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.

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 $8\times 8$ board is divided in dominoes (rectangles with dimensions $1 \times 2$ or $2 \times 1$). [list=a] [*] Prove that the total length of the border between horizontal and vertical dominoes is at most $52$. [*] Determine the maximum possible total length of the border between horizontal and vertical dominoes. [/list] [i]Proposed by B. Frenkin, A. Zaslavsky, E. Arzhantseva[/i]

We say that a polygon is rectangular when all of its angles are $90^\circ$ or $270^\circ$. Is it true that each rectangular polygon, which sides are with length equal to odd numbers only, [u]can't[/u] be covered with 2x1 domino tiles?

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 $\frac15 mn$.

A $1 \times 5n$ rectangle is partitioned into tiles, each of the tile being either a separate $1 \times 1$ square or a broken domino consisting of two such squares separated by four squares (not belonging to the domino). Prove that the number of such partitions is a perfect fifth power. [i](K. Kokhas)[/i]

Rectangles $1\times20$, $1\times 19$, ..., $1\times 1$ were cut out of $20\times20$ table. Prove that at least 85 dominoes(1×2 rectangle) can be removed from the remainder. Proposed by S. Berlov

Let $ n$ be a positive integer. A $4\times n$ rectangular grid is divided in$ 2\times 1$ or $1\times 2$ rectangles (as if it were completely covered with tiles of domino, no overlaps or gaps). Then all the grid points which are vertices of one of the $2\times 1$ or $1\times 2$ rectangles, are painted red. What is the least amount of red points you can get?

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]

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 ?

In a $50\times 50$ checkered square, each cell is colored in one of the 100 given colors so that all colors are used and there does not exist a monochromatic domino. Galia wants to repaint all the cells of one of the colors in a different color (from the given 100 colors) so that a monochromatic domino still won't exist. Is it true that Galia will surely be able to do this [i]Proposed by G. Sharafutdinova[/i]

All the cells of a $10\times10$ board are colored white initially. Two players are playing a game with alternating moves. A move consists of coloring any un-colored cell black. A player is considered to loose, if after his move no white domino is left. Which of the players has a winning strategy? [I]Proposed by A. Khrabrov[/i]

There are three types of piece shown as below. Today Alice wants to cover a $100 \times 101$ board with these pieces without gaps and overlaps. Determine the minimum number of $1\times 1$ pieces should be used to cover the whole board and not exceed the board. (There are an infinite number of these three types of pieces.) [asy] size(9cm,0); defaultpen(fontsize(12pt)); draw((9,10) -- (59,10) -- (59,60) -- (9,60) -- cycle); draw((59,10) -- (109,10) -- (109,60) -- (59,60) -- cycle); draw((9,60) -- (59,60) -- (59,110) -- (9,110) -- cycle); draw((9,110) -- (59,110) -- (59,160) -- (9,160) -- cycle); draw((109,10) -- (159,10) -- (159,60) -- (109,60) -- cycle); draw((180,11) -- (230,11) -- (230,61) -- (180,61) -- cycle); draw((180,61) -- (230,61) -- (230,111) -- (180,111) -- cycle); draw((230,11) -- (280,11) -- (280,61) -- (230,61) -- cycle); draw((230,61) -- (280,61) -- (280,111) -- (230,111) -- cycle); draw((280,11) -- (330,11) -- (330,61) -- (280,61) -- cycle); draw((280,61) -- (330,61) -- (330,111) -- (280,111) -- cycle); draw((330,11) -- (380,11) -- (380,61) -- (330,61) -- cycle); draw((330,61) -- (380,61) -- (380,111) -- (330,111) -- cycle); draw((401,11) -- (451,11) -- (451,61) -- (401,61) -- cycle); [/asy] [i]Proposed by amano_hina[/i]

A [i]domino[/i] is a $2 \times 1$ or $1 \times 2$ tile. Determine in how many ways exactly $n^2$ dominoes can be placed without overlapping on a $2n \times 2n$ chessboard so that every $2 \times 2$ square contains at least two uncovered unit squares which lie in the same row or column.

The infinite checkered plane is divided into dominoes. If we move any horizontal domino of the partition by 49 cells to the right or left, we will also get a domino of the partition. If we move any vertical domino of the partition up or down by 49 cells, we will also get a domino of the partition. Can this happen?

How many dominoes can be placed on a at least $3 \times 12$ board, such so that it is impossible to place a $1\times 3$, $3 \times 1$, or $ 2 \times 2$ tile on what remains of the board? Clarification: Each domino covers exactly two squares on the board. The chips cannot overlap.

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.

Rectangles $1\times20$, $1\times 19$, ..., $1\times 1$ were cut out of $20\times20$ table. Prove that from the remaining part of the table $36$ $1\times2$ dominos can be cut [I]Proposed by S. Berlov[/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.

Sofia and Viktor are playing the following game on a $2022 \times 2022$ board: - Firstly, Sofia covers the table completely by dominoes, no two are overlapping and all are inside the table; - Then Viktor without seeing the table, chooses a positive integer $n$; - After that Viktor looks at the table covered with dominoes, chooses and fixes $n$ of them; - Finally, Sofia removes the remaining dominoes that aren't fixed and tries to recover the table with dominoes differently from before. If she achieves that, she wins, otherwise Viktor wins. What is the minimum number $n$ for which Viktor can always win, no matter the starting covering of dominoes. [i]Proposed by Viktor Simjanoski[/i]

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?

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?

Let $x$ and $y$ be odd positive integers. A table $x$ x $y$ is given in which the squares with coordinates $(2,1)$, $(x - 2, y)$, and $(x, y)$ are cut. The remaining part of the table is covered in dominoes and squares [b]2 x 2[/b]. Prove that the dominoes in a valid covering of the table are at least $\frac{3}{2}(x+y)-6$

On a rectangular board with $m \times n$ squares ($m, n \ge 3$) there are dominoes ($2 \times 1$ or $1\times 2$ tiles), which do not overlap and do not extend beyond the board. Every domino covers exactly two squares of the board. Assume that the dominos cover the has the property that no more dominos can be added to the board and that the four corner spaces of the board are not all empty. Prove that at least $2/3$ of the squares of the board are covered with dominos.

You have a $3 \times 2021$ chessboard from which one corner square has been removed. You also have a set of $3031$ identical dominoes, each of which can cover two adjacent chessboard squares. Let $m$ be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps. What is the remainder when $m$ is divided by $19$?