[ Invariants and Semi-invariants ]

In the Martian alphabet, there are two letters - U and Y, and if any adjacent letters UY are removed from any word, the meaning of the word does not change. Similarly, the meaning does not change when the letter combinations YU or UUYU are added to any position in the word. Is it true that the words YUYUY and UYUYU have the same meaning?

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When transitioning from any word to a word with the same meaning, the difference between the number of letters У and Ы does not change.

## Solution

Assume the opposite - the words ЫУЫУЫ and УЫУЫУ have the same meaning. Then, from the word ЫУЫУЫ to the word УЫУЫУ, one can transition through several transformations described in the condition (i.e., removing adjacent letters УЫ and adding sequences of letters ЫУ or УУЫЫ in any position). With each of these transformations, the same number of letters У and Ы are added or removed. Thus, the difference between the number of letters У and Ы remains unchanged (i.e., it is an invariant of the considered transformation). However, in the word ЫУЫУЫ, there is one fewer У than Ы, while in the word УЫУЫУ, there is one more У. This contradicts our assumption.

## Answer

not true.

Author: Khachatryan M.A.

The figure "violinist" attacks the cell to the left on the side (with the elbow) and the cell to the right diagonally above (with the bow) if he is right-handed, and, conversely, the right cell on the side and the left cell diagonally above if he is left-handed (all violinists are facing us). Place as many "violinists" as possible in an "orchestra" of $8 \times 8$ cells so that they do not attack each other. (You can use any number of right-handed and left-handed violinists.)

![](https://cdn.mathpix.com/cropped/2024_05_06_f579c76dfb5bc127b163g-03.jpg?height=326&width=343&top_left_y=389&top_left_x=114)

This is how a right-handed violinist attacks

![](https://cdn.mathpix.com/cropped/2024_05_06_f579c76dfb5bc127b163g-03.jpg?height=326&width=340&top_left_y=792&top_left_x=116)

and this is how a left-handed violinist attacks

## Solution

Placing 32 violinists is not difficult: for example, you can fill four columns every other column (it doesn't matter whether they are right-handed or left-handed). However, more can be placed. A possible example with 34 violinists is shown in the diagram.

| Pr | $\Lambda$ |  |  | Pr | $\wedge$ |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  |  | Pr | $\Lambda$ |  |  | Pr | $\Lambda$ |
| Pr | $\Lambda$ |  |  | Pr | $\Lambda$ |  |  |
|  |  | Pr | $\Lambda$ |  |  | Pr | $\Lambda$ |
| Pr | $\Lambda$ |  |  | Pr | $\Lambda$ |  |  |
|  |  | Pr | $\Lambda$ |  |  | Pr | $\Lambda$ |
| Pr | $\Lambda$ |  |  | Pr | $\Lambda$ |  | Pr |
| $\Lambda$ |  | Pr | $\Lambda$ |  | Pr |  | $\Lambda$ |