{"id":"fill-cups-closed-form","text":"`min_seconds` computes the answer as `max(max(amount), ceil(sum(amount)/2))` in O(1) time rather than simulating the greedy filling process, because both lower bounds are provably achievable.","truth_value":"IN","source":"entries/2026/06/06/minimum-amount-of-time-to-fill-cups-solution.md","source_url":"","source_hash":"","justifications":[],"dependents":[],"metadata":{},"created_at":"","updated_at":"","reviewed_at":"","verified_at":"","retracted_at":"","explanation":{"steps":[{"node":"fill-cups-closed-form","truth_value":"IN","reason":"premise"}]}}