Extremal principle: lean on the most extreme element
Look at the largest, smallest, or most-something element first. Many combinatorial proofs collapse the moment you fix attention there.
Method · Extremal Principle
Intro
When you’re stuck on an existence, optimization, or counting problem — “does such an object exist?” or “what’s the max / min number of $X$?” — focus on the most extreme element: the maximum, the minimum, the longest path, the smallest distance. Extremality forces structure (the max-degree vertex must touch everyone; the smallest positive distance can’t be undercut), and that structure either contradicts a constraint (proving a bound) or constructs the answer. Olympiad-flavored but interview-direct: graph-degree puzzles, tournament problems, and tree arguments all collapse the moment you pick the right extreme.
✓ Intro · expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine — being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
✓ Try first · expand
Worked example
At a party with 10 guests, every guest shook hands with some (possibly zero) of the other guests. No one shakes their own hand and a given pair shakes at most once. What is the maximum number of guests who could all have different handshake counts?
✓ Worked example · expand
Practice 1 of 3Type a fraction, decimal, or expression — mathjs parses it.
✓ Practice · expand
Reflection
When you spot an existence or optimization problem, what cues tell you to reach for the <em>extreme</em> element rather than (say) pigeonhole or invariants? In your own words, why does picking the maximum or minimum so often crack the problem open?