In how many ways can you choose $ k $ squares on a chessboard $ n \times n $ ( $ k \leq n $) so that no two of the chosen squares lie in the same row or column?

How many squares different in size or location can be drawn on an $8 \times 8$ chess board? Each square drawn must consist of whole chess board’s squares.

A "labyrinth" is an $8 \times 8$ chessboard with walls between some neighboring squares. If a rook can traverse the entire board without jumping over the walls, the labyrinth is "good" ; otherwise it is "bad" . Are there more good labyrinths or bad labyrinths? (A Shapovalov)

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.

The $64$ cells of an $8 \times 8$ chessboard have $64$ different colours. A Knight stays in one cell. In each move, the Knight jumps from one cell to another cell (the $2$ cells on the diagonal of an $2 \times 3$ board) also the colours of the $2$ cells interchange. In the end, the Knight goes to a cell having common side with the cell it stays at first. Can it happen that: there are exactly $3$ cells having the colours different from the original colours?

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]

Let $ n$ and $ k$ be positive integers such that $ \frac{1}{2} n < k \leq \frac{2}{3} n.$ Find the least number $ m$ for which it is possible to place $ m$ pawns on $ m$ squares of an $ n \times n$ chessboard so that no column or row contains a block of $ k$ adjacent unoccupied squares.

Consider decompositions of an $8\times 8$ chessboard into $p$ non-overlapping rectangles subject to the following conditions: (i) Each rectangle has as many white squares as black squares. (ii) If $a_i$ is the number of white squares in the $i$-th rectangle, then $a_1<a_2<\ldots <a_p$. Find the maximum value of $p$ for which such a decomposition is possible. For this value of $p$, determine all possible sequences $a_1,a_2,\ldots ,a_p$.

positive real $C$ is $n-vengeful$ if it is possible to color the cells of an $n \times n$ chessboard such that: i) There is an equal number of cells of each color. ii) In each row or column, at least $Cn$ cells have the same color. a) Show that $\frac{3}{4}$ is $n-vengeful$ for infinitely many values of $n$. b) Show that it does not exist $n$ such that $\frac{4}{5}$ is $n-vengeful$.

(a) Decide whether the fields of the $8 \times 8$ chessboard can be numbered by the numbers $1, 2, \dots , 64$ in such a way that the sum of the four numbers in each of its parts of one of the forms [list][img]http://www.artofproblemsolving.com/Forum/download/file.php?id=28446[/img][/list] is divisible by four. (b) Solve the analogous problem for [list][img]http://www.artofproblemsolving.com/Forum/download/file.php?id=28447[/img][/list]

Markers in the colors violet, cyan, octarine and gamma were placed on all fields of a $41\times 5$ chessboard. Show that there are four squares of the same color that form the vertices of a rectangle whose edges are parallel to those of the board.

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]

In how many ways can $31$ squares be marked on an $8 \times 8$ chessboard so that no two of the marked squares have a common side? (R Zhenodarov)

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]

Given a chess-board $99\times 99$ with a set $F$ of fields marked on it (the set is different in three tasks). There is a beetle sitting on every field of the set $F$. Suddenly all the beetles have raised into the air and flied to another fields of the same set. The beetles from the neighbouring fields have landed either on the same field or on the neighbouring ones (may be far from their starting point). (We consider the fields to be neighbouring if they have at least one common vertex.) Consider a statement: [i]"There is a beetle, that either stayed on the same field or moved to the neighbouring one".[/i] Is it always valid if the figure $F$ is: a) A central cross, i.e. the union of the $50$-th row and the $50$-th column? b) A window frame, i.e. the union of the $1$-st, $50$-th and $99$-th rows and the $1$-st, $50$-th and $99$-th columns? c) All the chess-board?

A chessboard $1000 \times 1000$ is covered by dominoes $1 \times 10$ that can be rotated. We don't know which is the cover, but we are looking for it. For this reason, we choose a few $N$ cells of the chessboard, for which we know the position of the dominoes that cover them. Which is the minimum $N$ such that after the choice of $N$ and knowing the dominoed that cover them, we can be sure and for the rest of the cover? (Bulgaria)

The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to adjacent rooms. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess queen (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 2$)?

Two types of pieces, bishops and rooks, are to be placed on a $10\times 10$ chessboard (without necessarily filling it) such that each piece occupies exactly one square of the board. A bishop $B$ is said to [i]attack[/i] a piece $P$ if $B$ and $P$ are on the same diagonal and there are no pieces between $B$ and $P$ on that diagonal; a rook $R$ is said to attack a piece $P$ if $R$ and $P$ are on the same row or column and there are no pieces between $R$ and $P$ on that row or column. A piece $P$ is [i]chocolate[/i] if no other piece $Q$ attacks $P$. What is the maximum number of chocolate pieces there may be, after placing some pieces on the chessboard? [i]Proposed by José Alejandro Reyes González[/i]

A chess king tours an entire $8\times 8$ chess board, visiting each square exactly once and returning at last to his starting position. Prove that he made an even number of diagonal moves. (V Proizvolov)

A game is played on an ${n \times n}$ chessboard. At the beginning there are ${99}$ stones on each square. Two players ${A}$ and ${B}$ take turns, where in each turn the player chooses either a row or a column and removes one stone from each square in the chosen row or column. They are only allowed to choose a row or a column, if it has least one stone on each square. The first player who cannot move, looses the game. Player ${A}$ takes the first turn. Determine all n for which player ${A}$ has a winning strategy.

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]

In a $4\times 4$ chessboard, in how many ways can you place $3$ rooks and one bishop such that none of these pieces threaten another piece?

There are a board with $2n \cdot 2n \ (= 4n^2)$ squares and $4n^2-1$ cards numbered with different natural numbers. These cards are put one by one on each of the squares. One square is empty. We can move a card to an empty square from one of the adjacent squares (two squares are adjacent if they have a common edge). Is it possible to exchange two cards on two adjacent squares of a column (or a row) in a finite number of movements?

How many a) bishops b) horses can be positioned on a chessboard at most, so that no one threatens another?

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.