessboard $n \times n$, where $n\geq 4$ and $p=n+1$ is a prime number. A set of $n$ unit squares is called [i]tactical[/i] if after putting down queens on these squares, no two queens are attacking each other. Prove that there exists a partition of the chessboard into $n-2$ tactical sets, not containing squares on the main diagonals. Queens are allowed to move horizontally, vertically and diagonally.

A cell is cut from a chessboard $8\, x\, 8$, after which an open broken line was built, which vertices are the centers of the remaining cells. Each segment of the broken line has a length $\sqrt{17}$ or $\sqrt{65}$. When is the number of such broken lines bigger – when the cut cell is $(1,2)$ or $(3,6)$? (The rows and columns on the board are numerated consecutively from 1 to 8.)

Each square of a chessboard is painted either blue or red . Prove that the squares of one colour possess the property that the chess queen can perform a tour of all of them. The rules are that the queen may visit the squares of this colour not necessarily only once each , and may not be placed on squares of the other colour, although she may pass over them ; the queen moves along any horizontal , vertical or diagonal file over any distance. (A . K . Tolpugo , Kiev)