Jane wrote another number on the board. This time it was a two-digit number and again it did not include digit 5. Jane then decided to include it, but the number was written too close to the edge, so she decided to t the 5 in between the two digits. She noticed that the resulting number is 11 times larger than the original. What is the sum of digits of the new number?
a) Find the biggest 6-digit integer number such that each digit, except for the two on the left, is equal to the sum of its two left neighbours.
b) Find the biggest integer number such that each digit, except for the rst two, is equal to the sum of its two left neighbours. (Compared to part (a), we removed the 6-digit number restriction.)
Tile a \(5\times6\) rectangle in an irreducible way by laying \(1\times2\) rectangles.
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.
On the second day Robinson Crusoe stretched the rope between two pegs, put a ring on the rope, and tied the goat with another rope to the ring. What shape did the goat graze in this case?