Question 51
A natural number $k$ is called $n$-squared if, however the cells of a $2n \times k$ chessboard are coloured with $n$ different colours, there exist 4 distinct cells of the same colour whose centres are the vertices of a rectangle with sides parallel to the sides of the chessboard. Determine, as a function of $n$, the smallest natural number $k$ that is $n$-squared.