The checker is standing on the corner field of a $n\times n$ chess-board. Each of two players moves it in turn to the neighbour (i.e. that has the common side) field. It is forbidden to move to the field, the checker has already visited. That who cannot make a move losts. a) Prove that for even $n$ the first can always win, and if $n$ is odd, than the second can always win. b) Who wins if the checker stands initially on the neighbour to the corner field?

On each non-boundary unit segment of an $8\times 8$ chessboard, we write the number of dissections of the board into dominoes in which this segment lies on the border of a domino. What is the last digit of the sum of all the written numbers?

A $2N\times2N$ board is covered by non-overlapping dominos of $1\times2$ size. A lame rook (which can only move one cell at a time, horizontally or vertically) has visited each cell once on its route across the board. Call a move by the rook longitudinal if it is a move from one cell of a domino to another cell of the same domino. What is: [list=a] [*]the maximum possible number of longitudinal moves? [*]the minimum possible number of longitudinal moves? [/list]

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]

Alberto and Bianca play a game on a square board. Alberto begins. On their turn, players place a $1 \times 2$ or $2 \times 1$ domino on two empty squares on the board. The player who cannot put a domino loses. Determine who has a winning strategy (and prove it) if the board is: i) $3 \times 3$ ii) $3 \times 4$

The checkered plane is divided into dominoes. What is the maximum $k{}$ for which it is always possible to choose a $100\times 100$ checkered square containing at least $k{}$ whole dominoes? [i]Proposed by S. Berlov[/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. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.

You are given an odd number $n\ge 3$. For every pair of integers $(i, j)$ with $1\le i \le j \le n$ there is a domino, with $i$ written on one its end and with $j$ written on another (there are $\frac{n(n+1)}{2}$ domino overall). Amin took this dominos and started to put them in a row so that numbers on the adjacent sides of the dominos are equal. He has put $k$ dominos in this way, got bored and went away. After this Anton came to see this $k$ dominos, and he realized that he can't put all the remaining dominos in this row by the rules. For which smallest value of $k$ is this possible? [i]Oleksii Masalitin[/i]

Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.

Let $m$ and $n$ be natural numbers with $mn$ even. Jetze is going to cover an $m \times n$ board (consisting of $m$ rows and $n$ columns) with dominoes, so that every domino covers exactly two squares, dominos do not protrude or overlap, and all squares are covered by a domino. Merlin then moves all the dominoe color red or blue on the board. Find the smallest non-negative integer $V$ (in terms of $m$ and $n$) so that Merlin can always ensure that in each row the number squares covered by a red domino and the number of squares covered by a blue one dominoes are not more than $V$, no matter how Jetze covers the board.

The next board is completely covered with dominoes in an arbitrary manner. [img]https://cdn.artofproblemsolving.com/attachments/8/9/b4b791e55091e721c8d6040a65ae6ba788067c.png[/img] a) Prove that we can paint $21$ dominoes in such a way that there are not two dominoes painted forming a $S$-tetramino. b) What is the largest positive integer $k$ for which it is always possible to paint $k$ dominoes (without matter how the board is filled) in such a way that there are not two painted dominoes forming a $S$-tetramine? Clarification: A domino is a $1 \times 2$ or $2 \times 1$ rectangle; the $S$-tetraminos are the figures of the following types: [img]https://cdn.artofproblemsolving.com/attachments/d/f/8480306382d6b87ddb8b2a7ca96c91ee45bc6e.png[/img]

A $300\times 300$ board is arbitrarily filled with $2\times 1$ dominoes with no overflow, underflow, or overlap. (Tokens can be placed vertically or horizontally.) Decide if it is possible to paint the tiles with three different colors, so that the following conditions are met: $\bullet$ Each token is painted in one and only one of the colors. $\bullet$ The same number of tiles are painted in each color. $\bullet$ No piece is a neighbor of more than two pieces of the same color. Note: Two dominoes are [i]neighbors [/i]if they share an edge.

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.

Let $m$ and $n$ be natural numbers with $mn$ even. Jetze is going to cover an $m \times n$ board (consisting of $m$ rows and $n$ columns) with dominoes, so that every domino covers exactly two squares, dominos do not protrude or overlap, and all squares are covered by a domino. Merlin then moves all the dominoe color red or blue on the board. Find the smallest non-negative integer $V$ (in terms of $m$ and $n$) so that Merlin can always ensure that in each row the number squares covered by a red domino and the number of squares covered by a blue one dominoes are not more than $V$, no matter how Jetze covers the board.

Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.

Is it possible to use $2 \times 1$ dominoes to cover a $2k \times 2k$ checkerboard which has $2$ squares, one of each colour, removed ?

Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.

An $m \times n$ chessboard with $m,n \ge 2$ is given. Some dominoes are placed on the chessboard so that the following conditions are satisfied: (i) Each domino occupies two adjacent squares of the chessboard, (ii) It is not possible to put another domino onto the chessboard without overlapping, (iii) It is not possible to slide a domino horizontally or vertically without overlapping. Prove that the number of squares that are not covered by a domino is less than $\frac15 mn$.