11.5. On some cells of a rectangular board of size 101 by 99, there is one turtle each. Every minute, each of them simultaneously crawls to one of the cells adjacent to the one they are in, by side. In doing so, each subsequent move is made in a direction perpendicular to the previous one: if the previous move was horizontal - to the left or right, then the next one will be vertical - up or down, and vice versa. What is the maximum number of turtles that can move around the board indefinitely such that at each moment, no more than one turtle is in any cell?
Answer: 9800.

Solution. Examples of unlimited movement on the board of 9800 turtles.

Example 1. Place the reptiles in the cells of a rectangle consisting of cells at the intersection of the 98 lower horizontals and 100 leftmost verticals. They will all move in the same way: first all to the right, then all up, then all to the left, and then all down.

After 4 moves, the situation will coincide with the initial one, so the movement can continue in a permitted manner indefinitely.

Example 2. The initial placement of turtles is the same as in Example 1, but the movement is organized such that they are divided into 2x2 square cells, in each of which they simultaneously move clockwise.

We will prove that it is impossible to properly place more than 9800 turtles on a 101x99 board. Suppose the contrary, that the turtles are placed on the board in some way such that they have the possibility to move around the board indefinitely in the manner specified in the condition and not end up on the same cell in a quantity of two or more simultaneously. Color the cells of the board in a checkerboard pattern, with the lower left cell, as usual, being black. This will result in 5000 black and 4999 white cells. Among the black cells, we will call the cells with both vertical and horizontal coordinates being odd "odd" cells, and those with both coordinates being even "even" cells. Note that for white cells, one of these coordinates is even, and the other is odd. There will be $51 \cdot 50=2550$ odd black cells and $50 \cdot 49=2450$ even black cells. Clearly, at any moment, the number of turtles on even black cells should not exceed 2450.

Note that when two consecutive moves are performed, the parity of both coordinates of each turtle changes, so after two moves, all turtles from odd black cells will move to even black cells and vice versa, while turtles from white cells will move back to white cells. Consequently, at any time, the number of turtles on all black cells does not exceed $2 \cdot 2450=4900$.

Finally, when one move is performed, turtles from black cells move to white cells and vice versa, so the total number of turtles on the entire board cannot exceed $2 \cdot 4900=9800$.

Remark. In fact, we have proven that if there are more than 9800 turtles on the board, they will be able to make no more than two moves without ending up on the same cell in a quantity of two or more simultaneously.

Grading criteria. Proof of the maximality of the number 9800: 6 points. Construction of an example for 9800 turtles: 1 point. Attempt to prove any other estimate except 9800: 0 points.