6. (B. Novikov) On each cell of an $8 \times 8$ board, a guard is placed. Each guard can look in one of four directions (along the lines of the board) and watch all the guards on the line of his sight. For what largest $k$ can the guards' gazes be directed so that each guard is watched by at least $k$ other guards?
Solution: Answer: 5. We will prove that $k \leq 5$. For this, assume that $k \geqslant 6$. Consider the guards standing at the corners of the board. Each of them is watched by at least 6 guards, and these guards must stand at the edge of the board. Moreover, if a guard sees one of the corner guards, they do not see the other corner guards. Thus, at least 24 guards standing at the edge of the board are looking along the sides of the board. Then, inside the board, not at the corner guards, there are no more than four guards standing at the boundaries.

Now consider the guards standing in the central $6 \times 6$ square. Let's count the maximum possible number of "incoming gazes" for them. (We will not count gazes directed at guards on the boundary of the board). This number does not exceed $184=24+100+48+12$. (24 - from four guards on the boundary, 100 - 5 gazes from each of the 20 guards on the boundary of the $6 \times 6$ square, 48 - 4 gazes from each of the 12 guards on the boundary of the $4 \times 4$ square, 12 - 3 gazes from each of the 4 guards in the central $2 \times 2$ square.) Thus, for 36 guards, there are $184=36 \cdot 5+4<36 \cdot 6$ gazes. Therefore, among the guards, there are those who received fewer than 6 gazes.

Examples for $k=5$ can be arranged differently. One of the variants is shown in the figure (long arrows mean that several guards in a row are looking in the same direction, guards in the center can look in any direction).

![](https://cdn.mathpix.com/cropped/2024_05_06_e91b4554d9335e8daacbg-6.jpg?height=408&width=410&top_left_y=1306&top_left_x=562)

## Criteria.

Arithmetic errors (or errors in counting the number of certain cells) do not affect the evaluation if they did not affect the solution process. “+” - a correct and justified solution is provided;

“土” - a correct example and evaluation with minor (easily fixable) flaws that do not simplify the evaluation;

“干” - a correct example or a correct evaluation is provided;

“-.“ - an evaluation with minor flaws is provided;

“-” - all other cases, including if an evaluation with any significant flaws is provided, the evaluation is greater than 5, or the example is less than 5.