Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure. [asy] unitsize(0.5 cm); draw((0,0)--(1,0)); draw((0,1)--(1,1)); draw((2,1)--(3,1)); draw((0,2)--(3,2)); draw((0,3)--(3,3)); draw((0,0)--(0,3)); draw((1,0)--(1,3)); draw((2,1)--(2,3)); draw((3,1)--(3,3)); [/asy] Determine all $ m\times n$ rectangles that can be covered without gaps and without overlaps with hooks such that - the rectangle is covered without gaps and without overlaps - no part of a hook covers area outside the rectangle.

A V-tromino is a diagram formed by three unit squares.(As attachment.) (a)Is it possible to cover a $3\times 2013$ table by $3\times 671$ V-trominoes? (b)Is it possible to cover a $5\times 2013$ table by $5\times 671$ V-trominoes?

Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure. [asy] unitsize(0.5 cm); draw((0,0)--(1,0)); draw((0,1)--(1,1)); draw((2,1)--(3,1)); draw((0,2)--(3,2)); draw((0,3)--(3,3)); draw((0,0)--(0,3)); draw((1,0)--(1,3)); draw((2,1)--(2,3)); draw((3,1)--(3,3)); [/asy] Determine all $ m\times n$ rectangles that can be covered without gaps and without overlaps with hooks such that - the rectangle is covered without gaps and without overlaps - no part of a hook covers area outside the rectangle.

We have tiles (which are build from squares of side length 1) of following shapes: [asy] unitsize(0.5 cm); draw((1,0)--(2,0)); draw((1,1)--(2,1)); draw((1,0)--(1,1)); draw((2,0)--(2,1)); draw((0,1)--(1,1)); draw((0,2)--(1,2)); draw((0,1)--(0,2)); draw((1,1)--(1,2)); draw((0, 0)--(1, 0)); draw((0, 0)--(0, 1)); draw((5,0)--(6,0)); draw((5,1)--(6,1)); draw((5,0)--(5,1)); draw((6,0)--(6,1)); draw((4,1)--(5,1)); draw((5,2)--(6,2)); draw((5,1)--(5,2)); draw((6,1)--(6,2)); draw((4, 0)--(5, 0)); draw((4, 0)--(4, 1)); draw((6,2)--(7,2)); draw((7,1)--(7,2)); draw((6,1)--(7,1)); draw((11,0)--(12,0)); draw((11,1)--(12,1)); draw((11,0)--(11,1)); draw((12,0)--(12,1)); draw((10,1)--(11,1)); draw((10,2)--(11,2)); draw((10,1)--(10,2)); draw((11,1)--(11,2)); draw((10, 0)--(11, 0)); draw((10, 0)--(10, 1)); draw((9, 2)--(9, 1)); draw((9,1)--(10, 1)); draw((9,2)--(10,2)); [/asy] For each odd integer $n \ge 7$, determine minimal number of these tiles needed to arrange square with side of length $n$. (Attention: Tiles can be rotated, but they can't overlap.)

By a $\emph{tile}$ we mean a polyomino (i.e. a finite edge-connected set of cells in the infinite grid). There are many ways to place a tile in the infinite table (rotation is allowed but we cannot flip the tile). We call a tile $\textbf{T}$ special if we can place a permutation of the positive integers on all cells of the infinite table in such a way that each number would be maximum between all the numbers that tile covers in at most one placement of the tile. 1. Prove that each square is a special tile. 2. Prove that each non-square rectangle is not a special tile. 3. Prove that tile $\textbf{T}$ is special if and only if it looks the same after $90^\circ$ rotation.