You are given a $3\times n$ checkerboard, covered with $n$ red, $n$ white, and $n$ blue checkers. Call a board patriotic if every column has a red, white and blue checker. You want to make the given board patriotic. To do this, you are allowed to do a series of moves, where a move consists of changing the places of two checkers which are in the same row. Can you succeed?
In the case that you can, you should demonstrate why, and in the case you can't, you should give an example of a starting board which can't achieve patriotism.
Here are some examples when $n=7$. On the left is a starting position which is very unpatriotic: none of the columns feature all three colors. It can, however, be rearranged to the patriotic board on the right.