# Problem of the Week: The Islands

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, March 1 and solutions are due on Friday, March 8 by 5:00 p.m.

THE PROBLEM:
Picture a ﬁnite collection Γ of small islands connected by some network of bridges. Assume that it is possible to get from any island to any other island (not necessarily directly!) using the bridges. Now suppose an evil magician applies a spell L that turns each bridge into an island, and puts a bridge between two “new” islands if and only if the two bridges that became the islands connected to a common “old” island.
THE QUESTION:
Find some possible collections Γ for which the spell L leaves Γ unchanged in the sense that, after the spell is performed, we have the same number of islands, connected the same way. Can you ﬁnd any collections Γ for which doing L once changes Γ, but doing L twice restores Γ to how it was originally?

Author: Gettysburgian Staff