How can you solve pigeonhole questions using contradiction?

Answer 1

The easiest way might to answer that might be to show you an example.

Let's look at a Chess board with two of the diagonally opposite corners removed. Is it possible to cover the board with pieces of domino whose size is exactly two board squares?

The reason this is a pigeonhole problem is because the two diagonal square on a chess board

are the same color. So when you remove them you have 2 more square of one color than you do of the other.

So assume by contradiction that you can cover the board with pieces of domino whose size is exactly two board squares. Now every piece of domino must cover exactly two squares and these will be squares of different colors because adjacent square on the chess board are different colors. So for every domino piece I place, I set up a 1 to 1 correspondence between the set of one color square and the set of the other color squares. We now know the cardinality of the two sets is different since we removed those corners. So the pigeonhole principle tells us we can not have a 1 to 1 correspondence between two sets with different cardinalities. We conclude that it can't be done.

The idea in all cases where you want to use the pigeonhole principle and prove by contradiction is to assume it works and then let the pigeonhole principle prove it can't work.