Suppose a race track in the shape of a simple closed curve (i.e. it does not intersect itself), and suppose that distributed along it, in any way whatsoever, are N gas stations. Suppose that a race car needs L liters of gas to go completely around the track, and that the sum of the gas available at the N stations is exactly L.
Consider that the car cannot move without gas (*assume it can't travel by momentum alone, thus it MUST need gas to move*), and that it has constant mileage (consumption of gas is directly proportional to the distance the car moves, and depends on nothing else).
Prove that there exists at least one gas station such that, starting from it, the car can do a full lap.
Idea #1: The problem states: "Prove that there exists at least one gas station such that, ...." SUPPOSE NOT.
Assume the statement is false and show that it leads to a condradiction. I think this has already been shown. This is basically a restatement of what Danny, Charlie and others have already said.
Idea #2: Inductive reasoning.
If N=1, just one gas station, the statement is clearly true, just start at the gas station and do a lap. Perhaps someone more clever than I am can show that IF it's true for N-1 gas stations, then it's also true for N. Hmm, how about this: say you have N stations; pick any station such that if you started at it (and called it station i), you could at least make it to the next station i+1. Now imagine starting over, take all the gas at station i, transfer it to station i+1, and delete station i from the course. Now you have a track with only N-1 stations which we know we can do successfully because we assumed it as part of inductive reasoning. Now go back to N stations, recreate station i, transfer the gas back from i+1 to i. Surely the ability to go around the track successfully can't be hindered by getting the gas earlier than we would have if we were travelling around the N-1 station course. QED
Posted by Larry
on 2004-06-04 00:17:09