50 Trucks with Payload

Problem

  Given a fleet of 50 trucks, each with a full fuel tank and a range of 100 miles, how far can you deliver a payload? You can transfer the payload from truck to truck, and you can transfer fuel from truck to truck. Assume all the payload will fit in one truck.

Solution

We want to use as little fuel as possible so we try minimize the number of trucks we use as we go along. Let’s say we start with all 50 trucks with full fuel (5000 miles range).

For each mile, we lose 50 miles in range. After two miles, we lose 100 miles leaving us with 4900 miles. This can be supported by 49 trucks so we drop one truck.
As you can see for every 100 miles we lose in range, we drop a truck.

50 trucks: 100/50
49 trucks: 100/49
…

Total distance = 100/50 + 100/49 + 100/48 + … + 100/2 + 100/1 (harmonic series) = 449.920533833