How Many Planes Are Needed to Fly Around the World with Limited Fuel?

[Werg22]

A group of airplanes is based on a small island. The tank of each plane holds just enough fuel to take it halfway around the world. Any desired amount of fuel can be transferred from the tank of one plane to the tank of another while the planes are in flight. The only source of fuel is on the island. Assume no time is lost while refueling either in the air or on the ground, and that the planes have the same constant ground speed and rate of fuel consumption. At any time, a plane can reverse course and head back to the island base. What is the smallest number of planes that will ensure the flight of one plane around the world and that all planes return safely to the island base?

 

[Jimmy Snyder]

I don't know if this is smallest, but I can do it with

Spoiler

6 planes. 3 planes take off going west. After 1/8 of the circumference, each of the planes has 3/4 of a tank. One of the planes gives a 1/4 tank to each of the other planes, leaving it with 1/4 tank, just enough to get home which it now does. The remaining two planes now have full tanks. When they have gone 1/4 of the circumference, they each have 3/4 of a tank. One plane gives 1/4 tank to the other leaving it with 1/2 tank, just enough to get home which it now does. The remaining plane has a full tank. It travels to a point 1/2 around the circumference and still has a half tank. At that point, 3 planes take off going east. After 1/8 of the circumference, each of the planes has 3/4 of a tank. One of the planes gives a 1/4 tank to each of the other planes, leaving it with 1/4 tank, just enough to get home which it now does. The remaining two planes now have full tanks. When they have gone 1/4 of the circumference, they each have 3/4 of a tank. And the plane that had traveled west just meets up with them it's tank just empty. Each of the eastward planes gives the westward plane 1/4 tank. Now all three planes have 1/2 tank, just enough to get home, which they all proceed to do.

 

[some_one]

disregard that i forgot that they all should come back safely

 

[Werg22]

Nice try, jimmysnyder. But there are ways of doing with fewer planes.

 

[dipinsingh]

it can be done with 3 planes

let the three planes be A,B & C
let the distance to be traveled to go round the world once be x units

now each plane can travel x/2 units

Now the the 3 planes travel x/8 units.C gives enough fuel (1/4 tank) to A,B to make up for the distance they had traveled till now and comes back to the island.A & B travel an additional distance x/8 unitsand B gives A enough fuel( 1/4 tank) to A to make up for the disatnce x/8 units traveled and returns to the island withfuel of 1/2 tank.Now A has full tank and he can travel X/e distance more. When A has traveled x/2 units i.e. half way around the word, it had 1/2 tank fuel with it for going up to x/4 distance i.e. up to 3/4 around the world.and By that time B & C have come back to Their Base.

At that time( When A has traveled x/2 units) , B and C refilled their tanks in the island.They move in the opposite direction to meet A i.e move in the direction exactly opposite to the direction they had traveled previously.In this way they have to just travel just x/4 units to meet A.
B &C travel x/8 units.C gives B enough fuel ( 1/4 tank ony as B has got that much empty space in tank) to make up for the distance they have traveled and goes back.B travels x/8 units more where it meets A and gives it enough fuel ( 1/4 tank) to move x/8 units & returns to the island with 1/2 tank fuel. By the time A travels x/8 units,C travels x/8 units from Base & gives A enough fuel ( 1/4 tank) to return to the island and returns to the island along with A.