Puzzles/Physics puzzles/Flying around the World/Solution
The minimum is three. Two planes obviously do not suffice. 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 and 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 safely.
The error in the above thinking is that the first tranfser 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.
Solution 3. Split the whole circumference in to 12 units. Each plane has a full tank capacity to reach 6 units. Start two planes A and B travel 2 units , transfer the 2 unit fuel to plane A and use the remaining 2 units to come back. now plane A would be able to travel 8 units. using Plane B and C start the other direction at 2 unit distance transfer 2 unit fuel to plane C and plane B returns back home for refueling. plane B travels two more unit transfers the 2 units to plane A and now plan A and C would travel together to come back to two units short of destination. Now plane B can travel two units refuel A and C with one unit each. So plane A and C can now come one unit closer. then do a refueling from Plane B
Regarding solution 3:
The solution is incorrect. When A and C are 1 unit away from home with empty tanks, B is at the same location with them with 1 unit of fuel left. There is no extra time for B to fly home and come back to refuel A and C.