Math Chat: Crossing a Rickety Bridge at Night By Flashlight

Old bridge-crossing challenge (Tiku Majum-der and Conrad Weiser)

"There are four people who need to cross a river at night. There is a bridge that can only hold up to two people at a time. There is one flashlight that must be used when crossing. (It is extremely dark, and someone must bring the flashlight back to the others; no throwing anything, no halfway crosses, etc.).

"The four people take different amounts of time to cross the river. If two people cross together, they travel at the slower person's rate. The times are 10 minutes, 5 minutes, 2 minutes, and 1 minute for each of the four individuals." How fast can they complete the passage? Can they do it in 19 minutes? in 17 minutes?

Answer

They certainly can do it in 19 minutes, by having 1-minute "Speedy" escort each of the other three across the bridge. That's 2 + 5 + 10 = 17 minutes crossing, plus Speedy's two 1-minute return trips, for a total of 19 minutes.

Amazingly, Aubrey Dunne, James Fahs, Hany Farid, Jocelyn Granger, Heather Anne Harrison, Chad Heeter, Sam Ragucci, Erik Randolph, John Robertson, and Mike Bevan found a 17-minute solution. First 1 and 2 cross, 1 returns, 5 and 10 cross, 2 returns, 1 and 2 cross, for a total of 2 + 1 + 10 + 2 + 2 = 17 minutes.

Princeton freshmen Amanda Fulmer, David Greco, Michael Lindahl, and Graham Meyer explain that this method wins essentially because 2 - 1 < 5 - 2. If it took the second person 4 minutes instead of 2, the best strategy would be to have Speedy escort the other three, because 4 - 1 > 5 - 4.

Kenneth Eggert suggests the more imaginative solution of having Speedy carry each of the other three across in a total time of 5 minutes flat.

New intelligence challenge

The famous mathematician Steve Smale has listed 18 important math problems for the next century ("The Mathematical Intelligencer," Spring 1998). No. 1 is the notorious Riemann Hypothesis, related to prime numbers. No. 18 (this week's question) looks more like philosophy than math but is part of the great conversation of modern mathematics "What are the limits of intelligence, both artificial and human?"

* Send answers and new questions by e-mail to: Frank.Morgan@williams.edu