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Aeroplane


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#1 rookie1ja

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Posted 30 March 2007 - 03:43 PM

Airplanes - Back to the Cool Math Games
A distant planet “X” has only one airport located at the planet’s North Pole. There are only 3 airplanes and lots of fuel at the airport. Each airplane has just enough fuel capacity to get to the South Pole. The airplanes can transfer their fuel to one another.
Your mission is to fly around the globe above the South Pole with at least one airplane, and in the end, all the airplanes must return to the airport.

This old topic is locked since it was answered many times. You can check solution in the Spoiler below.
Pls visit New Puzzles section to see always fresh brain teasers.


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Spoiler for old wording

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#2 kmoore

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Posted 23 April 2007 - 01:29 AM

Love your puzzles!

I don't think this one can be solved as given. In the first step, at the time one plane reaches the 2/3rds mark, there are 2 planes back at the start which are going to shuttle fuel to it. But it must remain in the air, burning fuel until the supply plane comes. That plane will be running out of fuel when the "supply plane" arrives.
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#3 rookie1ja

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Posted 23 April 2007 - 03:00 PM

This one might be more about logic then real practice, since time plays a role and the chosen aeroplane can not just hang in the air without using any fuel. But if you use cars instead and assume constant consumption and a road around the globe, then it could work, couldn't it?
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#4 Paul

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Posted 23 April 2007 - 07:51 PM

I believe this is a solution (that doesn't require planes to hover):

Let the fraction of fuel-tank-filled for planes A, B, and C (respectively) be represented as:
[1, 1, 1]

airplane_puzzle.gif[/attachment:6a5ea]

1. All 3 planes go 1/4 the way toward the south pole. [3/4, 3/4, 3/4]
2. At that point plane C gives 1/4 tank to EACH of the other planes, leaving them full, and plane C with 1/4 tank to return to the north pole. [1, 1, 1/4]
3. At the equator, plane B gives plane A (the "full-circle plane") 1/4 tank, thus filling plane A; plane B has 1/2 tank left to return to the north pole. (Plane C arrives at airport) [1, 1/2, 1]
(Plane A now has enough fuel to pass the south pole and reach the equator on the other side.)
4. When plane B arrives at the airport, both B and C must instantly refuel and leave going the other direction. [1/2, 1, 1]
5. At 1/4 the way from the north pole, plane C gives plane B 1/4 tank, filling it up, while leaving itself with 1/2 tank to get back with (plenty). [1/4, 1, 1/2]
6. Plane B meets plane A at the equator as plane A is running out of fuel. Plane B, which has 3/4 tank left, gives half its fuel to plane A, leaving 3/8 tank in each plane. Plane C reaches the airport at this same time. [3/8, 3/8, 1]
7. Plane C instantly refuels and goes back to meet planes A and B at 1/4 the way from the north pole, with plenty of fuel for all three to return safely. [1/8, 1/8, 3/4] --> [1/3, 1/3, 1/3]

It sounds a bit messy, and I assume things happen instantly, but it works, doesn't it??

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#5 rookie1ja

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Posted 23 April 2007 - 11:06 PM

It works great. Wonderful and clear description - good job
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#6 dhuraal

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Posted 12 June 2007 - 07:21 AM

Maybe I'm not seeing it but i believe that the question never stipulates that the planes must travel from the north pole, to the south pole, back to the north. In which case you could simply fly one plane fly south a "little" then you could technically circle the globe (horizontally instead of vertically) and return to the airport in little time and only use a little fuel.
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#7 Numenor

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Posted 20 June 2007 - 11:13 PM

Maybe I'm not seeing it but i believe that the question never stipulates that the planes must travel from the north pole, to the south pole, back to the north. In which case you could simply fly one plane fly south a "little" then you could technically circle the globe (horizontally instead of vertically) and return to the airport in little time and only use a little fuel.

Actually, it does:

...Your mission is to fly round the globe with at least one aeroplane (above the south pole)...


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#8 newton

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Posted 09 July 2007 - 10:29 PM

Nice solution Paul. I actually came up with the same solution, though my ending was slightly different, it all works out.
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#9 PDR

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Posted 28 July 2007 - 07:39 PM

you can do this with 2 planes.

think of it as a circle instead of a globe. top of the circle is north pole, bottom is south pole.
Both planes start at top and go 1/3 of the way clockwise
Plane 1 transfers 1/3 of it's tank to the other plane
Plane 2 is now full
Plane 1 returns to the airport.
Plane 2 continues clockwise, and ends up at a point 2/3 around the circle, which is the same as saying 1/3 counterclockwise.
Plane 1 refuels and flies 1/3 counterclockwise where he meets plane 2.
Plane 2 gives 1/3 of tank to plane 2. Both now have 1/3 tank, enough to return to the start
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#10 luckygold6

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Posted 31 July 2007 - 08:20 PM

Actually PDR, I think you're confusing the 1/3 from the North Pole to the South Pole and 1/3 around the circle. The planes only have enough fuel to go from the North Pole to the South Pole, which means they have enough to go 1/2 way around the circle. So when you say the first plane and second plane go 1/3 of the way clockwise, they only have 1/2 - 1/3 = 1/6 of their fuel left. Thus niether planes will ever return to the North Pole again since they have to travel at least 1/3 of the way counterclockwise to get back.
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