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A piratical plank-stacking puzzle bedevils Liane and Tyler.
June 01, 2005
URL:http://www.drdobbs.com/parallel/the-pirates-cantilever/184406125
Dennis is a professor of computer science at New York University. His most recent books are Dr. Ecco's Cyberpuzzles (2002) and Puzzling Adventures (2005), both published by W. W. Norton. He can be contacted at [email protected].
Solution to Last Month's Dr. Ecco
The tall man who came knocking at Ecco's apartment cut quite a figure. A titanium prosthetic had replaced his left leg below the knee, but this big man made his way around with a vigor that few fully legged men could match. He wore a dark bushy beard and shaggy hair. He looked quite scary except that his dark eyes twinkled and he sported very distinctive smile lines.
"Call me Scooper," he said by way of introduction. "After my, er, accident, I got this leg and started walking around with a wooden cane. Soon after, my buddies gave me an eye patch, joking that I should put a jolly roger on my jeep's antenna. I had some time on my hands, so I decided I would study pirates. I found some pretty modern ones, operating as late as the 1930s off the Carolina coasts and specializing in attacking yachts. They did this with stealth and intelligence. Never did they physically harm any one. Sometimes their booty was difficult to divide, however.
"In one heist for which I have partial records, almost half the value of the theft was embodied in a single, very well-known diamond—so well known in fact that they couldn't sell it right away. They decided to award the diamond to the pirate who could win the following contest. They were very mathematically adept pirates, so I'm convinced someone won. I want to know how.
"Here is the contest: There is a set of wooden planks P. Given P, each contestant is to construct a structure of planks that cantilever off a flat dock.
"Is someone going to walk the plank?" 11-year-old Tyler asked.
"No," Scooper replied with a smile. "They didn't want anyone to walk the plank yet, but they wanted to make the pile extend out as far as possible from the dock without requiring glue, ropes, or attachment. That is, they wanted to create a pile of planks and have them stay put assuming no vibrations or wind."
"If all the planks are the same size and weight, and each plank can sit on only one other plank and support only one plank, then this is the 'Book Stacking Problem' and is nicely analyzed on http://mathworld .wolfram.com/BookStackingProblem.html," Liane volunteered.
"Interesting," Scooper said. "Could you explain how that goes?"
"The basic idea is simple," Liane replied. "One plank can extend half-way out in a cantilever without tipping; if you have two planks, then the first one extends 1/4 the way out and the other extends 1/2 way out beyond the first as you can see in the drawing in Figure 1.
"Let's analyze this. Suppose each plank weighs W and is of length L. The net weight of each plank is at its center. The torque around the dock edge is therefore +(L/4)W due to the bottom plank and -(L/4)W because of the top plank. So the net torque is zero. Similarly, the center of the top plank rests on the outer edge of the bottom plank, so it will not flip over. More planks allow an arbitrarily long cantilever, given enough planks."
"Physics is always a surprise," Scooper said.
"1. Our pirates are not so bookish, however. They have 10 thick and rigid planks of lengths 1 through 10 meters and weighing 5 through 50 kilograms they had just captured from a French yacht. Further, according to their description of the contest, all planks should share the same orientation—they should lie horizontally and their lengths should be perpendicular to the dock edge. To start with, they required that a plank could lie on only one other plank (or on the dock) and could support only one other plank. In that case, how far from the dock can a plank extend without tipping for the best structure?
"2. Now suppose that the pirates allow two or even more planks to lie on a supporting plank. Can you create a cantilever that goes even farther out?" (Hint: Yes, by more than a couple of meters.)
Liane and Tyler worked on these for an hour before coming up with an answer.
"Nice work," Scooper said, looking at their drawings. "Here is the hardest problem. Suppose there are no constraints about how planks can lie on one another? They need not share the same orientation, many planks can support a single plank, and so on. Then how far can the cantilever go out using just these 10 planks?"
I never heard the answer to this one.
For the solution to last month's puzzle, see page 92.
DDJ