Olympiad problem-solving technique

Induction

Two finite steps climb an infinite ladder.

Prove the base case; prove that each case forces the next. The craft lives in the inductive step: find where the size- $(k+1)$ problem contains a size- $k$ copy of itself. Remove a disk, delete a vertex, strip the last term.

When the step will not go through, strengthen the statement. A stronger hypothesis gives the induction more to lean on — a beautiful paradox: proving more can be easier than proving less.

Where it appears on Lemma: Level 5, and beneath nearly every recursion you will ever write.

Train it on Lemma

Induction unlocks at Level 5 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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More techniques

Modular ThinkingWhen the numbers are too big, shrink the universe.
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