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I’ve heard and thought about variants of this puzzle several times, and they always remind me of the tragedy of Amelia Earhart. She and her navigator Fred Noonan disappeared in 1937, while trying to fly around the world in a Lockheed Electra. Did she run out of fuel? Did something go badly wrong with the Electra? There has been no trace of either the plane or its occupants in these 84 years.
They were last seen at Lae Airfield in Papua New Guinea. So in a tribute to Earhart and Noonan, this puzzle is set there.
Assume Lae has a large number of identical Lockheed Electras that fly at identical speeds. It also has a huge store of aircraft fuel. With a full tank of fuel, each Electra can fly exactly halfway around the world (assume a “great circle” route, meaning the shortest possible circumnavigation). What’s more, these Electras are different from Earhart’s model in that they can be refueled in flight by another Electra. Doing so, we’ll assume, is instantaneous, doesn’t slow either plane down, or use any extra fuel. If a plane has no fuel, it can keep flying if it is being refueled. But you should note that the only place you can get fuel is Lae itself, and refueling there is also instantaneous.
Question: You want one Electra to do the full circumnavigation of Planet Earth, without landing anywhere else, and return to Lae. What’s the minimum number of Electras you need to accomplish this? How will they do it? Needless to say, all Electras that take off must return safely to Lae.
Hint: Try three Electras, for example. Call them Amelia, Fred and (why not) Ramanujan. Fill all three with fuel and send them off with plenty of fanfare. Do you need more Electras?
Scroll down for the answer.
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Answer:
Break the journey into eighths:
1/8 complete: All three have used up 1/4 of their fuel. Fred refuels Amelia and Ramanujan with 1/4 tank each. Both are now full. Fred has 1/4 tank, enough to return to Lae.
1/4: Amelia and Ramanujan, both 3/4 tank. Ramanujan to Amelia, 1/4 tank. Amelia full. Ramanujan 1/2, returns to Lae. Fred lands at Lae, refuels and waits.
3/8: No change.
1/2: Amelia 1/2 tank. Ramanujan at Lae, refuels. Fred and Ramanujan take off, flying the other way around the great circle.
5/8: Amelia 1/4 tank. Fred and Ramanujan are at the 7/8 point, 3/4 tank each. Fred to Ramanujan, 1/4 tank. Ramanujan full. Fred 1/2 tank, returns to Lae.
3/4: Amelia 0, meets Ramanujan, 3/4 tank. Ramanujan to Amelia, 1/4 tank. Ramanujan 1/2 tank, turns for Lae. Fred at Lae, refuels and takes off again.
7/8: Amelia 0, meets Fred, 3/4 tank. Fred to Amelia, 1/4 tank. All three head for Lae.
Lae: All three land safely. Amelia has done the circumnavigation. Three planes are enough!
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