How many hipster airplanes does it take to fly around the world?

by malmskog

Yeah, you probably don’t know, because, um, you’ve probably never heard of these airplanes anyway, because besides being really obscure I also made them up.

It doesn't take this many planes, I'll tell you that.

This week’s puzzle is about airplanes.  The story starts with a team of scientists discovering an amazing new fuel source.  This fuel produces no greenhouse gasses or pollutants of any type.  It is renewable.  One catch is that the only known source of this fuel is a tiny, very obscure island in the middle of the Pacific Ocean.  So the scientists have made their base on this island.  They are getting ready to announce their findings to the world.  Hooray! All of the world’s energy problems are solved.

However, these scientists are so out of touch with reality that they think they actually need a publicity stunt to go along with this announcement.  They decide that they will fly an airplane around the world using this amazing fuel.   One airplane must make a complete, uninterrupted circuit around the earth.  Yes, the long way around.  The pilot must fly a great circle, like the equator (though it could be a different great circle, for example including both the north and south poles).  Anyway, everybody agrees that this is a great idea (heh), but the problem is that when the scientists retrofit the airplanes they’ve got to run on this fuel, each plane can only hold enough fuel to take it half way around the world.

So it seems that the scientists are out of luck.  However, one scientist notices that there may be some hope if they use another plane as a helper.  Perhaps one plane could transfer fuel to another in flight.  They try this out and find that it works!  In fact, the pilots, planes, and fuel are all so amazing that fuel can be transferred from one plane to another in midflight, instantaneously and without any complications.  The two planes don’t even have to be going the same direction.  And another thing about these planes is that they change direction instantaneously, too, so you don’t even need to worry about that.

So everything seems to be back on.  However, the scientists soon realize that one helper plane won’t quite do the job.  To see why, lets consider what happens in the most basic scenario.  We’ll call the plane that is going to hopefully make the whole trip around the world plane A.  The helper plane will be plane B.  Let’s call the units of fuel worlds (Ws).  So we say that each plane can hold at most ½ W of fuel at a time. Considering the situation for a second, you see that our only possible hope is if both planes take off at the same time going the same direction with ½ W each of fuel.  The best that we can do here (without plane B running out of fuel on the way back and crashing into the ocean, which we don’t want) is to have B go 1/6 of the way around the world with A, then transfer 1/6 W of fuel to A, then go back to base with its remaining 1/6 W of fuel.  This will get plane A 4/6 of the way around the world, but if B refuels and takes off to meet A on the other side, B won’t be able to carry enough fuel to get them both home safely.  Ugh.

So the scientists clearly need to get some more planes involved.  The question is, how many planes to they need and how can they do it?

 

If you know the answer, send me an email at mathmostly@gmail.com!  The best answers I receive by Wednesday, March 28, will win amazing math mostly/somewhat science t-shirts for their authors.

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Congratulations to last week’s winners, Brian and Eli!  I got some great answers this week so it was a hard choice.  If you didn’t win last week, but you had the right answer, I’ll remember you and like you more next time.  So keep trying.  Thank you thank you to everyone who wrote in.

The solution to last week’s puzzle is simply this—the two kids go first, taking 2 minutes.  The fastest kid goes back, taking 1 minute.  Grandma and Mom cross, taking 10 minutes.  Pretty fast kid goes back, taking 2 minutes.  Both kids cross over, taking 2 more minutes.  Grand total: 2+1+10+2+2=17 minutes!  Awesome!

Reading the solution over, this may not seem challenging at all.  However, most people only come up with this solution after struggling for a long time trying out scenarios in which the fast kid always runs the flashlight back across the bridge.  This seems like the fastest way, right?  However, we know that this is doomed to failure because it means that each other person must cross with the fast kid, for a total of 17 minutes crossing time just going one way! That leaves the kid no time to ever cross back with the flashlight.

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