On a $999\times 999$ board a [i]limp rook[/i] can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A [i]non-intersecting route[/i] of the limp rook consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called [i]cyclic[/i], if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over. How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit? [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

A $100 \times100$ chessboard has a non-negative real number in each of its cells. A chessboard is [b]balanced[/b] if and only if the numbers sum up to one for each column of cells as well as each row of cells. Find the largest positive real number $x$ so that, for any balanced chessboard, we can find $100$ cells of it so that these cells all have number greater or equal to $x$, and no two of these cells are on the same column or row. [i]Proposed by CSJL.[/i]

An $8 \times 8$ chessboard consists of $64$ square units. In some of the unit squares of the board, diagonals are drawn so that any two diagonals have no common points. What is the maximum number of diagonals that can be drawn?

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]

$1\times 2$ dominoes are placed on an $8 \times 8$ chessboard without overlapping. They may partially stick out from the chessboard but the center of each domino must be strictly inside the chessboard (not on its border). Place on the chessboard in such a way: a) at least $40$ dominoes, (3 points) b) at least $41$ dominoes, (3 points) c) more than $41$ dominoes. (6 points) [i](Mikhail Evdokimov)[/i]

$16$ pieces of the shape $1\times 3$ are placed on a $7\times 7$ chessboard, each of which is exactly three fields. One field remains free. Find all possible positions of this field.

For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

$200 \times 200$ square is colored in chess order. In one move we can take every $2 \times 3$ rectangle and change color of all its cells. Can we make all cells of square in same color ?

On a chessboard with dimensions 1000 by 1000 and squares colored in the usual way in white and black, there is a set A consisting of 1000 squares. Any two fields of set A can be connected by a sequence of fields of set A so that subsequent fields have a common side. Prove that there are at least 250 white fields in set A.

On a chessboard a square is chosen . The sum of the squares of distances from its centre to the centre of all black squares is designated by $a$ and to the centre of all white squares by $b$. Prove that $a = b$. (A. Andj ans, Riga)

Konstantin moves a knight on a $n \times n$- chess board from the lower left corner to the lower right corner with the minimal number of moves. Then Isabelle takes the knight and moves it from the lower left corner to the upper right corner with the minimal number of moves. For which values of $n$ do they need the same number of moves?

An $8\times 8$ square board is divided into $64$ unit squares. A ’skew-diagonal’ of the board is a set of $8$ unit squares no two of which are in the same row or same column. Checkers are placed in some of the unit squares so that ’each skew-diagonal contains exactly two squares occupied by checkers’. Prove that there exist two rows or two columns which contain all the checkers.

The white fields of $3x3$ chess-board are filled with either $+1$ or $-1$. For every field, let us calculate the product of neighbouring numbers. Then let us change all the numbers by the respective products. Prove that we shall obtain only $+1$'s, having repeated this operation finite number of times.

We consider the division of a chess board $8 \times 8$ in p disjoint rectangles which satisfy the conditions: [b]a)[/b] every rectangle is formed from a number of full squares (not partial) from the 64 and the number of white squares is equal to the number of black squares. [b]b)[/b] the numbers $\ a_{1}, \ldots, a_{p}$ of white squares from $p$ rectangles satisfy $a_1, , \ldots, a_p.$ Find the greatest value of $p$ for which there exists such a division and then for that value of $p,$ all the sequences $a_{1}, \ldots, a_{p}$ for which we can have such a division. [color=#008000]Moderator says: see [url]https://artofproblemsolving.com/community/c6h58591[/url][/color]

A rook starts moving on an infinite chessboard, alternating horizontal and vertical moves. The length of the first move is one square, of the second – two squares, of the third – three squares and so on. a) Is it possible for the rook to arrive at its starting point after exactly $2013$ moves? b) Find all $n$ for which it possible for the rook to come back to its starting point after exactly $n$ moves.

$a)$ 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 $m \times n$, $m,n \in \mathbb{N}$

Ayman wants to color the cells of a $50 \times 50$ chessboard into black and white so that each $2 \times 3$ or $3 \times 2$ rectangle contains an even number of white cells. Determine the number of ways Ayman can color the chessboard.

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.

A knight stands on an infinite chess board. Find all places it can reach in exactly $2n$ moves.

A corner with arm $n$ is a figure made of $2n-1$ unit squares, such that 2 rectangles $1$ x $(n-1)$ are connected to two adjacent sides of a square $1$ x $1$, so that their unit sides coincide. The squares or a chessboard $100$ x $100$ are colored in 15 colors. We say that a corner with arm 8 is [i]“multicolored”[/i], if it contains each of the colors on the board. What’s the greatest number of corners with arm 8 which could be [i]“mutlticolored”[/i]?

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?

Find the smallest $n$ such that in an $8\times 8$ chessboard any $n$ cells contain two cells which are at least $3$ knight moves apart from each other.

A knight is modified so that it moves $p$ fields horizontally or vertically and $q$ fields in the perpendicular direction. It is placed on an infinite chessboard. If the knight returns to the initial field after $n$ moves, show that $n$ must be 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](5 points)[/i]

An $n\times n$ chessboard is given, where $n$ is an even positive integer. On every line, the unit squares are to be permuted, subject to the condition that the resulting table has to be symmetric with respect to its main diagonal (the diagonal from the top-left corner to the bottom-right corner). We say that a board is [i]alternative[/i] if it has at least one pair of complementary lines (two lines are complementary if the unit squares on them which lie on the same column have distinct colours). Otherwise, we call the board [i]nonalternative[/i]. For what values of $n$ do we always get from the $n\times n$ chessboard an alternative board?\\ \\ [i](Alexandru Petrescu and Andra Elena Mircea)[/i]