You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2\times 1$ and $3\times 1$. The playing field should be seamless with such dominoes and without overlapping, with no domino protruding beyond the playing field may. Furthermore, all dominoes must be aligned in the same way, i. e. their long sides must be parallel to each other. How many such coverings are possible? (Walther Janous)

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.

What sizes of squares with integer sides can be completely covered without overlap by identical tiles consisting of three squares with side $1$ joined together in one $L$ shape? [center][img]https://cdn.artofproblemsolving.com/attachments/3/f/9fe95b05527857f7e44dfd033e6fb01e5d25a2.png[/img][/center]

An angle of size $n \times m$, where $m, n \ge 2$, is called a figure, resulting from a rectangle of size $n \times m$ cells by removing the rectangle size $(n - 1) \times (m - 1)$ cells. Two players take turns making moves consisting in painting in a corner an arbitrary non-zero number of cells forming a rectangle or square.

Let $n > 1$ be an odd natural number. The squares of an $n \times n$ chessboard are alternately colored white and black so that the four corner squares are black. An $L$-triomino is an $L$-shaped piece that covers exactly three squares of the board. For which values ​​of $n$ is it possible to cover all black squares with $L$-triominoes, so that no two $L$-triominos overlap? For these values ​​of $n$ determine the smallest possible number of $L$-triominoes that are necessary for this.

Let's call a corner the figure that is obtained by removing one cell from a $2 \times 2$ square. Cut the $6 \times 6$ square into corners so that no two of them form a $2 \times 3$ or $3 \times 2$ rectangle together.

Prove that you can't split a square into finitely many hexagons, whose inner angles are all less than $180^o$.

A torpedo set consists of $2$ pieces of $1 \times 4$, $4$ pieces of $1 \times 3$, $6$ pieces of $1 \times 2$ and $ 8$ pieces of $1 \times 1$ ships. a) Can one put the whole set to a $10 \times 10$ table so that the ships do not even touch with corners? (The ships can be placed both horizontally and vertically.) b) Can we solve this problem if we change $4$ pieces of $1 \times 1$ ships to $3$ pieces of $1 \times 2$ ships? c) Can we solve the problem if we change the remaining $4$ pieces of $1 \times 1$ ships to one piece of $1 \times 3$ ship and one piece of $1 \times 2$ ship? (So the number of pieces are $2, 5, 10, 0$.)

An $8 \times 11$ rectangle of unit squares somehow becomes disassembled into $21$ contiguous parts . Prove that at least two of these parts, except for rotations and reflections have the same shape.

Daniel has an unlimited supply of tiles labeled “$2$” and “$n$” where $n$ is an integer. Find (with proof) all the values of $n$ that allow Daniel to fill an $8 \times 10$ grid with these tiles such that the sum of the values of the tiles in each row or column is divisible by $11$.

Given a square board with dimensions $1 995 \times 1 995$. These cells are painted with black and white paints in checkerboard order like this. that the corner cells are black. Two black and one white cells were randomly cut out of the board. Prove that the rest of the board can be divided into rectangles of size $1 \times 2$ .

The $U$-tile is made up of $1 \times 1$ squares and has the following shape: [img]https://cdn.artofproblemsolving.com/attachments/8/7/5795ee33444055794119a99e675ef977add483.png[/img] where there are two vertical rows of a squares, one horizontal row of $b$ squares, and also $a \ge 2$ and $b \ge 3$. Notice that there are different types of tile $U$ . For example, some types of $U$ tiles are as follows: [img]https://cdn.artofproblemsolving.com/attachments/0/3/ca340686403739ffbbbb578d73af76e81a630e.png[/img] Prove that for each integer $n \ge 6$, the board of $n\times n$ can be completely covered with $U$-tiles , with no gaps and no overlapping clicks. Clarifications: The $U$-tiles can be rotated. Any amount can be used in the covering of tiles of each type.

We consider tilings of a rectangular $m \times n$-board with $1\times2$-tiles. The tiles can be placed either horizontally, or vertically, but they aren't allowed to overlap and to be placed partially outside of the board. All squares on theboard must be covered by a tile. (a) Prove that for every tiling of a $4 \times 2010$-board with $1\times2$-tiles there is a straight line cutting the board into two pieces such that every tile completely lies within one of the pieces. (b) Prove that there exists a tiling of a $5 \times  2010$-board with $1\times 2$-tiles such that there is no straight line cutting the board into two pieces such that every tile completely lies within one of the pieces.

