The pigeonhole principle is a basic counting technique. It is illustrated in its simplest form as follows: We have n + 1 pigeons and n holes. We put all the pigeons in holes (in any way we want). The principle tells us that there must be at least one hole with at least two pigeons in it. Why is that true? Try to visualize the example of n = 2; therefore, we have 3 pigeons and 2 holes. Let’s try...