- #26

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I see the mistake now!I like the gist of this but I didn't follow:

It is not that each cannister must be ##\gt L## it is merely that problems can occur if there is even one of those cases-- for any selected sample path, if you hit one of those 'extra long' gaps and don't have enough gas in the tank at the time you encounter it, then you fail.

The clarification from @mfb is something I really didn't considerate. I'm having trouble my initial logic, so I'll try a different approach:

Considering U = units of distance and F = units of fuel. Since the total amount of fuel is always enough for the car to complete one cycle, we can assume that the amount of units travelled is the same amount of units of fuel used, which means that, for the complete circuit, ##X = U## (from starting point A to final point A): ##U_{a,a} = F_{a,a}##

(I'm not sure if it's necessary to add but that basically means that we can find a relation of U = kF, where k is a real constant. We are assuming k = 1 for simplification purposes)

Since the car starts without gas, it necessarily has to pick the starting point at one of the canisters.

Assuming N = 2, we have two canisters A and B. Assuming A to be the starting point. The car will fail only if the amount of units travelled (from A to B) is greater than the amount of fuel he has available on that interval, which means ##U_{a,b} > F_{a,b}##. Having in mind that ##U_{a,a} = U_{a,b} + U_{b,a}## and ##F_{a,a} = F_{a,b} + F_{b,a}##, and we know that the whole circle must wield ##U_{a,a} = F_{a,a}##: if the route AB makes the car run out of fuel before reaching canister B, then route BA makes the car arrive in canister A with more than enough fuel: ##U_{b,a} < F_{b,a}## to compensate. Therefore the car could simply start from point B.

This always applies, no matter how many canisters we have. We know it's possible to divide all the fuel available and also divide all the lenght of the track in the same number of partitions. No matter how many canisters we have, there is always a possible starting point. If there is a portion of the track between two canisters (1 and 2) that would make the car run out of fuel before reaching the second canister, it will be compensated by another portion of the track where the car will reach the second canister with more fuel than necessary (that "extra" amount is exactly how much the car lacked in the first situation).

(I had some trouble taking these out of my head; I apologize if it got too confusing)