Count Open Lockers After 100 Rounds

A hallway contains **100 lockers**, numbered 1 through 100. All begin closed. - In round 1, open every locker. - In round 2, close every second locker. - In round 3, toggle every third locker: an open locker closes, and a closed locker opens. - Continue in this way through round 100, toggling every locker whose number is a multiple of the round number. After all 100 rounds, how many lockers are open? Explain why your count follows from the operations rather than relying on a round-by-round simulation. ### What a Strong Answer Covers - Which rounds affect one particular locker. - How the parity of its number of toggles determines its final state. - A counting argument that accounts for all lockers from 1 through 100, including both endpoints. ### Follow-up Questions - Why do most toggle rounds for a given locker occur in pairs, and when does that pairing leave an unpaired round?