Locker Puzzle Strategy for Winning Wallets Game
Explore a strategic approach to winning a challenging wallet-finding game in a locker room scenario. Can a team of 100 players ensure that each member finds their own wallet by following a specific strategy without communication? Dive into the mathematical puzzle and discover the probability of success with clever tactics and optimal locker examination methods.
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Presentation Transcript
The Locker Puzzle Eugene Curtin, Max Warshauer The Mathematical Intelligencer, Volume 28, Issue 1, December 2006, Pages 28-31 Presenter: Guan-Jie Chen Date: May. 9, 2023
Issue (1 / 2) Suppose I take the wallets from you and ninety-nine of your closest friends. We play the following game with them: I randomly place the wallets inside one hundred lockers in a locker room, one wallet in each locker, and then I let you and your friends inside, one at a time. Each of you is allowed to open and look inside of up to fifty of the lockers. You may inspect the wallets you find there, even checking the driver's license to see whose it is, in an attempt to find your wallet.
Issue (2 / 2) Whether you succeed or not, you leave all hundred wallets exactly where you found them, and leave all hundred lockers closed, just as they were when you entered the room. You exit through a different door, and never communicate in any way with the other people waiting to enter the room. Your team of 100 players wins only if every team member finds his or her own wallet. If you discuss your strategy beforehand, can you win with a probability that isn't vanishingly small?
Predigestion - - 10 players, each one can examine 5 out of 10 lockers wallet number
Improvement - - Players 1-5 examine lockers 1-5 Players 6-10 examine lockers 6-10
Strategy Player i opens locker i first - - finds number i in locker i, i = i, success finds number k in locker i, k i then opens locker k & checks whether the number in locker k is equal to i, recursively
Example (1 / 2) 6, 8, 9, 7, 2, 4, 1, 5, 10, 3 Locker 1 6 4 7 Player 1 Number 6 4 7 1 Player 4, player 6, player 7 are in the same cyclic order (4 tries) None of the other players will waste any tries on these lockers
Example (2 / 2) 6, 8, 9, 7, 2, 4, 1, 5, 10, 3 - If the cycles are (6, 4, 7, l), (8, 5, 2), (9, 10, 3) success - If there exists a cycle with 6 or more elements failure
Probability of Success with the Strategy (1 / 5) - a cycle with 6 elements
Probability of Success with the Strategy (2 / 5) - a cycle with 7 elements
Probability of Success with the Strategy (3 / 5) 10 players:
Probability of Success with the Strategy (4 / 5) 100 players:
Probability of Success with the Strategy (5 / 5) 2N players: 1 - ln 2 0.306853
