question
You are on an island with an airport. The airport is home to an unlimited number of identical planes. Each plane has enough fuel capacity to fly half way around the world. The planes can refuel in-flight, instantly, from another planes' fuel tank. The island has unlimited fuel.

What is the fewest number of planes necessary to get one plane around the world assuming all planes need to return safely to the island?
Planes can depart in the opposite direction from where they initially took off as well.
Note: In the answer below I've left some of the fractions unsimplified so that its easier to follow.

You will need a minimum of 3 planes to fly around the world.

Suppose we have 3 planes A, B, & C start at the same time in the same direction. After going 1/8 around the circumference of the earth, plane C refuels planes A & B with 1/4 of its tank to each, leaving it 1/4 tank to fly back to the island. Planes A & B continue flying, now with a full tank of fuel. After reaching 2/8 distance around the world, plane B refuels A with 1/4 of its tank. Having used 1/4 already and transferred 1/4 to A, B has 2/4 of its tank to make the journey back to the island. Plane A now has a full tank of fuel and is 2/8 distance around the world.

Plane A continues to fly and reaches 6/8 distance across the world. It meets plane C at that point, that had headed back to the island, refueled, and set out in the other direction around the world. Plane C refuels A with 1/4 of its tank, leaving just 1/4 for itself as well. Plane B which had also reached the island, refueled, and headed in the same direction as C, meets planes A & C at 7/8 distance around the world. B refuels A & C with 1/4 of its tank to each leaving 1/4 for itself. All three planes now have 1/4 tank and can make the journey back home.

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