A Gas Station Problem

Prove that for any number of gas stations arranged on a circular track such that all the gas stations combined have just enough gas to get you all the way around the circle that there must exist a valid gas station that you can start at with an empty tank and go all the way around the circle.

Well of course...You start at the empty gas station....then you fill up your car's tank, at which point there is now an empty gas tank....under the ground perhaps, but none the less it is empty...now you may complete your lovely, sunny trip around the circle and return to the gas station...oh wow!...I just started empty and ended empty!

Interesting solution Mr. Ruff, but I regret to inform you that you did not traverse the circle.

More correctly, you did not prove that you could traverse the circle from that gas station.

Prove to me that I have not traversed the circle. My point still stands.

I asked for the solution for any number of gas stations.
Your solution only holds for a the circular track of exactly 1 gas station. In order for your solution to be correct, it would have to be provable that only the circular track of 1 gas station existed.

Let us, for example, apply your solution to the case of 2 gas stations. We'll call one gas station A and another gas station B. Let's say that A has 0 gas. Therefore, by definition of the circular track, B must have 1.00 gas (We're letting the amount required to traverse the circle be represented by a percentage). Your solution claims that A would be the correct solution. So I start at A with an empty tank, and fill my tank from the empty gas station, start driving around the circle only to not go anywhere.

Therefore your solution does not hold for 2 gas stations, and the argument can be expanded similarly to any number of gas stations larger than 1. Hence my statement that you did not traverse the circle; you did not explain to me how to traverse the circle in all cases, only one very specific case.

However, if you meant that there was no observable difference between starting at the initial gas station and ending at the initial gas station on a circular track then you would be correct. That does not changed the fact that you have not actually gone anywhere.

Before even reading your entire response, I couldn't help but notice that you are trying to tell me that I need to prove the existence of this singular, circular track and the single gas station on this track. I don't need to prove that it exist, as that is completely off topic, seeing that it is the very basis that your puzzle is based on. Asking someone to prove that something exist, when the point is to be finding a solution to a puzzle is absolutely ridiculous. In your puzzle, there is a circular track, with some # of gas stations. I have done this, and yet you are persistent in pointing out pointless facts or variables. I am simply answering the puzzle with a possible solution, and you are trying your darndest to alter the original puzzle.

Note to the forum: this question is still considered unsolved. Solutions will continue to be accepted.

Thank you for your submission Kenny Ruff. Your participation is greatly appreciated.