Aircraft refueling problem

Conditions:(1) Each plane has only one fuel tank,
(2) between the planes can refuel each other (note is mutual, no refueling machine)
(3) A tank of oil for a plane to fly around the earth half a circle,
Problem:
At least a few planes are needed to get at least one plane round the Earth back to the airport at takeoff. (all aircraft take off from the same airport and must be safely returned to the airport, not allowed to land halfway, no airfield in the middle)


Answer:Three aircraft, taking off five sorties.


parsing:This problem has been circulating on the internet for a long time, there are various kinds of answers, there are three, said five, and said six.
My understanding is:
If you say the number of takeoff, then it takes five times. But the actual number of aircraft required is 3.


The main point of the breakthrough is:(1) Because the requirement is that an aircraft can fly a circle, so other refueling aircraft can fly clockwise, can also fly counterclockwise. If you do not think of this, then you will come to the answer: Impossible to complete a lap of flight.
(2) The aircraft responsible for refueling can imagine that its fuel tank is connected to a plane to be flown (named aircraft a), and that all oil is supplied by refueling aircraft before the refueling plane leaves. The crux of the matter is when the refueling aircraft should return.


through the above two key points analysis:
Our main ideas are as follows:
A few planes were taken off at the same time, the aircraft a was given a certain distance, and then returned, when the last plane returned, aircraft A was still full of oil.
After the flight A has been sent out, the returned aircraft will fly in the opposite direction, connecting aircraft A at the point where aircraft a is exhausted, to ensure that aircraft a can return to the airport.


Then the first time to fly, in addition to aircraft a, there should be several planes take off together, responsible for refueling.
Set the length of the fly round to s
If only two planes fly (plane A, B)

b The place to return should be:

Considering that all 2 planes use this aircraft's oil, the plane can fly farthest from the X most must return, so there is the following equation. (Aircraft oil volume is S/2).

S/2 = 2*x + X--and x = S/6 That means a plane can only send aircraft a to the S/6. Aircraft a can reach S/6 + S/2 = 2S/3 at the end of oil. The airport is also S/3, and the airport is sent only S/6, should also be sent farther.


If there are three planes flying (aircraft A, B, C)

b The place to return should be (x1):

Considering that all 3 planes use the oil of this aircraft, the plane can fly farthest x1 The most must return, so there is the following equation. (Aircraft oil volume is S/2). S/2 = 3*x1 + x1 and x1 = S/8, B returns at S/8

c the place to return should be (X1+X2):

Considering that all 2 planes use the oil of this aircraft, the plane can fly farthest x2 the most must return, so there is the following equation. (Aircraft oil volume is S/2).
S/2 = 2*x2 + x2 + x1--x2 = S/8, C returns at (S/8+S/8)
At this point the aircraft a has flown out of S/8 + S/8 = S/4, after its consumption of all the oil, can fly to S/4 + S/2 = 3s/4 place. There are also s/4 from the airport, which is already equal to the departure time, so consider two planes to refuel and then pick up the plane a.
Aircraft a run out of oil at the airport and S/4, then we send a plane B to S/4, aircraft A and aircraft B in S/4 meet can again fly S/8, then out of the plane C to S/8. Plane C left the oil in S/8 for 3S/8, just enough to fly three planes back to the airport.


So, to sum up, a total of 3 aircraft, take off five times.