Determine for which natural numbers $n$ you can cover a $2n \times 2n$ chessboard with non-overlapping $L$ pieces. An $L$ piece covers four spaces and has appearance like the letter $L$. The piece may be rotated and mirrored at will.

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.

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)

The workers laid a floor of size $n \times n$ with tiles of two types: $2 \times 2$ and $3 \times 1$. It turned out that they were able to completely lay the floor in such a way that the same number of tiles of each type was used. Under what conditions could this happen? (You can’t cut tiles and also put them on top of each other.)

For what positive integers $n$ is it possible to tile an equilateral triangle of side $n$ with trapezoids each of which has sides $1, 1, 1, 2$? (NB Vassiliev)

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.

Pieces are placed in some squares of an $8 \times 8$ board sothat: a) There is at least one token in any rectangle with sides $2 \times 1$ or $1\times 2$. b) There are at least two neighboring pieces in any rectangle with sides $7\times 1$ or $1\times 7$. Find the smallest number of tokens that can be taken to fulfill with both conditions.

With $76$ tiles, of which some are white, other blue and the remaining red, they form a rectangle of $4 \times 19$. Show that there is a rectangle, inside the largest, that has its vertices of the same color.

You have a $7\times 7$ board divided into $49$ boxes. Mateo places a coin in a box. a) Prove that Mateo can place the coin so that it is impossible for Emi to completely cover the $48$ remaining squares, without gaps or overlaps, using $15$ $3\times1$ rectangles and a cubit of three squares, like those in the figure. [img]https://cdn.artofproblemsolving.com/attachments/6/9/a467439094376cd95c6dfe3e2ad3e16fe9f124.png[/img] b) Prove that no matter which square Mateo places the coin in, Emi will always be able to cover the 48 remaining squares using $14$ $3\times1$ rectangles and two cubits of three squares.

For what smallest $n$ can a $n \times n$ square be cut into squares $40 \times 40$ and $49 \times 49$ so that squares of both types are present?

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 $k$.

A rectangle is tiled with rectangles of size $6\times 1$. Prove that one of its side lengths is divisible by $6$.

A $50 \times 50$ square board is tiled by the tetrominoes of the following three types: [img]https://cdn.artofproblemsolving.com/attachments/2/9/62c0bce6356ea3edd8a2ebfe0269559b7527f1.png[/img] Find the greatest and the smallest possible number of $L$ -shaped tetrominoes In the tiling. (Folklore)

Consider a $ 2018\times 2018$. board. An "LC-tile" is a tile consisting of $9$ unit squares, having the shape as in the gure below. What is the maximum number of "LC-tiles" that can be placed on the board without superposing them? (Each of the $9$ unit squares of the tile must cover one of the unit squares of the board; a tile may be rotated, turned upside down, etc.) [img]https://cdn.artofproblemsolving.com/attachments/7/4/a2f992bc0341def1a6e5e26ba8a9eb3384698a.png [/img] Alexandru Girban

(a) Prove that a square with sides $1000$ divided into $31$ squares tiles, at least one of which has a side length less than $1$. (b) Show that a corresponding decomposition into $30$ squares is also possible. [i](Walther Janous)[/i]