Answer: Tuesday Teaser #14
The first thing to note with this puzzle is that in order to minimize the number of planes required we want to keep as many planes as possible close to the island. For this reason, the best strategy is going to involve sending my plane around the far side of the world alone, so the question is how to get my plane to the 1/4 point with a full tank and how to fuel my plane back from an empty tank at the 3/4 point.
The best way of doing this is to send two extra planes at the same time as my plane. The first plane is sent an 1/8 of the way around the world. At this point it’s used 1/4 of its tank and needs another 1/4 to get back to the island, so it can pass on 1/4 of its tank to each of the other two planes before heading back to the island. This leaves us with two planes that both have a full tank. Continuing to the point 1/4 of the way around the world, the second plane has used 1/4 of its tank and needs another 1/2 to get back to the island, so it can pass on 1/4 of its tank to my plane. This means that my plane has a full tank at the 1/4 point and can make it round to the 3/4 point before running out of fuel.
Getting my plane back from the 3/4 point is now essentially an analogous problem in that we can follow almost exactly the same method. We can send two planes, with one refuelling the other so that a plane arrives at the 3/4 point with 3/4 of a tank remaining. This can then refuel my plane with enough fuel to reach the point 7/8 of the way around. At this point, we can send the second helper plane again (it will have had time to return to the island), so that extra fuel is available for my plane at the point 7/8 of the way around.
Now, this may seem like four planes are needed: two at the start and two at the end, but actually, because refuelling is instantaneous, the two planes used at the start have time to return to the island and be used at the end too.
All of which means that the answer is THREE.
In case my explanation of this is unclear (more than likely), I’ve drawn a quick diagram below to illustrate the solution:
