(a) On each of the $1 \times 1$ squares of the top row of an $8 \times 8$ chessboard there is a black pawn, and on each of the $1 \times 1$ squares of the bottom row of this chessboard there is a white pawn. On each move one can shift any pawn vertically or horizontally to any adjacent empty $1 \times 1$ square. What is the smallest number of moves that are needed to move all white pawns to the top row and all black pawns to the bottom one? (b) The same question for a $7 \times 7$ board. (A Shapovalov_

On fields of $n \times n$ chessboard $n^2$ different integers have been arranged, one in each field. In each column, field with biggest number was colored in red. Set of $n$ fields of chessboard name [i]admissible[/i], if no two of that fields aren't in the same row and aren't in the same column. From all admissible sets, set with biggest sum of numbers in it's fields has been chosen. Prove that red field is in this set.

(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]

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 rooms adjacent by side. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess rook (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 1$)?

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.

Stoyan and Nikolai have two $100\times 100$ chess boards. Both of them number each cell with the numbers $1$ to $10000$ in some way. Is it possible that for every two numbers $a$ and $b$, which share a common side in Nikolai's board, these two numbers are at a knight's move distance in Stoyan's board (that is, a knight can move from one of the cells to the other one with a move)? [i]Nikolai Beluhov[/i]

Alice plays the following game of solitaire on a $20 \times 20$ chessboard. She begins by placing $100$ pennies, $100$ nickels, $100$ dimes, and $100$ quarters on the board so that each of the $400$ squares contains exactly one coin. She then chooses $59$ of these coins and removes them from the board. After that, she removes coins, one at a time, subject to the following rules: - A penny may be removed only if there are four squares of the board adjacent to its square (up, down, left, and right) that are vacant (do not contain coins). Squares “off the board” do not count towards this four: for example, a non-corner square bordering the edge of the board has three adjacent squares, so a penny in such a square cannot be removed under this rule, even if all three adjacent squares are vacant. - A nickel may be removed only if there are at least three vacant squares adjacent to its square. (And again, “off the board” squares do not count.) - A dime may be removed only if there are at least two vacant squares adjacent to its square (“off the board” squares do not count). - A quarter may be removed only if there is at least one vacant square adjacent to its square (“off the board” squares do not count). Alice wins if she eventually succeeds in removing all the coins. Prove that it is impossiblefor her to win.

Some squares of an $n \times n$ chessboard have been marked ($n \in N^*$). Prove that if the number of marked squares is at least $n\left(\sqrt{n} + \frac12\right)$, then there exists a rectangle whose vertices are centers of marked squares.

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?

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

A natural number is written in each square of an $ m \times n$ chess board. The allowed move is to add an integer $ k$ to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.

On a chessboard $5 \times 9$ squares, the following game is played. Initially, a number of frogs are randomly placed on some of the squares, no square containing more than one frog. A turn consists of moving all of the frogs subject to the following rules: $\bullet$ Each frog may be moved one square up, down, left, or right; $\bullet$ If a frog moves up or down on one turn, it must move left or right on the next turn, and vice versa; $\bullet$ At the end of each turn, no square can contain two or more frogs. The game stops if it becomes impossible to complete another turn. Prove that if initially $33$ frogs are placed on the board, the game must eventually stop. Prove also that it is possible to place $32$ frogs on the board so that the game can continue forever.

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

Consider a $7\times7$ chessboard that starts out with all the squares white. We start painting squares black, one at a time, according to the rule that after painting the first square, each newly painted square must be adjacent along a side to only the square just previously painted. The final figure painted will be a connected “snake” of squares. (a) Show that it is possible to paint $31$ squares. (b) Show that it is possible to paint $32$ squares. (c) Show that it is possible to paint $33$ squares.