Pigeonhole Principle

More pigeons than holes means a shared hole.

If $n+1$ objects are placed in $n$ boxes, some box gets two. Trivial to state, endlessly deep to apply — because the problem never tells you what the boxes are. Choosing the boxes IS the solution.

Classic box choices: residues mod $n$ , unit subsquares of a region, intervals of a range, pairs that sum to a constant. When asked to prove two objects are equal, close, or similar, ask what partition would force the collision.

Where it appears on Lemma: Level 3, and silently inside Ramsey theory at Level 8.

Train it on Lemma

Pigeonhole Principle unlocks at Level 3 of Lemma's eight-level ladder, with lessons that teach it and drills that make it stick. 78 lessons, 624 curated problems and unlimited generated practice at six difficulties. Free to start — no card, no trial clock.

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