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.

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Pigeonhole Principle unlocks at Level 3 of Lemma's eight-level ladder, with lessons that teach it and drills that make it stick. 125 lessons, 1206 curated problems and unlimited generated practice at six difficulties. Free to start, no card, and every paid plan opens with 3 free days.

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