[rookie1ja]

Airplanes - Back to the Logic Puzzles
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.

Spoiler for Solution:

Aeroplane - solution
Divide the way from pole to pole to 3 thirds (from the North Pole to the South Pole 3 thirds and from the South Pole to the North Pole 3 thirds).

1st step - 2 aeroplanes to the first third, fuel up one aeroplane which continues to the second third and the first aeroplane goes back to the airport.

2nd step - 2 aeroplanes fly again from the airport to the first third, fuel up one aeroplane which continues to the second third and the first aeroplane goes back to the airport.

3rd step - So there are 2 aeroplanes on the second third, each having 2/3 of fuel. One of them fuels up the second one and goes back to the first third, where it meets the third aeroplane which comes from the airport to support it with 1/3 of fuel so that they both can return to the airport. In the meantime, the aeroplane at the second third having full tank flies as far as it can (so over the South Pole to the last third before the airport).

4th step - The rest is clear – one (of the two) aeroplane from the airport goes to the first third (the opposite direction as before), shares its 1/3 of fuel and both aeroplanes safely land back at the airport.

Spoiler for old wording:

There is only one airport on a fictional planet, and that is on the north pole. There are only 3 aeroplanes and lots of fuel at the airport. A full tank of an aeroplane lasts exactly to the south pole, however the aeroplanes can transfer their fuel to one another.
Your mission is to fly round the globe with at least one aeroplane (above the south pole) and in the end all aeroplanes must return to the airport.

[kmoore]

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.

[rookie1ja]

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?

[Paul]

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??

Attached File(s)

•  airplane_puzzle.gif (4.76K)
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[rookie1ja]

It works great. Wonderful and clear description - good job

[dhuraal]

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.

[Numenor]

Quote

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:

Quote

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

[newton]

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

[PDR]

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

[luckygold6]

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.