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)

Investigate whether a chess knight can traverse a $4 \times 4$ mini-chessboard so that it reaches each of the $16$ squares only once. Note: the drawing below shows the endpoints of the eight possible moves of the knight $(C)$ on a chessboard of size $8 \times 8$. [asy] unitsize(0.4 cm); int i; fill(shift((2,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((4,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((1,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((5,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((1,5))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((5,5))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((2,6))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((4,6))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); for (i = 0; i <= 8; ++i) { draw((i,0)--(i,8)); draw((0,i)--(8,i)); } label("C", (3.5,4.5), fontsize(8)); [/asy]

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]

How many ways can $ 8$ mutually non-attacking rooks be placed on the $ 9\times9$ chessboard (shown here) so that all $ 8$ rooks are on squares of the same color? (Two rooks are said to be attacking each other if they are placed in the same row or column of the board.) [asy]unitsize(3mm); defaultpen(white); fill(scale(9)*unitsquare,black); fill(shift(1,0)*unitsquare); fill(shift(3,0)*unitsquare); fill(shift(5,0)*unitsquare); fill(shift(7,0)*unitsquare); fill(shift(0,1)*unitsquare); fill(shift(2,1)*unitsquare); fill(shift(4,1)*unitsquare); fill(shift(6,1)*unitsquare); fill(shift(8,1)*unitsquare); fill(shift(1,2)*unitsquare); fill(shift(3,2)*unitsquare); fill(shift(5,2)*unitsquare); fill(shift(7,2)*unitsquare); fill(shift(0,3)*unitsquare); fill(shift(2,3)*unitsquare); fill(shift(4,3)*unitsquare); fill(shift(6,3)*unitsquare); fill(shift(8,3)*unitsquare); fill(shift(1,4)*unitsquare); fill(shift(3,4)*unitsquare); fill(shift(5,4)*unitsquare); fill(shift(7,4)*unitsquare); fill(shift(0,5)*unitsquare); fill(shift(2,5)*unitsquare); fill(shift(4,5)*unitsquare); fill(shift(6,5)*unitsquare); fill(shift(8,5)*unitsquare); fill(shift(1,6)*unitsquare); fill(shift(3,6)*unitsquare); fill(shift(5,6)*unitsquare); fill(shift(7,6)*unitsquare); fill(shift(0,7)*unitsquare); fill(shift(2,7)*unitsquare); fill(shift(4,7)*unitsquare); fill(shift(6,7)*unitsquare); fill(shift(8,7)*unitsquare); fill(shift(1,8)*unitsquare); fill(shift(3,8)*unitsquare); fill(shift(5,8)*unitsquare); fill(shift(7,8)*unitsquare); draw(scale(9)*unitsquare,black);[/asy]

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?

The numbers from $1$ to $64$ are written on the squares of a chessboard (from $1$ to $8$ from left to right on the first row , from $9$ to $16$ from left to right on the second row , and so on). Pluses are written before some of the numbers, and minuses are written before the remaining numbers in such a way that there are $4$ pluses and $4$ minuses in each row and in each column . Prove that the sum of the written numbers is equal to zero.

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.

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?

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 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.

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.

On a square of a chessboard there is a pawn . Two players take turns to move it to another square, subject to the rule that , at each move the distance moved is strictly greater than that of the previous move. A player loses when unable to make a move on his turn. Who wins if the players always choose the best strategy? (The pawn is always placed in the centre of its square. ) ( F . L . Nazarov)

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?

A chessboard is covered with $32$ dominos. Each domino covers two adjacent squares. Show that the number of horizontal dominos with a white square on the left equals the number with a white square on the right.

A game is played on an $m \times n$ chessboard. At the beginning, there is a coin on one of the squares. Two players take turns to move the coin to an adjacent square (horizontally or vertically). The coin may never be moved to a square that has been occupied before. If a player cannot move any more, he loses. Prove: [list] [*] If the size (number of squares) of the board is even, then the player to move first has a winning strategy, regardless of the initial position. [*] If the size of the board is odd, then the player to move first has a winning strategy if and only if the coin is initially placed on a square whose colour is not the same as the colour of the corners. [/list]

On a chessboard $8\times 8$, $n>6$ Knights are placed so that for any 6 Knights there are two Knights that attack each other. Find the greatest possible value of $n$.

On a chess board, the boundaries of the squares are assumed to be black. Draw a circle of the greatest possible radius lying entirely on the black squares.

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]

(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]

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.

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]

[u]Chessboards [/u] Joe B. is playing with some chess pieces on a $6\times 6$ chessboard. Help him find out some things. [b]p1.[/b] Joe B. first places the black king in one corner of the board. In how many of the $35$ remaining squares can he place a white bishop so that it does not check the black king? [b]p2.[/b] Joe B. then places a white king in the opposite corner of the board. How many total ways can he place one black bishop and one white bishop so that neither checks the king of the opposite color? [b]p3.[/b] Joe B. now clears the board. How many ways can he place $3$ white rooks and $3$ black rooks on the board so that no two rooks of opposite color can attack each other? [b]p4.[/b] Joe B. is frustrated with chess. He breaks the board, leaving a $4\times 4$ board, and throws $3$ black knights and $3$ white kings at the board. Miraculously, they all land in distinct squares! What is the expected number of checks in the resulting position? (Note that a knight can administer multiple checks and a king can be checked by multiple knights.) [b]p5.[/b] Suppose that at some point Joe B. has placed $2$ black knights on the original board, but gets bored of chess. He now decides to cover the $34$ remaining squares with $17$ dominos so that no two overlap and the dominos cover the entire rest of the board. For how many initial arrangements of the two pieces is this possible? Note: Chess is a game played with pieces of two colors, black and white, that players can move between squares on a rectangular grid. Some of the pieces move in the following ways: $\bullet$ Bishop: This piece can move any number of squares diagonally if there are no other pieces along its path. $\bullet$ Rook: This piece can move any number of squares either vertically or horizontally if there are no other pieces along its path. $\bullet$ Knight: This piece can move either two squares along a row and one square along a column or two squares along a column and one square along a row. $\bullet$ King: This piece can move to any open adjacent square (including diagonally). If a piece can move to a square occupied by a king of the opposite color, we say that it is checking the king. If a piece moves to a square occupied by another piece, this is called attacking.

A straight line is drawn across an $8\times 8$ chessboard. It is said to [i]pierce [/i]a square if it passes through an interior point of the square. At most how many of the $64$ squares can this line [i]pierce[/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)

Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?