How do you choose the "right" method for solving the problem at hand? The trouble is, you are spoilt for choice: Descartes' method, proof by contradiction, proof by induction, the search for invariants, the extremum principle, Fermat's infinite descent, coloring proofs... In fact, it all depends on the context and the problem you are tackling. One category of combinatorial problems, several applications of which are explored here (and you are invited to ponder the statements for yourself before rushing to read their solutions!), relies on the famous and powerful pigeonhole principle. In French, this is known as the principe des tiroirs, the "drawer principle"; the English name evokes the image of pigeons distributed among the openings of a dovecote. The first version was stated in elementary terms by the German mathematician Johann Peter Gustav Lejeune Dirichlet in 1834, under the name Schubfachprinzip, literally "drawer principle":
if n + 1 objects are put into n drawers,
then at least one drawer contains at least two objects.
More generally, if kn + 1 objects are put into n drawers, where k is a positive integer, then at least one drawer will contain at least k + 1 objects. Equivalently, if p objects are put into n drawers, then at least one drawer contains at least q objects, where q is the integer obtained by rounding the quotient p / n up if necessary.
As easy as pie --------------------