How far can a jeep cross a fuel-less desert if it can cache gas along the way?
A jeep's tank holds enough fuel to travel exactly 1 unit of distance, and it can carry no more. Fuel is freely available at base, and the jeep can make multiple trips, dropping fuel caches in the desert to draw on later, but it must survive on only the fuel it's carrying or finds cached. Using unlimited fuel at base and unlimited trips, how far can a single crossing ultimately reach?
Reveal the answer
There's no limit: with enough supply trips laying down caches, the maximum reachable distance grows without bound, though painfully slowly, tracking half the harmonic series, so reaching even twice the tank's range already takes several trips, and ten times the range takes an astronomically large number. It was first analysed rigorously by mathematician Nathan Fine in 1947.
— Nathan J. Fine, The Jeep Problem — The American Mathematical Monthly, 1947