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?

There is a hole in the roof with dimensions $23 \times 19$ cm. Can August fill the the roof with tiles of dimensions $5 \times 24 \times 30$ cm?

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.

Let $Z$ shape be a shape such that it covers $(i,j)$, $(i,j+1)$, $(i+1,j+1)$, $(i+2,j+1)$ and $(i+2,j+2)$ where $(i,j)$ stands for cell in $i$-th row and $j$-th column on an arbitrary table. At least how many $Z$ shapes is necessary to cover one $8 \times 8$ table if every cell of a $Z$ shape is either cell of a table or it is outside the table (two $Z$ shapes can overlap and $Z$ shapes can rotate)?

Find all quadrilaterals which can be covered (without overlappings) with squares with side $ 1$ and equilateral triangles with side $ 1$. (Emanuele Tron)

A cube of edge $2$ with one of the corner unit cubes removed is called a [i]piece[/i]. Prove that if a cube $T$ of edge $2^n$ is divided into $2^{3n}$ unit cubes and one of the unit cubes is removed, then the rest can be cut into [i]pieces[/i].

In a $10$ by $10$ square grid (which we call “the bay”) you are requested to place ten “ships”: one $1$ by $4$ ship, two $1$ by $3$ ships, three $1$ by $2$ ships and four $1$ by $1$ ships. The ships may not have common points (even corners) but may touch the “shore” of the bay. Prove that (a) by placing the ships one after the other arbitrarily but in the order indicated above, it is always possible to complete the process; (b) by placing the ships in reverse order (beginning with the smaller ones), it is possible to reach a situation where the next ship cannot be placed (give an example). (KN Ignatjev)

A $45 \times 45$ grid has had the central unit square removed. For which positive integers $n$ is it possible to cut the remaining area into $1 \times n$ and $n\times 1$ rectangles?

We are given tiles in the form of right angled triangles having perpendicular sides of length $1$ cm and $2$ cm. Is it possible to form a square from $20$ such tiles? ( S . Fomin , Leningrad)

What is the smallest number of $3$-cell corners that you need to paint in a $5 \times5$ square so that you cannot paint more than one corner of one it? (Shaded corners should not overlap.)

Let $a$ and $b$ be positive integers. We tile a rectangle with dimensions $a$ and $b$ using squares whose side-length is a power of $2$, i.e. the tiling may include squares of dimensions $1\times 1, 2\times 2, 4\times 4$ etc. Denote by $M$ the minimal number of squares in such a tiling. Numbers $a$ and $b$ can be uniquely represented as the sum of distinct powers of $2$: $a=2^{a_1}+\cdots+2^{a_k},\; b=2^{b_1}+\cdots +2^{b_\ell}.$ Show that $$M=\sum_{i=1}^k \;\sum_{j=1}^{\ell} 2^{|a_i-b_j|}.$$

Rectangle on a checked paper with length of a unit square side being $1$ Is divided into domino figures( two unit square sharing a common edge). Prove that you colour all corners of squares on the edge of rectangle and inside rectangle with $3$ colours such that for any two corners with distance $1$ the following conditions hold: they are coloured in different colour if the line connecting the two corners is on the border of two domino figures and coloured in same colour if the line connecting the two corners is inside a domino figure.

Can an arbitrary polygon be cut into isosceles trapezoids?

Infinitely many equidistant parallel lines are drawn in the plane. A positive integer $n \geqslant 3$ is called frameable if it is possible to draw a regular polygon with $n$ sides all whose vertices lie on these lines, and no line contains more than one vertex of the polygon. (a) Show that $3, 4, 6$ are frameable. (b) Show that any integer $n \geqslant 7$ is not frameable. (c) Determine whether $5$ is frameable. [i]Proposed by Muralidharan[/i]

A $7\times 7$ square is cut into pieces following types: [img]https://cdn.artofproblemsolving.com/attachments/e/d/458b252c719946062b655340cbe8415d1bdaf9.png[/img] Show that exactly one of the pieces is of type (b). [img]https://cdn.artofproblemsolving.com/attachments/4/9/f3dd0e13fed9838969335c82f5fe866edc83e8.png[/img]

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]

Jiiri and Mari both wish to tile an $n \times n$ chessboard with cards shown in the picture (each card covers exactly one square). Jiiri wants that for each two cards that have a common edge, the neighbouring parts are of different color, and Mari wants that the neighbouring parts are always of the same color. How many possibilities does Jiiri have to tile the chessboard and how many possibilities does Mari have? [img]https://cdn.artofproblemsolving.com/attachments/7/3/9c076eb17ba7ae7c000a2893c83288a94df384.png[/img]

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 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]

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]

A checkered polygon $A$ is drawn on the checkered plane. We call a cell of $A$ [i]internal[/i] if all $8$ of its adjacent cells belong to $A$. All other (non-internal) cells of $A$ we call [i]boundary[/i]. It is known that $1)$ each boundary cell has exactly two common sides with no boundary cells; and 2) the union of all boundary cells can be divided into isosceles trapezoid of area $2$ with vertices at the grid nodes (and acute angles of the trapezoids are equal $45^\circ$). Prove that the area of the polygon $A$ is congruent to $1$ modulo $4$.

(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)

A [i]simple polygon[/i] is a polygon whose perimeter does not self-intersect. Suppose a simple polygon $\mathcal P$ can be tiled with a finite number of parallelograms. Prove that regardless of the tiling, the sum of the areas of all rectangles in the tiling is fixed.\\ [i]Note:[/i] Points will be awarded depending on the generality of the polygons for which the result is proven.

The sword is a figure consisting of $6$ unit squares presented in the picture below (and any other figure obtained from it by rotation). [img]https://cdn.artofproblemsolving.com/attachments/4/3/08494627d043ea575703564e9e6b5ba63dc2ef.png[/img] Determine the largest number of swords that can be cut from a $6\times 11$ piece of paper divided into unit squares (each sword should consist of six such squares).

Pinocchio claims that he can take some non-right-angled triangles , all of which are similar to one another and some of which may be congruent to one another, and put them together to form a rectangle. Is Pinocchio lying? (A Fedotov)