Solution

Imagine having a big tank with enough gas for a round trip and enough room for going through the motions of emptying every gas station on your way. Start at any station and mind to record the amount of gasoline on reaching gas stations on your way around. At the end of the trip, when you pull into the station of departure with the original amount of gas, check your list. The station marked with the least number is the one where you want to start on an empty tank.

Why? Is that obvious?

Assume the least amount in question was, say, K gallons. Assume also you can start your second trip with this amount of gasoline. As before, as you go, record the amounts of fuel on reaching the gas stations on your way around the trek. The list of numbers will be the same as on the first trip but shifted circularly. Thus if you remove K from each of the numbers on the list, none of the numbers will become negative.

References