One of the most useful tools for proving mathematical statements is the Pigeonhole principle. Here is one example: suppose that a flock of \(10\) pigeons flies into a set of \(9\) pigeonholes to roost. Because there are \(10\) pigeons but only \(9\) pigeonholes, at least one of these \(9\) pigeonholes must have at least two pigeons in it.