A lightly damaged rook moves around on a $m \times n$ chessboard by taking turns moves to a horizontal or vertical field. For which $m$ and $n$, is it possible for him to have visited each field exactly once? The starting field counts as visited, squares skipped during a move, however, are not.

An $8\times 8$ chessboard is made of unit squares. We put a rectangular piece of paper with sides of length 1 and 2. We say that the paper and a single square overlap if they share an inner point. Determine the maximum number of black squares that can overlap the paper.

In an $m\times n$ rectangular chessboard,there is a stone in the lower leftmost square. Two persons A,B move the stone alternately. In each step one can move the stone upward or rightward any number of squares. The one who moves it into the upper rightmost square wins. Find all $(m,n)$ such that the first person has a winning strategy.

2500 chess kings have to be placed on a $100 \times 100$ chessboard so that [b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex); [b](ii)[/b] each row and each column contains exactly 25 kings. Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.) [i]Proposed by Sergei Berlov, Russia[/i]

Let be given an $n \times n$ chessboard, $n \in N$. We wish to tile it using particular tetraminos which can be rotated. For which $n$ is this possible if we use (a) $T$-tetraminos (b) both kinds of $L$-tetraminos?

What is the minimum number of movements that a horse must carry out on chess, on an $8\times 8$ board, to reach the upper right square starting at the lower left? Remember that the horse moves in the usual $L$-shaped manner.

Sixteen points are placed in the centers of a $4 \times 4$ chess table in the following way: • • • • • • • • • • • • • • • • (a) Prove that one may choose $6$ points such that no isoceles triangle can be drawn with the vertices at these points. (b) Prove that one cannot choose $7$ points with the above property.

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.

Let $n \geq 2$ be an integer. Consider an $n\times n$ chessboard with the usual chessboard colouring. A move consists of choosing a $1\times 1$ square and switching the colour of all squares in its row and column (including the chosen square itself). For which $n$ is it possible to get a monochrome chessboard after a finite sequence of moves?

Oleksiy placed positive integers in the cells of the $8\times 8$ chessboard. For each pair of adjacent-by-side cells, Fedir wrote down the product of the numbers in them and added all the products. Oleksiy wrote down the sum of the numbers in each pair of adjacent-by-side cells and multiplied all the sums. It turned out that the last digits of both numbers are equal to $1$. Prove that at least one of the boys made a mistake in the calculation. For example, for a square $3\times 3$ and the arrangement of numbers shown below, Fedir would write the following numbers: $2, 6, 8, 24, 15, 35, 2, 6, 8, 20, 18, 42$, and their sum ends with a digit $6$; Oleksiy would write the following numbers: $3, 5, 6, 10, 8, 12, 3, 5, 6, 9, 9, 13$, and their product ends with a digit $0$. \begin{tabular}{| c| c | c |} \hline 1 & 2 & 3 \\ \hline 2 & 4 & 6 \\ \hline 3 & 5 & 7 \\ \hline \end{tabular} [i]Proposed by Oleksiy Masalitin and Fedir Yudin[/i]

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]

A $T$-tetromino is a non-convex as well as non-rotationally symmetrical tetromino, which has a maximum number of outside corners (popularly also "Tetris Stone "called). Find all natural numbers $n$ for which, a $n \times n$ chessboard is found that can be covered only with such $T$-tetrominos.

A [i]lazy[/i] rook can only move from a square to a vertical or a horizontal neighbour. It follows a path which visits each square of an $8 \times 8$ chessboard exactly once. Prove that the number of such paths starting at a corner square is greater than the number of such paths starting at a diagonal neighbour of a corner square. [i](7 points)[/i]

$a)$ Is it possible, on modified chessboard $20 \times 30$, to draw a line which cuts exactly $50$ cells where chessboard cells are squares $1 \times 1$ $b)$ What is the maximum number of cells which line can cut on chessboard $m \times n$, $m,n \in \mathbb{N}$

The white fields of $1983\times 1984 $1983x1984 are filled with either $+1$ or $-1$. For every black field, the product of neighbouring numbers is $+1$. Prove that all the numbers are $+1$.

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](5 points)[/i]

On an $8 \times 8$ chessboard, $17$ cells are marked. Prove that one can always choose two cells among the marked ones so that a Knight will need at least three moves to go from one of the chosen cells to the other. (R Zhenodarov)

$N^2$ pieces are placed on an $N \times N$ chessboard. Is it possible to rearrange them in such a way that any two pieces which can capture each other (when considered to be knights) after the rearrangement are on adjacent squares (i.e. squares having at least one common boundary point)? Consider two cases: (a) $N = 3$. (b) $N = 8$ (S Stefanov)

On a chessboard (with $64$ squares) there are situated $32$ white and $32$ black pools. We say that two pools form a mixed pair when they are with different colors and they lie on the same row or column. Find the maximum and the minimum of the mixed pairs for all possible situations of the pools. [i]K. Dochev[/i]

Let us put pieces on some squares of $2n \times 2n$ chessboard in such a way that on every horizontal and vertical line there is an odd number of pieces. Prove that the whole number of pieces on the black squares is even.

We visit all squares exactly once on a $n\times n$ chessboard (colored in the usual way) with a king. Find the smallest number of times we had to switch colors during our walk. [i]Proposed by Dömötör Pálvölgyi, Budapest[/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.

In an $8\times 8$ chessboard, the rows are numbers from $1$ to $8$ and the columns are labelled from $a$ to $h$. In a two-player game on this chessboard, the first player has a White Rook which starts on the square $b2$, and the second player has a Black Rook which starts on the square $c4$. The two players take turns moving their rooks. In each move, a rook lands on another square in the same row or the same column as its starting square. However, that square cannot be under attack by the other rook, and cannot have been landed on before by either rook. The player without a move loses the game. Which player has a winning strategy?

2500 chess kings have to be placed on a $100 \times 100$ chessboard so that [b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex); [b](ii)[/b] each row and each column contains exactly 25 kings. Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.) [i]Proposed by Sergei Berlov, Russia[/i]

Consider a $8\times 8$ chessboard where all $64$ unit squares are at the start white. Prove that, if any $12$ of the $64$ unit square get painted black, then we can find $4$ lines and $4$ rows that have all these $12$ unit squares.