# Find the minimum number of aircraft

On Bagshot Island, there is an airport. The airport is the homebase of an unlimited number of identical airplanes. Each airplane has a fuel capacity to allow it to fly exactly 1/2 way around the world, along a great circle. The planes have the ability to refuel in flight without loss of speed or spillage of fuel. Though the fuel is unlimited, the island is the only source of fuel.

What is the fewest number of aircraft necessary to get one plane all the way around the world assuming that all of the aircraft must return safely to the airport? How did you get to your answer?

*Notes:*

(a) Each airplane must depart and return to the same airport, and that is the only airport they can land and refuel on ground.

(b) Each airplane must have enough fuel to return to airport.

(c) The time and fuel consumption of refueling can be ignored. (so we can also assume that one airplane can refuel more than one airplanes in air at the same time.)

(d) The amount of fuel airplanes carrying can be zero as long as the other airplane is refueling these airplanes. What is the fewest number of airplanes and number of tanks of fuel needed to accomplish this work? (we only need airplane to go around the world)

**fewest number of aircraft is 3**!

Imagine 3 aircraft (A, B and C). A is going to fly round the world. All three aircraft start at the same time in the same direction. After 1/6 of the circumference, B passes 1/3 of its fuel to C and returns home, where it is refueled and starts immediately again to follow A and C.

C continues to fly alongside A until they are 1/4 of the distance around the world. At this point C completely fills the tank of A which is now able to fly to a point 3/4 of the way around the world. C has now only 1/3 of its full fuel capacity left, not enough to get back to the home base. But the first ‘auxiliary’ aircraft reaches it in time in order to refuel it, and both ‘auxiliary’ aircraft are the able to return safely to the home base.

Now in the same manner as before both B and C fully refuelled fly towards A. Again B refuels C and returns home to be refuelled. C reaches A at the point where it has flown 3/4 around the world. All 3 aircraft can safely return to the home base, if the refuelling process is applied analogously as for the first phase of the flight.

**3**

** Different method**

**P is main craft now X and P goes till 1/6 and X refuels P and return back and refuelled and go together with y (opposite direction ) with get refuel from X at point 5/6 (X mean while goes back refueld and come back with Z)then Y goes to 4/ 6 and refuel P then both come to 5/6 where they get fueled from X and Z to come back to start.Genius is back xd **

Two planes needed and three tanks of fuel.

Plane 1 and 2 start off together. At the 1/4 mark in the flight around the world, plane 2 refuels plane 1 with half of its tank, and returns to the airport. At this point, plane 1 has a full tank and can make it to the 3/4 mark around the world. Plane 2 has just enough fuel to make it back to the airport.

When plane 1 reaches the halfway point around the world, plane 2 has reached the airport and can refuel and fly out again, in the opposite direction.

Plane 1 and 2 reach the 3/4 mark around the world at the same time (plane 2 only had to fly 1/4 of the way around the world since it flew in the opposite direction). Plane 2 again refuels plane 1 with half of its tank, and both have just enough fuel to make it back to the airport.

You made a mistake here: When Plane 1 and 2 reach 3/4 mark around the world at same time, plane 2 has already consumed 1/2 tank of fuel ( recall: Each airplane has a fuel capacity to allow it to fly exactly 1/2 way around the world). If plane 2 gave the rest half tank to plane 1, plane 2 will fall into the ocean.

Same applies to the first part. If plane 2 gives half its tank at the 1/4 mark, then it is empty.

The minimum is three.

Start three planes from the island (yours and two others). After going 1/8 around, one plane refuels both yours and the other plane with 1/4 of a tank each, leaving it with 1/4 of a tank which is precisely enough to take it home.

Two planes continue with a full tank again.

Reaching 1/4 around the other plane transfers 1/4 tank of fuel to yours, leaving it with 1/2 tank, to take it precisely home again.

Your plane is now 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 your plane at its 3/4 position, where your 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!

### Your Answer

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