Around the World

There is an island on a planet and infinitely many planes on it. You need to make one of these planes fly all around the world and land back to the island. However, each of the planes can carry fuel which is enough to travel just half of the way, and fuel cannot be stored anywhere on the planet, except for the island and in the planes. Assuming the planes can refuel each other mid-air, and all of them must eventually arrive safely back on the island, how many of them do you need to accomplish the task?

SOLUTION

3 planes are enough, label them A, B, C. They leave the island simultaneously in a clockwise direction, and after 1/8 of the way, A refuels B and C completely, then turns back towards the island. B and C continue to fly until they get to 1/4 of the way, where B refuels C completely and turns back towards the island. When C gets mid-way, A and B leave the island counter-clockwise, and after 1/8 of the way, A refuels B completely and turns back towards the island. B continues towards C, and when the two planes meet, they share their fuel, then fly together towards the island. In the meantime A arrives on the island, refuels completely, and starts flying again counter-clockwise towards B and C, so that it can meet them and give them enough fuel, so that all of them arrive safely on the island.
It is easy to see that 2 planes are not enough.