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.

In how many ways can you rearrange the numbers 1, 2, ..., 100 so the neighbouring numbers differ by not more than 1?

A strange wonderland creature is called a painting chameleon. If the queen puts the painting chameleon on a chess-like board then he moves one square at a time along the board either horizontally or vertically. When he moves, he either changes his colour to the colour of the square he moves to, or he paints the square he moves to into his own colour. The queen puts a white painting chameleon on an all-black board \(8\times8\) and orders the chameleon to paint the board into a chessboard. Can he succeed?

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

A big square was cut into smaller squares. Sebastian used all the pieces and constructed two squares with different side lengths by glueing the pieces together. Show an example of how he could do that.

It was Sebastian’s younger brother who cut the big square in Example 2. Now you need to help him to cut one of the squares (which Sebastian obtained after glueing the pieces) into smaller congruent triangles. But please make sure the elder brother can do the same thing as before: to divide the resulting congruent triangles into two groups and to glue the pieces of each group together to make two squares with different side lengths.