What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 13

I hadn’t heard of the crossing-the-desert problem until I read Gamma (page 127). From Wikipedia:

There are n units of fuel stored at a fixed base. The jeep can carry at most 1 unit of fuel at any time, and can travel 1 unit of distance on 1 unit of fuel (the jeep’s fuel consumption is assumed to be constant). At any point in a trip the jeep may leave any amount of fuel that it is carrying at a fuel dump, or may collect any amount of fuel that was left at a fuel dump on a previous trip, as long as its fuel load never exceeds 1 unit…

The jeep must return to the base at the end of every trip except for the final trip, when the jeep travels as far as it can before running out of fuel…

[T]he objective is to maximize the distance traveled by the jeep on its final trip.

The answer is, if n fuel dumps are used, the jeep can go a distance of

H_n = \displaystyle 1 + \frac{1}{3} + \frac{1}{5} + \dots + \frac{1}{2n-1}.

Since the right-hand side approaches infinity as n gets arbitrarily large, it is possible to cross an arbitrarily long desert according the rules of this problem.

green line

When I researching for my series of posts on conditional convergence, especially examples related to the constant \gamma, the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.

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