pigeonhole

Pigeonhole is a term with several related uses. As a noun, it denotes a small, compartmentalized slot in a desk, filing cabinet, or mail room used for sorting papers, letters, or documents. Pigeonholes are usually arranged in a grid or row and are sometimes labeled to indicate their contents.

As a verb, to pigeonhole means to assign someone or something to a fixed category or to treat them in a simplified, overly rigid way. The term can carry a neutral sense of classification or a negative sense of stereotyping someone.

Etymology: The noun derives from the small compartments resembling spaces used for pigeons in dovecotes; the modern term dates from the 17th century in English.

In mathematics, the pigeonhole principle, also called Dirichlet's principle, is a simple yet powerful idea. It states that if n items are distributed into m containers and n > m, then at least one container must contain more than one item. Though intuitive, the principle has many uses in proofs and problem solving. Examples: with 367 people and 366 possible birthdays, two share a birthday; with 6 documents and 5 drawers, some drawer holds at least two documents. The principle establishes existence guarantees rather than constructive distributions, and it generalizes to more complex settings, including functions, sequences, and higher dimensions.

Because of its broad applicability, the pigeonhole principle is a standard tool in combinatorics, computer science, probability, and logic.