$\underline{\text { Krotov S.S. }}$

a) In all cells of a $20 \times 20$ square, there are soldiers. Vanya names a number $d$, and Petya rearranges the soldiers so that each one moves a distance of at least $d$ (the distance is taken between the centers of the old and new cells). For which $d$ is this possible?

b) The same problem for a $21 \times 21$ square.
The greatest distance a soldier can move, starting from the central cell, is

$10 \sqrt{2}$. We will show that this is indeed the maximum value of $d$.

a) Divide the square into four $10 \times 10$ squares and swap the opposite squares (each cell will move by the specified distance).

b) Divide the square as shown in the figure (for simplicity, a $7 \times 7$ square is depicted).

![](https://cdn.mathpix.com/cropped/2024_05_06_43a9f723b796fae20c45g-40.jpg?height=452&width=463&top_left_y=394&top_left_x=818)

Soldiers $x, y, z$ perform a cyclic permutation. Figures $A$ and $B$ (rectangles with removed corners) swap places, and squares $C$ and $D$ also swap places.

## Answer

a), b) For $d \leq 10 \sqrt{2}$.