# The Airplane Riddle

Professor Fukanō, the famous scientist, has embarked on a new challenge – piloting around the world in a plane of his own design. There’s just one problem: there’s not enough fuel to complete the journey. Luckily, there are two other planes to help. Can you help the professor fly for the whole trip and achieve his dream, without anyone running out of fuel and crashing? Judd A. Schorr shows how.

### Transcript:

Professor Fukanō, the famous eccentric scientist and adventurer, has embarked on a new challenge: flying around the world nonstop in a plane of his design. Able to travel consistently at the incredible speed of one-degree longitude around the equator per minute, the plane would take six hours to circle the world.

There’s just one problem: the plane can only hold 180 kiloliters of fuel, only enough for exactly half the journey. Let’s be honest. The professor probably could have designed the plane to hold more fuel, but where’s the fun in that? Instead, he’s devised a slightly more elaborate solution: building three identical planes for the mission.

In addition to their speed, the professor’s equipped them with a few other incredible features. Each of the planes can turn on a dime and instantly transfer any amount of its fuel to any of the others in midair without slowing down, provided they’re next to each other. The professor will pilot the first plane, while his two assistants Fugōri and Orokana, will pilot each other.

However, only one airport, located on the equator, has granted permission for the experiment, making it the starting point, the finish line, and the only spot where the planes can land, take off, or refuel on the ground. How should the three planes coordinate so the professor can fly continuously for the whole trip and achieve his dream without anyone running out of fuel and crashing?

According to the professor’s calculations, they should be able to pull it off by a hair. The key is to maximize the support each assistant provides, not wasting a single kiloliter of fuel. It also helps us think symmetrically to make shorter trips in either direction while setting the professor up for a long unsupported stretch in the middle.

Here’s his solution. All three planes take off at noon, flying west, each fully loaded with 180 kiloliters. After 45 minutes, or one-eighth of the way around, each plane has 135 kiloliters left. Orokana gives 45 to the professor and 45 to Fugōri, fully refueling them both. With her remaining 45, Orokana returns to the airport and heads to the lounge for a well-deserved break.

Forty-five minutes later, with one-quarter of the trip complete, the professor and Fugōri are both at 135 kiloliters again. Fugōri transfers 45 into the professor’s tank, leaving himself with the 90 he needs to return. Professor Fukanō stretches and puts on his favorite album. He’ll be alone for a while.

In the meantime, Orokana has been anxiously awaiting Fugōri’s return, her plane fully refueled and ready to go. As soon as his plane touches the ground, she takes off, this time flying east. At this point, exactly 180 minutes have passed, and the professor is at the halfway point of his journey with 90 kiloliters of fuel left. For the next 90 minutes, the professor and Orokana’s planes fly towards each other, meeting at the three-quarter mark.

Just as the professor’s fuel is about the run-out, he sees Orokana’s plane. She gives him 45 kiloliters of her remaining 90, leaving them with 45 each. But that’s just half of what they need to make it to the airport. Fortunately, this is precisely when Fugōri, having refueled, takes off. Forty-five minutes later, just as the other two planes are about to run empty, he meets them at the 315-degree point and transfers 45 kiloliters of fuel to each, leaving 45 for himself. All three planes land at the airport just as their fuel gauges reach zero.

As the reporters and photographers cheer, the professor promises his planes will soon be available for commercial flights, just as soon as they figure out how to keep their inflight meals from spilling everywhere.

### Ali Kaya

This is Ali. Bespectacled and mustachioed father, math blogger, and soccer player. I also do consult for global math and science startups.