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A wide variety of questions in mathematics starts with the question ’Is it possible...?’. In such problems you would either present an example, in case the described situation is possible, or rigorously prove that the situation is impossible, with the help of counterexample or by any other means. Sometimes the border between what seems should be possible and impossible is not immediately obvious, therefore you have to be cautious and verify that your example (or counterexample) satisfies the conditions stated in the problem. When you are asked the question whether something is possible or not and you suspect it is actually possible, it is always useful to ask more questions to gather additional information to narrow the possible answers. You can ask for example "How is it possible"? Or "\(\bf Which\) properties should the correct construction satisfy"?

A bagel is cut into sectors with ten cuts. How many pieces were produced?

Which triangle has the largest area? The dots form a regular grid.

Replace the letters with digits in a way that makes the following sum as big as possible: \[SEND +MORE +MONEY.\]

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;

(b) \(6\times 6\) rectangle?

Irreducibly tile a floor with \(1\times2\) tiles in a room that is

(a) \(5\times8\); (b) \(6\times8\).

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: