{"id":"pigeonhole-principle","domain":"discrete","type":"Concept","title":"Pigeonhole Principle","slug":"pigeonhole-principle","url":"/mathematics/discrete/pigeonhole-principle/","summary":"If n+1 items are placed in n containers, some container has 2+. Simple but powerful proof technique across number theory, combinatorics, cryptography.","also_known_as":"Dirichlet's box principle / drawer principle","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Graham, R.L., Knuth, D.E. and Patashnik, O. (1994). Concrete Mathematics. 2nd ed."},{"tier":1,"citation":"Erdős, P. and Szekeres, G. (1935). A combinatorial problem in geometry. Compositio Mathematica, 2, 463-470."},{"tier":2,"citation":"Graham, R.L., Rothschild, B.L. and Spencer, J.H. (1990). Ramsey Theory. 2nd ed."},{"tier":3,"citation":"Brualdi, R.A. (2010). Introductory Combinatorics. 5th ed."}],"relationships":[{"type":"related_to","target":"combinatorics","label":"Combinatorics — Ramsey theory"},{"type":"proves","target":"rational-number","label":"Rational Number — repeating decimals"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.9"}