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)

Let $n$ be a natural number. A tiling of a $2n \times 2n$ board is a placing of $2n^2$ dominos (of size $2 \times 1$ or $1 \times 2$) such that each of them covers exactly two squares of the board and they cover all the board.Consider now two [i]sepearate tilings[/i] of a $2n \times 2n$ board: one with red dominos and the other with blue dominos. We say two squares are red neighbours if they are covered by the same red domino in the red tiling; similarly define blue neighbours. Suppose we can assign a non-zero integer to each of the squares such that the number on any square equals the difference between the numbers on it's red and blue neighbours i.e the number on it's red neigbhbour minus the number on its blue neighbour. Show that $n$ is divisible by $3$ [i] Proposed by Tejaswi Navilarekallu [/i]

A man disposes of sufficiently many metal bars of length $2$ and wants to construct a grill of the shape of an $n \times n$ unit net. He is allowed to fold up two bars at an endpoint or to cut a bar into two equal pieces, but two bars may not overlap or intersect. What is the minimum number of pieces he must use?

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

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?

Square $600\times 600$ is divided into figures of four types, shown in figure. In the figures of the two types, shown on the left, in painted black, the cells recorded number $2^k$, where $k$ is the number of the column, where is this cell (columns numbered from left to right by numbers from $1$ to $600$). Prove that the sum of all recorded numbers are divisible by $9$. [asy] // Set up the drawing area size(10cm,0); defaultpen(fontsize(10pt)); unitsize(0.8cm); // A helper function to draw a single unit square // c = coordinates of the lower-left corner // p = fill color (default is white) void drawsq(pair c, pen p=white) { fill(shift(c)*unitsquare, p); draw(shift(c)*unitsquare); } // --- Shape 1 (left) --- // 2 columns, 3 rows, black square in the middle-left drawsq((1,1), black); // middle-left black drawsq((2,0)); // bottom-right drawsq((2,1)); // middle-right drawsq((2,2)); // top-right // --- Shape 2 (next to the first) --- // 2 columns, 3 rows, black square in the middle-right drawsq((4,0)); drawsq((4,1)); drawsq((4,2)); drawsq((5,1), black); // middle-right black // --- Shape 3 (the "T" shape, 3 across the bottom + 1 in the middle top) --- drawsq((7,0)); drawsq((8,0)); drawsq((9,0)); drawsq((8,1)); // --- Shape 4 (the "T" shape, 3 across the top + 1 in the middle bottom) --- drawsq((11,1)); drawsq((12,1)); drawsq((13,1)); drawsq((12,0)); [/asy]

Find the smallest number of squares on an $8\times8$ board that should be colored so that every $L$-tromino on the board contains at least one colored square.

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

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]

Two right isosceles triangles of legs equal to $1$ are glued together to form either an isosceles triangle - called [i]t-shape[/i] - of leg $\sqrt2$, or a parallelogram - called [i]p-shape[/i] - of sides $1$ and $\sqrt2$. Find all integers $m$ and $n, m, n \ge 2$, such that a rectangle $m \times n$ can be tilled with t-shapes and p-shapes.

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]

A square board with dimensions $100 \times 100$ is divided into $10 000 $unit squares. One of the squares is cut out. Is it possible to cover the rest of the board by isosceles right angled triangles which have hypotenuses of length $2$, and in such a way that their hypotenuses lie on sides of the squares and their other two sides lie on diagonals? The triangles must not overlap each other or extend beyond the edges of the board. (S Fomin, Leningrad)

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$ 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?

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.

Can $77$ blocks each $3 \times 3 \times1$ be assembled to form a $7 \times 9 \times 11$ block?

A grid measuring $10 \times 11$ is given. How many "crosses" covering five unit squares can be placed on the grid? (pictured right) so that no two of them cover the same square? [img]https://cdn.artofproblemsolving.com/attachments/a/7/8a5944233785d960f6670e34ca7c90080f0bd6.png[/img]

Let $k$ and $n$ be integers such that $k\ge 2$ and $k \le n \le 2k-1$. Place rectangular tiles, each of size $1 \times k$, or $k \times 1$ on a $n \times n$ chessboard so that each tile covers exactly $k$ cells and no two tiles overlap. Do this until no further tile can be placed in this way. For each such $k$ and $n$, determine the minimum number of tiles that such an arrangement may contain.

Is it possible to divide a $6 \times 6$ chessboard into $18$ rectangles, each either $1 \times 2$ or $2 \times 1$, and to draw exactly one diagonal on each rectangle such that no two of these diagonals have a common endpoint? (A Shapovalov)

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)

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.

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.

Given is a floor plan composed of $n$ unit squares. Albert and Berta want to cover this floor with tiles, with all tiles having the shape of a $1\times 2$ domino or a $T$-tetromino. Albert only has tiles from one color, while Berta has two-color dominoes and tetrominoes available in four colors. Albert can use this floor plan in $a$ ways to cover tiles, Berta in $ b$ ways. Assuming that $a \ne 0$, determine the ratio $b/a$.

Let $m, n$ be positive integers with $m<n$ and consider an $n\times n$ board from which its upper left $ m\times m$ part has been removed. An example of such board for $n=5$ and $m=2$ is shown below. Determine for which pairs $(m, n)$ this board can be tiled with $3\times 1$ tiles. Each tile can be positioned either horizontally or vertically so that it covers exactly three squares of the board. The tiles should not overlap and should not cover squares outside of the board.

A black unit equilateral triangle is drawn on the plane. How can we place nine tiles, each a unit equilateral triangle, on the plane so that they do not overlap, and each tile covers at least one interior point of the black triangle? (Folklore)