10.3. Find all natural $n$, for which on a square grid of size $n$ by $n$ cells, it is possible to mark $n$ cells, each in different rows and different columns, that can be sequentially visited by a knight's move in chess, starting from some cell, without landing on the same cell twice, and returning to the starting cell. The knight can only land on the marked cells.
Answer. $n=4$.

Solution. An example for $n=4$ is not difficult:

![](https://cdn.mathpix.com/cropped/2024_05_06_9b63a86e4c147040aef2g-2.jpg?height=132&width=128&top_left_y=1042&top_left_x=1044)

We will prove that for $n \neq 4$, the required set of cells does not exist.

Suppose that for a given $n$, it is possible to mark $n$ cells in different rows and different columns of an $n \times n$ board, which can be sequentially traversed by a knight's move and return to the initial marked cell. Consider the horizontal displacements of the knight during its sequential moves to the marked cells in the columns of the board. The knight either moves to a cell in column a (the first from the left) from a cell in column b (the second from the left), and then moves to a cell in column c (the third from the left), or vice versa. In the first case (the second case is symmetric to the first), to avoid repeating columns, the knight can only move to column b from column d, and can only move from column c to column e. All preceding moves to column d and subsequent moves from column e are made with a horizontal displacement of 2, except for one with a displacement of 1, which will close the chain of the knight's movements and can only occur at the right edge of the board. In total, the knight will move horizontally 2 times by one cell and $n-2$ times by two cells. The same can be said about its vertical movements along the rows of the board. Each move of the knight involves a displacement of one cell in one direction and two cells in the other, so in the total set of all $2n$ displacements in both directions, the number of displacements by one cell must equal the number of displacements by two cells, from which $2n-4=4$ and $n=4$.

Grading criteria. The idea of considering moves along the three outermost columns (rows) of the board: 2 points.