Two types of tiles, depicted on the figure below, are given. [img]https://wiki-images.artofproblemsolving.com//2/23/Izrezak.PNG[/img] Find all positive integers $n$ such that an $n\times n$ board consisting of $n^2$ unit squares can be covered without gaps with these two types of tiles (rotations and reflections are allowed) so that no two tiles overlap and no part of any tile covers an area outside the $n\times n$ board. \\ [i]Proposed by Art Waeterschoot[/i]

The workers laid a floor of size $n\times n$ ($10 <n <20$) with two types of tiles: $2 \times 2$ and $5\times 1$. It turned out that they were able to completely lay the floor so that the same number of tiles of each type was used. For which $n$ could this happen? (You can’t cut tiles and also put them on top of each other.)

Does there exist a hexagon (not necessarily convex) with side lengths $1, 2, 3, 4, 5, 6$ (not necessarily in this order) that can be tiled with a) $31$ b) $32$ equilateral triangles with side length $1?$

A rectangular floor is covered by a certain number of equally large quadratic tiles. The tiles along the edge are red, and the rest are white. There are equally many red and white tiles. How many tiles can there be?

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 unit square is removed from the corner of an $n \times n$ grid, where $n \geq 2$. Prove that the remainder can be covered by copies of the figures of $3$ or $5$ unit squares depicted in the drawing below. [asy] import geometry; draw((-1.5,0)--(-3.5,0)--(-3.5,2)--(-2.5,2)--(-2.5,1)--(-1.5,1)--cycle); draw((-3.5,1)--(-2.5,1)--(-2.5,0)); draw((0.5,0)--(0.5,3)--(1.5,3)--(1.5,1)--(3.5,1)--(3.5,0)--cycle); draw((1.5,0)--(1.5,1)); draw((2.5,0)--(2.5,1)); draw((0.5,1)--(1.5,1)); draw((0.5,2)--(1.5,2)); [/asy] [b]Note:[/b] Every square must be covered once and figures must not go over the bounds of the grid.

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

$15\times 36$-checkerboard is covered with square tiles. There are two kinds of tiles, with side $7$ or $5.$ Tiles are supposed to cover whole squares of the board and be non-overlapping. What is the maximum number of squares to be covered?

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

Is it possible to cover an $n \times n$ chessboard which has its center square cut out with tiles shown in the picture (each tile covers exactly $4$ squares, tiles can be rotated and turned around) if a) $n = 5$, b) $n = 2003$? [img]https://cdn.artofproblemsolving.com/attachments/6/5/8fddeefc226ee0c02353a1fc11e48ce42d8436.png[/img]

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

We are given a supply of $a \times b$ tiles with $a$ and $b$ distinct positive integers. The tiles are to be used to tile a $28 \times 48$ rectangle. Find $a, b$ such that the tile has the smallest possible area and there is only one possible tiling. (If there are two distinct tilings, one of which is a reflection of the other, then we treat that as more than one possible tiling. Similarly for other symmetries.) Find $a, b$ such that the tile has the largest possible area and there is more than one possible tiling.

The checkered plane (infinitely large house paper) is given. For which pairs (a,, b) one can color each of the squares with one of $a \cdot b$ colors, so that each rectangle of size $ a \times b$ or $b \times a$, placed appropriately in the checkered plane, always contains a unit square with each color ?

A plane is tiled with regular hexagons of side $1$. $A$ is a fixed hexagon vertex. Find the number of paths $P$ such that: (1) one endpoint of $P$ is $A$, (2) the other endpoint of $P$ is a hexagon vertex, (3) $P$ lies along hexagon edges, (4) $P$ has length $60$, and (5) there is no shorter path along hexagon edges from $A$ to the other endpoint of $P$.

A square with the dimension $ 1 \times1 $ has been removed from a square board $ 3 ^n \times 3 ^n $ ($ n \in \mathbb {N}, $ $ n> 1 $). a) Prove that any defective board with the dimension $ 3 ^ n \times3 ^ n $ can be covered with shaped figures of shape 1 (the 3 squares' one) and of shape 2 (the 5 squares' one). Figures covering the board must not overlap each other and must not cross the edge of the board. Also the squares removed from the board must not be covered. (b) How many small figures in shape 2 must be used to cover the board? [img]https://cdn.artofproblemsolving.com/attachments/4/7/e970fadd7acc7fd6f5897f1766a84787f37acc.png[/img]

For which positive integers $n$ is it possible to cover a $(2n+1) \times (2n+1)$ chessboard which has one of its corner squares cut out with tiles shown in the figure (each tile covers exactly $4$ squares, tiles can be rotated and turned around)? [img]https://cdn.artofproblemsolving.com/attachments/6/5/8fddeefc226ee0c02353a1fc11e48ce42d8436.png[/img]

Prove that the unit square can be tiled with rectangles (not necessarily of the same size) similar to a rectangle of size $1\times(3+\sqrt[3]{3})$.

It is known that a cells square can be cut into $n$ equal figures of $k$ cells. Prove that it is possible to cut it into $k$ equal figures of $n$ cells.

A tiling of the plane with polygons consists of placing the polygons in the plane so that interiors of polygons do not overlap, each vertex of one polygon coincides with a vertex of another polygon, and no point of the plane is left uncovered. A unit polygon is a polygon with all sides of length one. It is quite easy to tile the plane with infinitely many unit squares. Likewise, it is easy to tile the plane with infinitely many unit equilateral triangles. (a) Prove that there is a tiling of the plane with infinitely many unit squares and infinitely many unit equilateral triangles in the same tiling. (b) Prove that it is impossible to find a tiling of the plane with infinitely many unit squares and finitely many (and at least one) unit equilateral triangles in the same tiling.

Given a square side $1$ and $2n$ positive reals $a_1, b_1, ... , a_n, b_n$ each $\le 1$ and satisfying $\sum a_ib_i \ge 100$. Show that the square can be covered with rectangles $R_i$ with sides length $(a_i, b_i)$ parallel to the square sides.

A rectangular $M \times N$ board is divided into $1 \times $ cells. There are also many domino pieces of size $1 \times 2$. These pieces are placed on a board so that each piece occupies two cells. The board is not entirely covered, but it is impossible to move the domino pieces (the board has a frame, so that the pieces cannot stick out of it). Prove that the number of uncovered cells is (a) less than $\frac14 MN$, (b) less than $\frac15 MN$.

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?

A $23 \times 23$ square is tiled with $1 \times 1, 2 \times 2$ and $3 \times 3$ squares. What is the smallest possible number of $1 \times 1$ squares?

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.)

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 $\frac15 mn$.