6. 6.1. The figure "lame rook" can move to an adjacent cell in one move. On a $20 \times 20$ board, crosses were placed in all cells that the "lame rook" can reach from the top-left corner in exactly 10 moves. How many cells were marked with a cross?
Answer: 36.

## Solution.

Let's set up a coordinate system so that the top-left cell has coordinates $(0,0)$. In one move, the sum of the rook's coordinates changes by 1. In 10 moves, the sum of the coordinates cannot exceed 10, and it will become an even number.

![](https://cdn.mathpix.com/cropped/2024_05_06_8a1d5b347fbc679ac0eeg-5.jpg?height=572&width=612&top_left_y=1953&top_left_x=800)

Thus, only the shaded cells on the diagram can be marked with a cross. It is clear that the shaded cells will be marked. Their number is 36. 6.2. The figure "crippled rook" can move to an adjacent cell in one move. On a $20 \times 20$ board, crosses were placed in all cells that the "crippled rook" can reach from the top-left corner in exactly 12 moves. How many cells were marked with a cross?

Answer: 49.