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Airplanes can hold enough fuel to fly halfway around the world. The only source of fuel is a single airport, which is also the only place planes can take off and land. However, planes can transfer fuel to each other midair. Assume that planes can change direction and refuel instantly, and all planes fly at the same constant rate.

Amelia Earhart wants to circumnavigate the globe (namely, start at the airport, fly to an opposite point on the earth, then back). She needs the help of other planes, but of course doesn't want any of them to crash. How many planes does she need, including her own?

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    $\begingroup$ This is just the travellers across a desert problem with a 4-day resource limit instead of 5 days, and a requirement of one person making it instead of two. $\endgroup$
    – user88
    Commented Apr 19, 2015 at 20:51
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    $\begingroup$ @JoeZ. There is another difference: the problem you mention is moving down a line, while mine is moving around a circle. This affords more options for assists. $\endgroup$
    – Mike Earnest
    Commented Apr 19, 2015 at 20:55
  • $\begingroup$ Oh, right, planes can fly back as well. $\endgroup$
    – user88
    Commented Apr 19, 2015 at 20:55
  • $\begingroup$ Alright, I guess it's not a duplicate then. $\endgroup$
    – user88
    Commented Apr 19, 2015 at 20:55
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    $\begingroup$ See Operation Blackbuck, en.wikipedia.org/wiki/Operation_Black_Buck , and the refueling scheme, for a real-world example $\endgroup$
    – DJohnM
    Commented Apr 20, 2015 at 4:18

3 Answers 3

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3 planes are needed.

They all take off at the same time and fly 1/8th the way around the world. One gives 1/4th of a tank to each of the other planes and goes back to refuel.

The other two continue another 1/8th of the way. The support plane gives 1/4th of a tank to Earhart and flies back to refuel.

When Earhart is running on fumes 6/8ths of the way around the world, she is met by a support plane that gives her 1/4th of a tank.

They fly to the 7/8ths mark and meet a second support plane which gives them both 1/4th of a tank.

As the planes from the first 4th and last 4th are reused, only 3 planes are needed.

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  • $\begingroup$ You were faster than me to post your answer :) $\endgroup$
    – A.D.
    Commented Apr 19, 2015 at 21:53
  • $\begingroup$ Close, but after flying 1/8th + 1/8th = 1/4 of the way, Earhart has 50% fuel, so giving her another 1/4 tank only brings it up to 75%. So she crashes 5/8ths of the way. $\endgroup$
    – geometrian
    Commented Apr 19, 2015 at 22:55
  • $\begingroup$ @imallett she is given 1/4 of a tank 4 times. Twice on the first half twice on the second. She has 100% fuel after given it the second time. $\endgroup$
    – kaine
    Commented Apr 19, 2015 at 23:16
  • $\begingroup$ "One gives 1/4th of a tank to the others" 1/8th of the way, so if Earhart gets it, then the other support plane still has 3/4 tank (down to 1/2 tank by the time they get 1/4 of the way). Transfer another 1/4 tank, and Earhart makes it, but the other support plane crashes on the way back. See my answer for a fix. $\endgroup$
    – geometrian
    Commented Apr 19, 2015 at 23:20
  • $\begingroup$ @imallet "One gives 1/4th of a tank to the others" means each of Amelia and the other support plane are given 1/4 of a tank. Therefore they both have full tanks while the one going back has just enough to make it back. $\endgroup$
    – Spencerkatty
    Commented Apr 19, 2015 at 23:23
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She needs three planes

Here comes the explanation

All planes will be numbered. Number 1 will be Amelia Earhart's plane.

1- So p1, p2, p3 take off at the same time and fly in the same direction.
2- After a fourth of the tank capacity, p2 gives a fourth of its tank capacity to the two other planes and returns to the airport. At this point, p1 and p3 have a full tank.
3- After a fourth of the tank capacity, p3 gives a fourth of its tank capacity to p1 and returns to the airport. At this point, p1 still has a full tank.
4- After a half of p1 tank capacity, p2 and p3 take off and fly toward p1 but in the opposite direction from the first take off.
5- After a fourth of tank capacity, p2 gives a fourth of its tank to p3 and returns to airport.
6- After a fourth of tank capacity, p3 meets p1 and gives it a half of its tank capacity. It means that p1 can head to the airport directly. But not p3. That's why in the meantime, p2 fly toward p3 and gives it the needed fuel to return to the airport.

I hope it is clear enough.

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    $\begingroup$ Similar problem. In step 2, p1 and p3 do not have a full tank, since they have together burned 1/4 + 1/4 = 1/2 a tank (they are burning at the same time). $\endgroup$
    – geometrian
    Commented Apr 19, 2015 at 22:57
  • $\begingroup$ I don't understand what you're saying. At step 2, p2 give one fourth to p1 and one fourth to p3. As p1 and p3 have burn each one fourth of their tank, now it is full again. $\endgroup$
    – A.D.
    Commented Apr 19, 2015 at 23:30
  • $\begingroup$ See discussion on kaine's answer. $\endgroup$
    – geometrian
    Commented Apr 19, 2015 at 23:32
  • $\begingroup$ Still don't understand what you mean. $\endgroup$
    – A.D.
    Commented Apr 19, 2015 at 23:36
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    $\begingroup$ Ambiguity in language. "p2 gives a fourth of its tank capacity to the two other planes" means most directly "p2 transfers a fourth of its tank capacity to the two other planes jointly", not "p2 transfers half of its tank capacity to the two other planes to split". In exactly the same way as kaine's answer, your answer is correct under this clarification. $\endgroup$
    – geometrian
    Commented Apr 19, 2015 at 23:40
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Like the other answers, the answer is:

Three planes

However, a correct explanation is more complicated. In particular:

The second support plane needs to do two out-and-back flights on each side of the planet. Here's a full breakdown (important new steps bolded):

There are three planes (AE, S1, and S2). They each start with a full tank.

1. They all fly 1/8th of the way.
2. AE, S1, and S2 all now have 3/4 tanks. S2 transfers 1/4 tank to S1 and turns back. AE now has 3/4 tank, S1 has a full tank, and S2 has 1/2 tank.
3. S1 and AE continue for another 1/8th of the way.
4. S1 has 3/4 tank while AE has 1/2 tank. Meanwhile S2 returns home with 1/4 tank to spare.
5. S1 transfers 1/2 tank to AE and turns back. S1 now has 1/4 tank while AE has a full tank. S2, refueled to a full tank, sets off again.
6. All planes fly another eighth of the way. AE is now 3/8ths of the way with 3/4 fuel, S1 and S2 rendezvous 1/8th of the way (S1 has just run out of fuel and S2 has 3/4 tank).
7. S2 transfers 1/4 tank to S1 and turns around. S2 now has 1/2 tank and S1 now has 1/4 tank.
8. The planes fly another eighth of the way: S1 arrives at home with an empty tank, S2 arrives at home with 1/4 tank. AE reaches the 4/8ths mark with 1/2 tank.
9. S1 and S2 refuel and set off again to do the whole thing in reverse on the other side of the planet.

[Edit: after discussion on the other answers, it was noted that:

If S2 gives 1/4 tank to both S1 and AE (thereby transferring 1/2 tank), then it also works (and S2 only needs to make one trip).

The phrasing of both other answers led to my confusion on this; they are both correct under this clarification. My answer is then just an alternate approach.]

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