# Tricky problem on pigeonhole principle

The question was asked in my today's quiz and I have no idea how to start with it. It's evident that we have to use pigeonhole principle somehow but how I am not getting. Question is " A 4×9 rectangular board is divided into squares each of which is coloured red or green or blue. Prove that the board contains a rectangle whose four corner squares are all the same colour. Thanks.

• I am getting that feeling. Can I approach by any different way. Please suggest. – Believer Sep 5 '19 at 21:33
• Also; it seems to be false. – Servaes Sep 5 '19 at 21:33
• If there were an answer (Donald's answer below shows there is not), you might start by noting that with three colors in four squares, each row will have at least two squares of the same color. Its a little bit counterintuitive in that the colors are the pidgeonholes and the squares are the pidgeons. – David Diaz Sep 5 '19 at 21:40

Counter Example ... for instance ... • Each column must have at least two squares that are the same colours. These pairs can be in $\binom{4}{2}=6$ different configurations. There are $3$ colours, so there are $3 \times 6 =18$ ways to avoid any duplication of pairs. So I suspect the question should have been about a $4$ by $19$ rectangle. – Donald Splutterwit Sep 5 '19 at 21:53