Problem of the Week: Espionage!
Editor’s Note: The Department of Mathematics at Gettysburg College hosts a problem of the week challenge to determine each semester’s Paul Mugabi problem-solving award recipient(s). Each week’s entries are scored by a faculty judge, and winner(s) from each week will receive a Problem Of the Week (P.O.W.) button. The Gettysburgian is not involved in or responsible for accepting or evaluating students’ submissions to this contest.
THE RULES:
The contest is open to all Gettysburg College students. Up to three people may work together on a submission. Make sure your name is on your submission and that any sources are properly cited. Send solutions to bkennedy@gettysburg.edu. This problem was posted on Friday, January 28 and solutions are due on Friday, February 4 by 5:00 p.m.
THE PROBLEM:
We are officers sending secret agents A, B, and C on a mission to Mystery Mountain. When A, B, and C meet at Mystery Mountain, they will need to enter three long numbers x, y, and z into three different locks to get in.
Before the mission starts, we have to decide which numbers to tell each agent. (For example, we could tell each agent all three numbers, or we could tell each agent just one of the numbers, or we could tell two of the agents all the numbers and the third nothing at all, and so on.)
Each of A, B, and C has a 1/3 probability of being captured by Dr. Sinister on the way to Mystery Mountain. Assume that, if any agent is captured, they will reveal all that they know, and will not arrive at the meeting at Mystery Mountain.
We want whichever agents arrive at Mystery Mountain to be able to get in, and we don’t want Dr. Sinister to learn how to get in.
THE QUESTION:
What information can we give to A, B, and C to maximize the chances of our desired outcome? Answer, and explain.
