TERMINATOR

**Objective: Identify the counterfeit coin and determine if it's heavier or lighter.** **I choose: Three weighings.** **Defense:** Two weighings plus a guess is insufficient. After two weighings, you've eliminated possibilities but face ambiguity in most scenarios. With 12 coins and two unknowns (which coin AND whether it's heavy or light), you need maximum information. Three weighings provide 3³ = 27 possible outcomes, sufficient to distinguish among 24 scenarios (12 coins × 2 weight directions). The standard algorithm: divide into groups of 4, systematically narrow through comparison. Two weighings guarantee only ~9 distinguishable outcomes—inadequate. Three weighings: certain success.

**I pick three weighings.** Restating the problem plainly: I need to identify which of 12 coins is different AND whether it's heavier or lighter. That's finding 1 answer from 24 possibilities (12 coins × 2 states). Three weighings give 3³ = 27 possible outcomes—just enough information theoretically. Two weighings plus a guess gives only 3² = 9 outcomes, leaving 15 possibilities uncovered. Even if I narrow it down significantly, guessing among remaining candidates is unreliable. Three weighings, using a systematic divide-and-conquer strategy (splitting into groups of 4), guarantees I'll find the answer with certainty.