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.

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]

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]

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 $a_1,a_2,\ldots ,a_p$.

The maximum possible number of knights are placed on a $5 \times 5$ chessboard so that no two attack each other. Prove that there is only one possible placement. (A Kanel)

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 $1$?

Given a chessboard that is infinite in all directions. Is it possible to place an infinite number of queens on it so that on each horizontally, on each vertical and on each diagonal of both directions (i.e. on a set of cells located at an angle of $45^o$ or $135^o$ to the horizontal) was exactly one queen?

Given an $n \times n$ board which is divided into $n^2$ squares of size $1 \times 1$, all of which are white. Then, Aqua selects several squares from this board and colors them black. Ruby then places exactly one $1\times 2$ domino on the board, so that the domino covers exactly two squares on the board. Ruby can rotate the domino into a $2\times 1$ domino. After Aqua colors, it turns out there are exactly $2024$ ways for Ruby to place a domino on the board so that it covers exactly $1$ black square and $1$ white square. Determine the smallest possible value of $n$ so that Aqua and Ruby can do this. [i]Proposed by Muhammad Afifurrahman, Indonesia [/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]

What is the largest number of horses that you can put on a chessboard without there being two horses that can beat each other? a. Describe an arrangement with that maximum number. b. Prove that a larger number is not possible. (A chessboard consists of $8 \times 8$ spaces and a horse jumps from one field to another field according to the line "two squares vertically and one squared horizontally" or "one square vertically and two squares horizontally") [asy] unitsize (0.5 cm); int i, j; for (i = 0; i <= 7; ++i) { for (j = 0; j <= 7; ++j) { if ((i + j) % 2 == 0) { if ((i - 2)^2 + (j - 3)^2 == 5) { fill(shift((i,j))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red); } else { fill(shift((i,j))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.8)); } } }} for (i = 0; i <= 8; ++i) { draw((i,0)--(i,8)); draw((0,i)--(8,i)); } label("$a$", (0.5,-0.5), fontsize(10)); label("$b$", (1.5,-0.5), fontsize(10)); label("$c$", (2.5,-0.5), fontsize(10)); label("$d$", (3.5,-0.5), fontsize(10)); label("$e$", (4.5,-0.5), fontsize(10)); label("$f$", (5.5,-0.5), fontsize(10)); label("$g$", (6.5,-0.5), fontsize(10)); label("$h$", (7.5,-0.5), fontsize(10)); label("$1$", (-0.5,0.5), fontsize(10)); label("$2$", (-0.5,1.5), fontsize(10)); label("$3$", (-0.5,2.5), fontsize(10)); label("$4$", (-0.5,3.5), fontsize(10)); label("$5$", (-0.5,4.5), fontsize(10)); label("$6$", (-0.5,5.5), fontsize(10)); label("$7$", (-0.5,6.5), fontsize(10)); label("$8$", (-0.5,7.5), fontsize(10)); label("$P$", (2.5,3.5), fontsize(10)); [/asy]

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)

A marble is placed on each $33$ unit square of a $10*10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on 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]

What is the least possible number of the checkers being required a) for the $8\times 8$ chess-board, b) for the $n\times n$ chess-board, to provide the property: [i]Every line (of the chess-board fields) parallel to the side or diagonal is occupied by at least one checker[/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]

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)

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]

We visit all squares exactly once on a $n\times n$ chessboard (colored in the usual way) with a king. Find the smallest number of times we had to switch colors during our walk. [i]Proposed by Dömötör Pálvölgyi, Budapest[/i]

In how many ways can $n$ rooks be placed on a $2n$ x $2n$ chessboard, so that they cover all the white fields?

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.

What is the maximum number of 1 × 1 boxes that can be colored black in a n × n chessboard so that any 2 × 2 square contains a maximum of 2 black boxes?

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]

A number is written in each square of a chessboard, so that each number not on the border is the mean of the $4$ neighboring numbers. Show that if the largest number is $N$, then there is a number equal to $N$ in the border squares.

Given a chessboard $n \times n$, where $n\geq 4$ and $p=n+1$ is a prime number. A set of $n$ unit squares is called [i]tactical[/i] if after putting down queens on these squares, no two queens are attacking each other. Prove that there exists a partition of the chessboard into $n-2$ tactical sets, not containing squares on the main diagonals. Queens are allowed to move horizontally, vertically and diagonally.

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.