# Proof by induction question about fuel depots

You are asked to drive a lunar rover around the moon (which is just a circle in this question). There are (finitely many) fuel depots on the way, with the total amount of fuel stored in them enough to get around the moon exactly once. Show that there exists a depot from which you can start driving and travel the whole way around the moon, picking up fuel at each depot as you pass, without running out of fuel between depots.

Can I have a hint for this? I've tried induction but I'm having trouble with the induction step.

• Induction is of little use (and gives little insight) here, though it is possible ... – Hagen von Eitzen Apr 21 '14 at 11:18

Start anywhere and assume you have some unspecified sufficient (i.e. you will never run out of fuel along the trip) amount $x$ of fuel in the tank initially. As you refill during the trip exactly as much as you consume, you will end up with the same amount $x$ of fuel after the full round. During the trip, the fuel continuously decreases, except at the depots, where it suddenly increases. Let $x_1,\ldots, x_n$ be the amounts of fuel in your tank when arriving at the $n$ depots. By assumption, $x$ was large enough so that all $x_i$ are $\ge 0$. Among all suitable choices of $x$, there is one we cannot go below: For some $x$, at least one of the $x_i$ will become $=0$. Now what if we start at this (or one of these, if there are several) depot with empty tank, fill from the depot and start?