A jeep carries fuel for exactly 100 km in a full tank and may leave fuel in the desert to pick up later. There are 4 tank loads of fuel at base and none anywhere else. Travelling one way, with no need to return, how far can the jeep get?

A jeep carries fuel for exactly 100 km in a full tank and may leave fuel in the desert to pick up later. There are 4 tank loads of fuel at base and none anywhere else. Travelling one way, with no need to return, how far can the jeep get?

Approach: Measure distance in tank ranges and ask how many crossings of a stretch are needed while n loads of fuel still have to be carried forward.

3520/21 km. Measure distance in tank ranges of 100 km. While n loads still have to be moved forward over a stretch of length d, the jeep makes n forward and n - 1 return crossings, leaving fuel at depots along the way, so the fuel burnt is (2n - 1)d. Setting (2n - 1)d = 1 uses up exactly one load and leaves n - 1, so the ground covered while n loads remain is 1/(2n - 1). With 4 loads the total is 1 + 1/3 + 1/5 + 1/7 = 176/105 tank ranges, which is 17600/105 = 3520/21, about 167.6 km. The naive answer of 400 km assumes fuel moves for free. Because the harmonic style series 1 + 1/3 + 1/5 + ... diverges, any distance is reachable, at a fuel cost growing exponentially with it.

Follow-up: How far can the jeep get with 4 loads if it must return to base afterwards?

Key concepts: fuel depots, harmonic style series, crossing count.