Fly plane around the world half fuel puzzle

There is an air force base on an island. You are given a plane to fly around the world without landing (except at the island where you started). The island lies on the equator and you will be flying around the equator, essentially a circle. The plane can carry enough fuel for half the journey. You can take help from other planes that can transfer fuel to your plane instantaneously while in flight. All planes fly at the same speed and carry the same amount of fuel, which again is just enough for a journey half way around the world. Find the minimum number of planes required so that one plane can circumnavigate the globe and all the refueling planes return to the island safely.
Credits: Martin Gardner

1 comment:

  1. A sequence for three planes is:
    Start three planes from the island. After going 1/8 around one plane refuels the two others with 1/4 of its tank, leaving it with 1/4 of the tank which is precisely enough to take that plane home. The other planes have a full tank again. Reaching 1/4 around one of the planes transfers 1/4 tank of fuel to the other one, leaving it with 1/2 tank again taking it precisely home. The remaining plane is the circling one and is full at 1/4 and can go to 3/4 with that fuel. The first returning plane refuels and start off the other way meeting the circling plane at its 3/4 position, where the circling plane is empty and the meeting one is half empty. Sharing their fuel they have both 1/4 tank left which can take them to the 7/8 position. The remaining plane has arrived there in the meanwhile with 3/4 tank giving 1/4 to the other now empty planes. All three planes have now 1/4 and can return home.
    What a trip!
    Two Planes( A & B suppose).Both planes will fly together,one plane(A) will transfer the half of its fuel to other(B) at 1/4 the distance and come back,the second(B) will complete the trip up to 3/4 then first(A) plane should start flying other way round and at 1/4 transfer any amount of fuel between 1/4 to 1/2 of the full tank to B and both planes will land at island safetly.
    The error in the above thinking is that the first tranfer amounts to a 1/4 tank, which allows plane B only to reach the 1/2 + 1/8 position which is not the 3/4 position as claimed. Recall here that a full tank with amount A (being a fraction between 0 and 1) will last for a fraction of A/2 of the trip distance. -sboehringer
    Regarding solution 2:
    Plane A cannot transfer any fuel to Plane B at 1/4 the distance, because that will leave it with less than half a tank, and it will not be able to return to the island. This problem is similar to the jeep problem, except that you can refuel from the other direction, and it can be shown that 2 planes will not suffice. (Plane A delivers 1/3 of a fuel tank to plane B after exhausting 1/3 of its fuel tank and returns to the island empty. Plane B is then able to fly for 4/3 of a tank, which only takes it to the 2/3 mark, while plane A can only perform the same refueling in the other direction for 1/3 of its tank, which is 1/6 of the flight, leaving a 1/6 gap.
    The first solution is correct.

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