IT-CESENATICO 2009 Problem 6

A natural number $k$ is said $n$ -squared if by colouring the squares of a $2n \times k$ chessboard, in any manner, with $n$ different colours, we can find $4$ separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of $n$ , the smallest natural $k$ that is $n$ -squared.