Problems

Each cell of a $2\times 2$ square can be painted either black or white. How many different patterns can be obtained?

There are $25$ bugs sitting on the squares of a $5\times 5$ board, $1$ at each square. When I clap my hands, each bug jumps to a square diagonally from where it was before. Show that after I clap my hands, at least $5$ squares will be empty.

Can you cover a $10\times 10$ board using only $T$-shaped tetrominos?

Can you cover a $10\times 10$ square with $1\times 4$ rectangles?

Two opposite corners were removed from an $8\times 8$ chessboard. Is it possible to cover this chessboard with $1\times 2$ rectangular blocks?

One unit square of a $10\times 10$ square board was removed. Is it possible to cover the rest of it with $3$-square $L$-shaped blocks?

A $7\times 7$ square was tiled using $1\times 3$ rectangular blocks in such a way that one of the squares has not been covered. Find all the squares that could be left without being covered.

Can you cover a $13\times 13$ square using $2\times 2$ and $3\times 3$ squares?

Is it possible to cover a $10\times 10$ board with the $L$-tetraminos without overlapping? The pieces can be flipped and turned.

On a $9\times 9$ board $65$ bugs are placed in the centers of some of the squares. The bugs start moving at the same time and speed to a square that shares a side with the one they were in. When they reach the center of that square, they make a $90$ degrees turn and keep walking (without leaving the board). Prove that at some moment of time there are two bugs in the same square. Note: When they turn it can be either to the right or to the left.