Problem 5. The figure "violinist" attacks the cell to the left on the side (with the elbow) and the cell to the upper right diagonally (with the bow), if he is right-handed, and, conversely, the right cell on the side and the upper left cell diagonally, 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 combination of right-handed and left-handed violinists.)

![](https://cdn.mathpix.com/cropped/2024_05_06_5d4a771360f9cc25c167g-5.jpg?height=162&width=164&top_left_y=145&top_left_x=919)

This is how a right-handed violinist attacks

![](https://cdn.mathpix.com/cropped/2024_05_06_5d4a771360f9cc25c167g-5.jpg?height=160&width=160&top_left_y=142&top_left_x=1115)

and this is how a left-handed violinist attacks

(M. Khachaturian)
Solution. Placing 32 violinists is not difficult: for example, you can fill four columns every other one (it doesn't matter whether they are right-handed or left-handed). However, it is possible to place more. An example with 34 violinists is shown in the figure.

| $\mathrm{p}$ | $\Lambda$ |  |  | $\mathrm{p}$ | $\Lambda$ |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  |  | $\mathrm{p}$ | $\Lambda$ |  |  | $\mathrm{p}$ | 1 |
| $\mathrm{p}$ | $\Lambda$ |  |  | $\mathrm{p}$ | $\Lambda$ |  |  |
|  |  | $\mathrm{p}$ | $\Lambda$ |  |  | $\mathrm{p}$ | $\Lambda$ |
| $\mathrm{p}$ | $\Lambda$ |  |  | $\mathrm{p}$ | $\Lambda$ |  |  |
|  |  | $\mathrm{p}$ | $\Lambda$ |  |  | $\mathrm{p}$ | $\Lambda$ |
| $\mathrm{p}$ | $\Lambda$ |  |  | $\mathrm{p}$ | $\wedge$ |  | $\mathrm{p}$ |
| $\Lambda$ |  | $\mathrm{p}$ | $\wedge$ |  | $\mathrm{p}$ |  | $\Lambda$ |

Comment. It can be proven that it is impossible to place more than 34 violinists. Indeed, consider a vertical strip two cells wide. Any violinist standing in this strip and not in the top row will hit one cell in the same strip. We can associate this empty cell with the given violinist. If it turns out that two violinists hit the same empty cell (there are two similar cases of this, one of which is shown in the figure), then the cell below the twice-hit one must also be empty, so it can be associated with one of the two violinists, say the lower one. If this cell is hit by a bow from a violinist standing even lower, we can associate this violinist with the adjacent cell horizontally (which is necessarily empty), and so on.

Thus, in the strip, there can be a maximum of 9 violinists - two in the top cells and seven more in the remaining 14 cells, because each violinist there corresponds to an empty cell, meaning that the occupied cells do not exceed half.

However, two strips with nine violinists cannot be adjacent, otherwise four violinists would sit in a row in the top row. Therefore, there can be no more than two such strips, and in the remaining two strips, there can be a maximum of 8 violinists each, so there are no more than $9+9+8+8=34$ musicians in the orchestra.