Dmitriev 0.

The mammoth figure moves like a bishop (diagonally), but only in three of the four possible directions (the missing direction can differ for different mammoths). What is the maximum number of non-attacking mammoths that can be placed on an $8 \times 8$ chessboard?
Evaluation. From each mammoth, we will release three arrows in the directions in which it can attack. We will match an arrow to a diagonal (not necessarily the main one) if the mammoth from which the arrow originates is on this diagonal, and the arrow travels along it. Then, no more than two arrows are matched to each diagonal: otherwise, two of them would be going in the same direction, and one of the mammoths would be attacking the other. Since there are 30 diagonals in total (15 in each direction), no more than 60 arrows are matched to them, and thus the total number of mammoths is no more than \(60 \div 3 = 20\).

Three possible examples of placing 20 mammoths that do not attack each other are shown in the figure.
![](https://cdn.mathpix.com/cropped/2024_05_06_7dfbf12cfa0901bdf2bfg-12.jpg?height=438&width=1280&top_left_y=1436&top_left_x=386)

## Answer

20 mammoths.