1. Can a $7 \times 7~$ square be tiled with the two types of tiles shown in the figure? (Tiles can be rotated and reflected but cannot overlap or be broken) 2. Find the least number $N$ of tiles of type $A$ that must be used in the tiling of a $1011 \times 1011$ square. Give an example of a tiling that contains exactly $N$ tiles of type $A$. [asy] size(4cm, 0); pair a = (-10,0), b = (0, 0), c = (10, 0), d = (20, 0), e = (20, 10), f = (10, 10), g = (0, 10), h = (0, 20), ii = (-10, 20), j = (-10, 10); draw(a--b--c--f--g--h--ii--cycle); draw(g--b); draw(j--g); draw(f--c); draw((30, 0)--(30, 20)--(50,20)--(50,0)--cycle); draw((40,20)--(40,0)); draw((30,10)--(50,10)); label((0,0), "$(A)$", S); label((40,0), "$(B)$", S); [/asy] [i]Proposed by Muralidharan Somasundaran[/i]

Describe a method to convert any triangle into a rectangle with side 1 and area equal to the original triangle by dividing that triangle into finitely many subtriangles.

$15\times 36$-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 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]

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) A square is cut into right triangles with legs of lengths $3$ and $4$. Prove that the total number of the triangles is even. (b) A rectangle is cut into right triangles with legs of lengths $1$ and $2$. Prove that the total number of the triangles is even. (A Shapovalov)

Show that we cannot tile a $10 x 10$ board with $25$ pieces of type $A$, or with $25$ pieces of type $B$, or with $25$ pieces of type $C$.

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]

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.

A crymble is a solid consisting of four white and one black unit cubes as shown in the picture. Find the side length of the smallest cube that can be exactly filled up with crymbles. [img]https://cdn.artofproblemsolving.com/attachments/b/0/b1e50f7abbfb7d356913d746d653fd3875f5ae.png[/img]

Let $n\geq 2$ be a natural number. A unit square is removed from a square $n$ x $n$ and the remaining figure is cut into squares 2 x 2 and 3 x 3. Determine all possible values of $n$.

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

In the illustration, a regular hexagon and a regular octagon have been tiled with rhombuses. In each case, the sides of the rhombuses are the same length as the sides of the regular polygon. (a) Tile a regular decagon ($10$-gon) into rhombuses in this manner. (b) Tile a regular dodecagon ($12$-gon) into rhombuses in this manner. (c) How many rhombuses are in a tiling by rhombuses of a $2002$-gon? Justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/8/a/8413e4e2712609eba07786e34ba2ce4aa72888.png[/img]

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]

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 $P$.

On an infinite sheet of tiles, an infinite number of $1 \times 2$ tile rectangles are placed, their edges follow the lines of the tiles, and they do not touch each other, not even the corners. Is it true that the remaining checkered sheet can be completely covered with $1 \times 2$ checkered rectangles? [hide=original wording]Uz bezgalīgas rūtiņu lapas ir novietoti bezgaglīgi daudzi 1 x 2 rūtiņu taisnstūri, to malas iet pa rūtiņu līnijām, un tie nesaskaras cits ar citu pat ne ar stūriem. Vai tiesa, ka atlikušo rūtiņu lapu var pilnībā noklāt ar 1 x 2 rūtiņu tainstūriem? [/hide]

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 call a tiling of an $m\times$ n rectangle with arabos (see figure below) [i]regular[/i] if there is no sub-rectangle which is tiled with arabos. Prove that if for some $m$ and $n$ there exists a [i]regular[/i] tiling of the $m\times n$ rectangle then there exists a [i]regular[/i] tiling also for the $2m \times 2n$ rectangle. [img]https://cdn.artofproblemsolving.com/attachments/1/1/2ab41cc5107a21760392253ed52d9e4ecb22d1.png[/img]

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.

Let $n \ge 3$ be a positive integer. We call a $3 \times 3$ grid [i]beautiful[/i] if the cell located at the center is colored white and all other cells are colored black, or if it is colored black and all other cells are colored white. Determine the minimum value of $a+b$ such that there exist positive integers $a$, $b$ and a coloring of an $a \times b$ grid with black and white, so that it contains $n^2 - n$ [i]beautiful[/i] subgrids.

On a $ 30 \times30 $ square board or placed figures of shape 1 (of 5 squares) (in all four possible positions) and shaped figures of shape 2 (of 4 squares) . The figures do not overlap, they do not pass through the edges of the board and the squares of which they are drawn lie exactly through the squares of the board. a) Prove that the board can be fully covered using $100$ figures of both shapes. b) Prove that if there are already $50$ shaped figures on the board of shape 1, then at least one more figure can be placed on the board. c) Prove that if there are already $28$ figures of both shapes on the board then at least one more figure of both shapes can be placed on the board. [img]https://cdn.artofproblemsolving.com/attachments/3/f/f20d5a91d61557156edf203ff43acac461d9df.png[/img]

Cut a rectangle into $18$ rectangles so that no two adjacent ones form a rectangle.

You have a large number of congruent equilateral triangular tiles on a table and you want to fit $n$ of them together to ma€ke 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 $13$. Determine if $19$ and $20$ are feasible .

A $9\times9$ chess board has two squares from opposite corners and its central square removed. Is it possible to cover the remaining squares using dominoes, where each domino covers two adjacent squares? Justify your answer.

A $5\times 5$ board is covered by eight hooks (a three unit square figure, shown in the picture) so that one unit square remains free. Determine all squares of the board that can remain free after such covering. [img]https://cdn.artofproblemsolving.com/attachments/6/8/a8c4e47ba137b904bd28c01c1d2cb765824e6a.png[/img]