Three-Sum Reduction: When to Use It

Start with the idea

A problem is a good match for Three-Sum Reduction when it asks for unique triples and k-sum reductions. Here the concrete task is: Return the number of unique triples summing to zero. The visible input [-1,0,1,2,-1,-4] is an example, not a definition of the whole problem. Ask what operations the solution must support and what the input guarantees. Check whether the input is sorted, whether you may reorder it, and whether moving a pointer preserves the answer you seek. The right choice follows from those guarantees, not from a familiar noun in the prompt.

The decisive invariant is: Sort, fix one value, and solve the remaining target with two pointers while skipping duplicates. Try to say why it applies to the task without quoting the implementation. Which fact becomes known after one update, and which possible answers remain? Two positions move according to a monotone fact: after one comparison, some candidates can be ruled out permanently. Name that fact. If that claim cannot be maintained on a legal input, the pattern is wrong or one of its preconditions is missing. A counterexample is more useful than a vague warning: change one property of the sample, trace the first two steps, and identify the exact claim that fails.

Compare costs before committing. The advertised bound is O(n^2) time · O(1) excluding output; identify the operation responsible for it and the extra memory it needs.