The numbers from $1$ to $64$ must be written on the small squares of a chessboard, with a different number in each small square. Consider the $112$ numbers you can make by adding the numbers in two small squares which have a common edge. Is it possible to write the numbers in the squares so that these $112$ sums are all different?

The cells of an $n \times n$ chessboard are coloured in several colours so that no $2\times 2$ square contains four cells of the same colour. A [i]proper path [/i] of length $m$ is a sequence $a_1,a_2,..., a_m$ of distinct cells in which the cells $a_i$ and $a_{i+1}$ have a common side and are coloured in different colours for all $1 \le i < m$. Show that there exists a proper path of length $n$.

Eight pawns are placed on a chessboard, so that there is one in each row and column. Show that an even number of the pawns are on black squares.

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.

Jiiri and Mari both wish to tile an $n \times n$ chessboard with cards shown in the picture (each card covers exactly one square). Jiiri wants that for each two cards that have a common edge, the neighbouring parts are of different color, and Mari wants that the neighbouring parts are always of the same color. How many possibilities does Jiiri have to tile the chessboard and how many possibilities does Mari have? [img]https://cdn.artofproblemsolving.com/attachments/7/3/9c076eb17ba7ae7c000a2893c83288a94df384.png[/img]

Eight rooks are placed on a chessboard so that no two rooks attack each other. Prove that one can always move all rooks, each by a move of a knight so that in the final position no two rooks attack each other as well. (In intermediate positions several rooks can share the same square).

Consider a board of $a \times b$, with $a$ and $b$ integers greater than or equal to $2$. Initially their squares are colored black and white like a chess board. The permitted operation consists of choosing two squares with a common side and recoloring them as follows: a white square becomes black; a black box turns green; a green box turns white. Determine for which values of $a$ and $b$ it is possible, by a succession of allowed operations, to make all the squares that were initially white end black and all the squares that were initially black end white. Clarification: Initially there are no green squares, but they appear after the first operation.

Consider arrangements of the numbers $1$ through $64$ on the squares of an $8\times 8$ chess board, where each square contains exactly one number and each number appears exactly once. A number in such an arrangement is called super-plus-good, if it is the largest number in its row and at the same time the smallest number in its column. Prove or disprove each of the following statements: (a) Each such arrangement contains at least one super-plus-good number. (b) Each such arrangement contains at most one super-plus-good number. Proposed by Gerhard J. Woeginger

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.

Consider a chessboard $6 \times 6$, made up of $36$ single squares. We want to place $6$ chess rooks on this board, one rook on each square, so that there are no two rooks on the same row, nor two rooks on the same column. Note that, once the rooks have been placed in this way, we have that, for every square where a rook has not been placed, there is a rook in the same row as it and a rook in the same column as it. We will say that such rooks are in line with this square. For each of those $30$ houses without rooks, color it green if the two rooks aligned with that same house are the same distance from it, and color it yellow otherwise. For example, when we place the $6$ rooks ($T$) as below, we have: (a) Is it possible to place the rooks so that there are $30$ green squares? (b) Is it possible to place the rooks so that there are $30$ yellow squares? (c) Is it possible to place the rooks so that there are $15$ green and $15$ yellow squares?

A chess-board $6\times 6$ is tiled with the $2\times 1$ dominos. Prove that you can cut the board onto two parts by a straight line that does not cut dominos.

The squares of a $10 \times 10$ chessboard are labelled with $1,2,...,100 $ in the usual way: the $i$-th row contains the numbers $10i -9,10i - 8,...,10i$ in increasing order. The signs of fifty numbers are changed so that each row and each column contains exactly five negative numbers. Show that after this change the sum of all numbers on the chessboard is zero.

Is it possible to cover an $n \times n$ chessboard which has its center square cut out with tiles shown in the picture (each tile covers exactly $4$ squares, tiles can be rotated and turned around) if a) $n = 5$, b) $n = 2003$? [img]https://cdn.artofproblemsolving.com/attachments/6/5/8fddeefc226ee0c02353a1fc11e48ce42d8436.png[/img]

The nonnegative integer $n$ and $ (2n + 1) \times (2n + 1)$ chessboard with squares colored alternatively black and white are given. For every natural number $m$ with $1 < m < 2n+1$, an $m \times m$ square of the given chessboard that has more than half of its area colored in black, is called a $B$-square. If the given chessboard is a $B$-square, find in terms of $n$ the total number of $B$-squares of this chessboard.

Sixteen squares of an $8\times 8$ chessboard are chosen so that there are exactly lwo in each row and two in each column. Prove that eight white pawns and eight black pawns can be placed on these sixteen squares so that there is one white pawn and one black pawn in each row and in cach colunm.

Show that, in a chessboard, it is possible to traverse to any given square from another given square using a knight. (A knight can move in a chessboard by going two steps in one direction and one step in a perpendicular direction as shown in the given figure)

The king have revised the chess-board $8\times 8$ having visited all the fields once only and returned to the starting point. When his trajectory was drawn (the centres of the squares were connected with the straight lines), a closed broken line without self-intersections appeared. a) Give an example that the king could make $28$ steps parallel the sides of the board only. b) Prove that he could not make less than $28$ such a steps. c) What is the maximal and minimal length of the broken line if the side of a field is $1$?

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.

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 ($n > 1$)?

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.

Distinct pebbles are placed on a $1001 \times 1001$ board consisting of $1001^2$ unit tiles, such that every unit tile consists of at most one pebble. The [i]pebble set[/i] of a unit tile is the set of all pebbles situated in the same row or column with said unit tile. Determine the minimum amount of pebbles that must be placed on the board so that no two distinct tiles have the same [i]pebble set[/i]. [hide=Where's the Algebra Problem?]It's already posted [url=https://artofproblemsolving.com/community/c6h2742895_simple_inequality]here[/url].[/hide]

Let $m$ and $n$ be positive integers. Some squares of an $m \times n$ board are coloured red. A sequence $a_1, a_2, \ldots , a_{2r}$ of $2r \ge 4$ pairwise distinct red squares is called a [i]bishop circuit[/i] if for every $k \in \{1, \ldots , 2r \}$, the squares $a_k$ and $a_{k+1}$ lie on a diagonal, but the squares $a_k$ and $a_{k+2}$ do not lie on a diagonal (here $a_{2r+1}=a_1$ and $a_{2r+2}=a_2$). In terms of $m$ and $n$, determine the maximum possible number of red squares on an $m \times n$ board without a bishop circuit. ([i]Remark.[/i] Two squares lie on a diagonal if the line passing through their centres intersects the sides of the board at an angle of $45^\circ$.)

In how many ways can the king in the chessboard reach the eighth rank in $7$ moves from its original square on the first row?

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.

Let $n$ be a positive integer. In how many ways can we mark cells on an $n\times n$ board such that no two rows and no two columns have the same number of marked cells? [i]Selim Bahadir, Turkey[/i]