Problem of the Week: Feeding Time

By Department of Mathematics

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 panel of faculty judges, 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 or put solutions in the marked envelope in the hallway outside Glatfelter 215. This problem was posted on Friday, February 1 and solutions are due on Friday, February 8 by 5:00 p.m.

THE PROBLEM:

Every night I feed my pet walrus exactly one of the following for dinner: fish, pizza, or tacos. My walrus loves pizza and tacos, but fish are better for him, so I adhere to the following rule: if I feed my walrus something other than fish on any particular night, I will feed him fish the following night. (If I feed my walrus fish on any particular night, I can give him anything the following night).

According to the above rule, then, the following is a valid “10-night menu”:

F, P, F, F, T, F, F, T, F, P

The following is not a valid “10-night menu”:

F, P, T, F, T, F, F, T, T, F

QUESTIONS:

How many valid 10-night menus are there? If n is a positive whole number, how many valid n-night menus are there?

If you’re feeling adventurous, generalize to the case where there are k “unhealthy” options (that is, k things other than fish that I might feed my walrus).

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Author: Gettysburgian Staff

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