Let $n$ be an integer greater than $3$. Show that it is possible to divide a square into $n^2 + 1$ or more disjointed rectangles and with sides parallel to those of the square so that any line parallel to one of the sides intersects at most the interior of $n$ rectangles. Note: We say that two rectangles are [i]disjointed [/i] if they do not intersect or only intersect at their perimeters.

Find the smallest positive integer $n$ such that it is possible to paint each of the $64$ squares of an $8 \times 8$ board of one of $n$ colors so that any four squares that form an $L$ as in the following figure (or congruent figures obtained through rotations and/or reflections) have different colors. [img]https://cdn.artofproblemsolving.com/attachments/a/2/c8049b1be8f37657c058949e11faf041856da4.png[/img]

Find the largest natural number $n$ such that any set of $n$ tetraminoes, each of which is one of the four shapes in the picture, can be placed without overlapping in a $20 \times 20$ table (no tetramino extends beyond the borders of the table), such that each tetramino covers exactly 4 cells of the 20x20 table. An individual tetramino is allowed to turn and flip at will. [img]https://cdn.artofproblemsolving.com/attachments/b/9/0dddb25c2aa07536b711ded8363679e47972d6.png[/img]

The plane is divided into unit squares. Each box should be be colored in one of $n$ colors , so that if four squares can be covered with an $L$-tetromino, then these squares have four different colors (the $L$-Tetromino may be rotated and be mirrored). Find the smallest value of $n$ for which this is possible.

You have a large number of congruent equilateral triangular tiles on a table and you want to fit $n$ of them together to make a convex equiangular hexagon (i.e. one whose interior angles are $120^o$) . Obviously, $n$ cannot be any positive integer. The first three feasible $n$ are $6, 10$ and $13$. Show that $12$ is not feasible but $14$ is.

Given are two positive integers $k$ and $n$ with $k \le n \le 2k - 1$. Julian has a large stack of rectangular $k \times 1$ tiles. Merlin calls a positive integer $m$ and receives $m$ tiles from Julian to place on an $n \times n$ board. Julian first writes on every tile whether it should be a horizontal or a vertical tile. Tiles may be used the board should not overlap or protrude. What is the largest number $m$ that Merlin can call if he wants to make sure that he has all tiles according to the rule of Julian can put on the plate?

For a positive integer $n$, we consider an $n \times n$ board and tiles with dimensions $1 \times 1, 1 \times 2, ..., 1 \times n$. In how many ways exactly can $\frac12 n (n + 1)$ cells of the board are colored red, so that the red squares can all be covered by placing the $n$ tiles all horizontally, but also by placing all $n$ tiles vertically? Two colorings that are not identical, but by rotation or reflection from the board into each other count as different.

The figure obtained by removing one small unit square from the $2\times 2$ grid table is called an $L$ ''shape". .Put $k$ L-shapes in an $8\times 8$ grid table. Each $L$-shape can be rotated, but each $L$ shape is required to cover exactly three small unit squares in the grid table, and the common area covered by any two $L$ shapes is $0$, and except for these $k$ $L$ shapes, no other $L$ shapes can be placed. Find the minimum value of $k$.

Let $n\geq 3$ a positive integer. In each cell of a $n\times n$ chessboard one must write $1$ or $2$ in such a way the sum of all written numbers in each $2\times 3$ and $3\times 2$ sub-chessboard is even. How many different ways can the chessboard be completed?

Compute the number of ways of tiling the $2\times 10$ grid below with the three tiles shown. There is an infinite supply of each tile, and rotating or reflecting the tiles is not allowed. [img]https://cdn.artofproblemsolving.com/attachments/5/a/bb279c486fc85509aa1bcabcda66a8ea3faff8.png[/img]

Basil has an unrestricted supply of straight bricks $1 \times 1 \times 3$ and Γ-shape bricks made of three cubes $1\times 1\times 1$. Basil filled a whole box $m \times n \times k$ with these bricks, where $m, n$ and $k$ are integers greater than $1$. Prove that it was sufficient to use only Γ-shape bricks. (Mikhail Evdokimov)

In a game, there are several tiles of different colors and scores. Two white tiles are equal to three yellow tiles, a yellow tile equals $5$ red chips, $3$ red tile are equal to $ 8$ black tiles, and a black tile is worth $15$. i) Find the values ​​of all the tiles. ii) Determine in how many ways the tiles can be chosen so that their scores add up to $560$ and there are no more than five tiles of the same color.