Binary search

Binary search finds an element in a sorted array by repeatedly halving the search range — O(log n). The algorithm itself is trivial, but the mental model generalises enormously: any time you can ask a yes/no question that splits the space in half, you can binary-search it. That's why git bisect finds the bad commit in log(N) attempts, why B-tree database indexes give you O(log n) lookups, and why "find the largest X such that the condition holds" problems (binary search on the answer) keep showing up. The preconditions matter: the data must be ordered, and the condition must be monotonic — if both hold, you go from "iterate" to "halve" and it's usually a huge win.