In a large box, 10 smaller boxes were placed. In each of the nested boxes, either 10 even smaller ones were placed, or nothing was placed. In each of the smaller ones, either 10 or none were placed again, and so on. After this, it turned out that there were exactly 2006 boxes with contents. How many are empty? #
Each box A, except for the very first one, is associated with a box B in which it lies (if the box B itself lies in some other box, we do not associate it with box A). Thus, each box, except for the first one, is associated with exactly one box, and each non-empty box is associated with exactly ten boxes. Let S denote the total number of boxes. On one hand, the number of "associations" is S - 1, and on the other hand, it is 10 × 2006 (since there are 2006 non-empty boxes). Therefore, it follows that S = 10 × 2006 + 1 - 2006 = 18055.

## Answer

18055.

Author: Raschina I.V.

If an octopus has an even number of legs, it always tells the truth. If it has an odd number of legs, it always lies. One day, a green octopus said to a dark blue one:

- I have 8 legs. And you only have 6.
- I have 8 legs, - the dark blue one was offended. - And you only have 7.
- The dark blue one really has 8 legs, - the purple one supported and boasted: - And I have 9!
- No one has 8 legs, - the striped octopus joined the conversation. - Only I have 8 legs!

Who among the octopuses had exactly 8 legs?

## Solution

If the purple octopus is telling the truth, then it has an even number of legs. But in that case, it cannot say it has 9 legs. Therefore, the purple octopus is lying. So, the dark blue octopus does not have 8 legs. Since the dark blue octopus says it has 8 legs, it is lying. Therefore, it has an odd number of legs. By saying the dark blue octopus has 6 legs, the green one lied. Therefore, it also lied the first time, and it does not have 8 legs. Thus, the first statement of the striped octopus is true. Therefore, the second statement is also true, and it indeed has 8 legs. The others have an odd number of legs.

## Answer

The striped octopus had 8 legs.