ELMO 2025 Problem 6

Let $n \ge 2$ be an integer. An $n \times n$ grid is filled with the numbers $1$ through $n$ such that each row contains every number exactly once and any two adjacent rows differ by exactly two elements. Find all $n$ such that for any valid labeling of the grid, there exists a set of $n$ cells, each with a different number, such that no two are on the same row or column.

Andrew Carratu