Question 72
The cells of an $n \times n$ chessboard are coloured in several colours so that no $2 \times 2$ square contains four cells of the same colour. A proper path of length $m$ is a sequence $a_1, a_2, \dots, a_m$ of distinct cells in which the cells $a_i$ and $a_{i+1}$ have a common side and are coloured in different colours for all $1 \le i \lt m$. Show that there exists a proper path of length $n$.