6. Several cells on a $14 \times 14$ board are marked. It is known that no two marked cells are in the same row and the same column, and also, that a knight can, starting from some marked cell, visit all marked cells in several jumps, visiting each exactly once. What is the maximum possible number of marked cells?
Answer: 13.

Since there is no more than one marked cell (field) in each row, there are no more than 14 marked cells. Suppose there are 14 marked cells. Number the rows and columns from 1 to 14 from bottom to top and from left to right, and color the cells in a checkerboard pattern, where the sum of the row and column numbers for white cells is odd, and for black cells, it is even. Let the path of the knight through the marked cells be $\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right),\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right),\left(\mathrm{x}_{3}, \mathrm{y}_{3}\right), \ldots,\left(\mathrm{x}_{14}, \mathrm{y}_{14}\right)$. Consider the sum $\left(\mathrm{x}_{1}+\mathrm{y}_{1}\right)+\left(\mathrm{x}_{2}+\mathrm{y}_{2}\right)+\left(\mathrm{x}_{3}+\mathrm{y}_{3}\right)+ \ldots+\left(\mathrm{x}_{14}+\mathrm{y}_{14}\right)$. Since the knight changes the color of the cell with each move, there will be seven odd terms in the sum, making it odd. On the other hand, the knight visits each row and column exactly once, so the sum is $2(1+2+3+\ldots+14)$ and is even. This is a contradiction.

An example for 13 marked cells is shown in the figure on the right.

Criteria. An example is provided, but there is no estimate: 2 points.

An estimate is made, but there is no example or it is incorrect: 4 points.

![](https://cdn.mathpix.com/cropped/2024_05_06_af042a60fd879bc8b32fg-3.jpg?height=582&width=578&top_left_y=813&top_left_x=2139)