Discussion: This student gets right to the core of the problem - with enough vertical lines intersection three horizontal lines there must be a repeating of one of the possibilities for red and blue and since two of the colors in a column must be the same in this way one obtains a rectangle all of whose vertices are the same color.

Written response: Good application of the pigeonhole principle. Note that you only need 7 patterns, since if no two patterns occur twice then one of the patterns has to be 3R or 3B, say 3R. Since there are only 4 patterns with fewer than 2R (BBB, RBB, BRB, BBR) there has to be at least one with 2R and then we get a rectangle with all vertices red. A similar problem to this states: Given a 3 x 7 chessboard show that there is some rectangle with all its corners the same color.