Problems

Does there exist an irreducible tiling with \(1\times2\) rectangles of a \(4\times 6\) rectangle?

Irreducibly tile a floor with \(1\times2\) tiles in a room that is a \(5\times8\) rectangle.

Having mastered tiling small rooms, Robinson wondered if he could tile big spaces, and possibly very big spaces. He wondered if he could tile the whole plane. He started to study the tiling, which can be continued infinitely in any direction. Can you help him with it?

Tile the whole plane with the following shapes:

Robinson Crusoe was taking seriously the education of Friday, his friend. Friday was very good at maths, and one day he cut 12 nets out of hardened goat skins. He claimed that it was possible to make a cube out of each net. Robinson looked at the patterns, and after some considerable thought decided that he was able to make cubes from all the nets except one. Can you figure out which net cannot make a cube?

It is known that it is possible to cover the plane with any cube’s net. (You will see it in the film that will be shown at the end of this session). But Robinson, unfortunately, lived on an uninhabited island in the 19th century, and did not know about the film. Try to help him to figure out how to cover the plane with nets \(\#2\), \(\#6\), and \(\#8\) from the previous exercise.

Remove a \(1 \times 1\) square from the corner of a \(4 \times 4\) square. Can this shape be dissected into \(3\) congruent parts?

Can a \(5\times5\) square checkerboard be covered by \(1\times2\) dominoes?

Can you cover a \(10 \times 10\) board using only \(T\)-shaped tetraminoes?

Peter marks several cells on a \(5 \times 5\) board. His friend, Richard, will win if he can cover all of these cells with non-overlapping corners of three squares, that do not overlap with the border of the square (you can only place the corners on the squares). What is the smallest number of cells that Peter should mark so that Richard cannot win?

A game takes place on a squared \(9 \times 9\) piece of checkered paper. Two players play in turns. The first player puts crosses in empty cells, its partner puts noughts. When all the cells are filled, the number of rows and columns in which there are more crosses than zeros is counted, and is denoted by the number \(K\), and the number of rows and columns in which there are more zeros than crosses is denoted by the number \(H\) (18 rows in total). The difference \(B = K - H\) is considered the winnings of the player who goes first. Find a value of B such that

1) the first player can secure a win of no less than \(B\), no matter how the second player played;

2) the second player can always make it so that the first player will receive no more than \(B\), no matter how he plays.