Aeroplane

[rookie1ja]

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.

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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.

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.

[Guest]

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?

[Guest]

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??

[rookie1ja]

It works great. Wonderful and clear description - good job

[Guest]

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.

[Guest]

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)...

[Guest]

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

[Guest]

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

[Guest]

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.

[Guest]

Actually PDR, I think you're confusing the 1/3 from the North Pole to the South Pole and 1/3 around the circle.

D'oh!

[Guest]

send two in one direction

and the third in the exact opposite

half way to the south pole transfer the remaining fuel on one aircraft

with a full tank this plane can finish the 1/4 to 3/4 legs of the journey

meeting with the third aircraft and refueling with the remainder

our plane can now finish the circumventing of this planet

cai

[Guest]

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

This, to me, is the ONLY good solution where AT LEAST ONE airplane litterally goes AROUND THE GLOBE as the puzzle demanded.

And if even Rookie agrees...

Rookie's own solution is quicker, and less messy, but it just flies back and forth across one side of the planet.

My own brain staggered on this one, I must admit. This isn't the kind of puzzle I'm 'designed' for

Really great thinking.

BurningFuel

[Guest]

...but gravity is stronger at the poles, thus using more fuel.

[Guest]

Couldn't you just fill the tank of the airplane full then put another tank of gass on the plane so that when the airplane's tank starts to run low, you could just refill it?

[Guest]

You guys are missing the simplest solution. 2 planes start to the south, at 1/2 way the #2 plane transfers the remaining 1/2 tank of fuel to plane #1, which continues on. Plane 2 ditches in the ocean. Plane #3 takes off in the opposite direction of #1 and at the 1/2 way point transfers 1/2 tank fuel (then also ditches) to #3 which continues on to return at airport. The only proviso is that 1 craft returns to airport, thus the other 2 are expendable. It is all in the timing. Mike

[Guest]

Thank you Mike! Everybody is giving these complicated answers, none of which involve ditching planes, and the most obvious/simple one was not mentioned until you posted. The problem stipulates nothing about the ditching of planes being frowned upon, although I do give props to those who worked around the destruction of aircraft.

[Guest]

Actually, the riddle does stipulate that all planes must return to the airport... My question is, how can there only be one airport? Usually with air travel there is an origin and a destination. If there are no other airports, where do the planes fly to? And why would they build at the north pole where jet fuel and other critical fluids would constantly be at risk of freezing? Not to mention the exhorborent costs involved in keeping the planes de-iced... This sounds more like an illogic puzzle to me...

[rookie1ja]

Actually, the riddle does stipulate that all planes must return to the airport... My question is, how can there only be one airport? Usually with air travel there is an origin and a destination. If there are no other airports, where do the planes fly to? And why would they build at the north pole where jet fuel and other critical fluids would constantly be at risk of freezing? Not to mention the exhorborent costs involved in keeping the planes de-iced... This sounds more like an illogic puzzle to me...

it's just a puzzle to tease your brain and have some fun ... don't take it too seriously

[Guest]

yeah, i know. just trying to impose a little humor. sorry.

[rookie1ja]

no probs, there are many such posts on my forum that made me laugh

[Guest]

Rookie, it seems that the only way your plan would work is for the planes to be able to land over the 2/3 mark and wait for an airplane to come fuel it up. Are the planes able to land? Or can they hover without burning fuel?

[Guest]

Lets see if i can solve this.

A1 = 1st Plane

A2 = 2nd Plane

A3 = 3rd Plane

They fly counter-clockwise. (NP-1-2-3-SP-4-5-6-NP)

North Pole

(Point 1) (Point 4)

(Point 2) (Point 5)

(Point 3) (Point 6)

South Pole

A1 starts

A2 starts

A1 gets to 75% fuel (Point 1)

A2 gets to 75% fuel (Point 1)

A1 gives 25% of fuel to A1

A1 has 100% fuel

A2 returns and refuels

A1 get to 75% fuel (Point 2)

A2 flies to (Point 2)

A2 gives 25% to A1

A1 has 100% fuel

A3 flies to (Point 1)

A2 flies to (Point 1)

A3 gives 25% fuel to A2

A2 has 25 % fuel

A3 returns and refuels

A2 returns and refuels

A1 flies to the South Pole

A2 flies to (Point 5)

A1 flies to (Point 5)

A2 gives 25% fuel to A1

A1 has 25% fuel

A2 has 25% fuel

A1 flies to (Point 6)

A2 flies to (Point 6)

A3 flies to (Point 6)

A3 gives 25% fuel to A1

A3 gives 25% fuel to A2

A1 has 25 % fuel

A2 has 25 % fuel

A3 has 25 % fuel

A1 flies to the North Pole

A2 flies to the North Pole

A3 flies to the North Pole

*CONGRATS* If you have got to this point you have nothing more to read.

This shows it can be done!

Like the solution?

[Guest]

I was also stuck on the concept of hovering.

[Guest]

No planes have to hover. The puzzle does not specify the speed that the planes have to fly at. Instead of flying at the same speed as the refuling planes and then waiting for them to catch up, the main plain can simply fly slower.