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Wizard
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Wizard
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December 3rd, 2017 at 12:27:07 PM permalink
An island has a fleet of identical airplanes. Each plane can hold enough fuel to make it half way around the earth. The planes can refuel each other mid-flight. There is an unlimited supply of fuel on the island. Assuming no time or fuel lost in refueling or making a u-turn, what is the fewest number of airplanes needed to fly around the earth and how can it be done? All planes must return safely to the island.

As usual, free beer to the first acceptable answer and solution.

The question for the poll is how many planes would you need?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
RS
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December 3rd, 2017 at 1:09:05 PM permalink
6

A and B leave simultaneously. At 1/6 around the world, each plane should have 2/3 a tank remaining. B gives A 1/3 a tank, letting B have enough fuel to return to his home island safely. This will let A get 4/6 around the world without additional help.

Going the other way around, C, D, E and F leave at the same time. C gives D 1/3 a tank at 2/6 around the world and E gives F 1/3 a tank at the same spot, C and E flip a U-Turn (like B did, above) and get home. D and F each have a full tank. D goes to A at A's 4/6 around the world and gives him 1/3 a tank. A and D return to F, where F gives them each 1/3 a tank. The 3 now each have 1/3 a tank, return to the island.

E and F would obviously be leaving later than C and D, such that they can all meet at the appropriate times.
OnceDear
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December 3rd, 2017 at 1:13:47 PM permalink
deleted. I had the 1st 1/6 of the journey covered as per RS solution, then my head exploded.
Psalm 25:16 Turn to me and be gracious to me, for I am lonely and afflicted. Proverbs 18:2 A fool finds no satisfaction in trying to understand, for he would rather express his own opinion.
Ayecarumba
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December 3rd, 2017 at 1:35:48 PM permalink
Quote: Wizard

An island has a fleet of identical airplanes. Each plane can hold enough fuel to make it half way around the earth. The planes can refuel each other mid-flight. There is an unlimited supply of fuel on the island. Assuming no time or fuel lost in refueling or making a u-turn, what is the fewest number of airplanes needed to fly around the earth and how can it be done? All planes must return safely to the island.

As usual, free beer to the first acceptable answer and solution.

The question for the poll is how many planes would you need?


One. The problem doesn't state that the plane can't land to refuel.
Simplicity is the ultimate sophistication - Leonardo da Vinci
OnceDear
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December 3rd, 2017 at 1:40:49 PM permalink
hmmmm

you can do it with three planes. A, B and C
3 set off together, say east to west.
at the 1/8th point, plane A transfers 1/4 of a tankful to C and transfers 1/4 of a tankful to B
Plane A has 1/4 tank remaining and returns home to refuel.
B&C, fully fuelled, continue a while to the 1/4 point where B transfers 1/4 a tankful to C.
C continues with a full tank, and B has 1/2 tank with which it flies home.
Plane A, refueled sets off in the opposite direction and times its journey to meet C at the 3/4 point where C is nearly empty.
at the 3/4 point A transfers 1/4 of a tankful to C.
A has already used 1/2 its tankful leaving it with 1/4 so turns back hoping for a miracle. Flying alongside C and waving to eachother to keep their hope alive.
A and C reach the 7/8 point and are flying on vapour when they meet B, which has refueled and set off west to east.
B transfers 1/4 tank to A and 1/4 tank to C. He's already used 1/4 tank and keeps 1/4 tank
All three fly together that last 1/8 of the way and all land just as they reach empty.


Let's hope KYU didn't figure it
$:o)

Edited a few times to amend cock-ups
Last edited by: OnceDear on Dec 3, 2017
Psalm 25:16 Turn to me and be gracious to me, for I am lonely and afflicted. Proverbs 18:2 A fool finds no satisfaction in trying to understand, for he would rather express his own opinion.
Wizard
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December 3rd, 2017 at 1:56:42 PM permalink
Quote: Ayecarumba

One. The problem doesn't state that the plane can't land to refuel.



Planes may not land to refuel. In other words, no caches of gas allowed.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
RS
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December 3rd, 2017 at 1:57:00 PM permalink
deleted
Wizard
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December 3rd, 2017 at 2:22:09 PM permalink
Quote: OnceDear

hmmmm


you can do it with three planes. A, B and C
3 set off together, say east to west.
at the 1/8th point, plane A transfers 1/4 of a tankful to C and transfers 1/4 of a tankful to B
Plane A has 1/4 tank remaining and returns home to refuel.
B&C, fully fuelled, continue a while to the 1/4 point where B transfers 1/4 a tankful to C.
C continues with a full tank, and B has 1/2 tank with which it flies home.
Plane A, refueled sets off in the opposite direction and times its journey to meet C at the 3/4 point where C is nearly empty.
at the 3/4 point A transfers 1/4 of a tankful to C.
A has already used 1/2 its tankful leaving it with 1/4 so turns back hoping for a miracle. Flying alongside C and waving to eachother to keep their hope alive.
A and C reach the 7/8 point and are flying on vapour when they meet B, which has refueled and set off west to east.
B transfers 1/4 tank to A and 1/4 tank to C. He's already used 1/4 tank and keeps 1/4 tank
All three fly together that last 1/8 of the way and all land just as they reach empty.



I have gone through every line of this and confirmed this works! Congratulations on a beer well-earned.


Time Fuel A Fuel B Fuel C Loc. A Loc. B Loc. C Comments
0 1 1 1 0 0 0 3 set off together, say west.
0.0625 0.75 0.75 0.75 0.125 0.125 0.125 at the 1/8th point, plane A transfers 1/4 of a tankful to C and transfers 1/4 of a tankful to B
0.0625 0.25 1 1 0.125 0.125 0.125 Plane A has 1/4 tank remaining and returns home to refuel. B&C continue
0.125 0 0.75 0.75 0 0.25 0.25 A refuels, B transfers 1/4 to C. B returns home
0.25 1 0.5 1 0 0.25 0.25 A continues on
0.5 0.5 1 1 0.5 0 0 C leaves east
0.75 0 1 0.5 0.75 0 0.75 A transfers 1/4 of a tankful to C. 
0.75 0.25 1 0.25 0.75 0 0.75 B leaves east
0.875 0 0.75 0 0.875 0.875 0.875 B transfers 1/4 tank to A and 1/4 tank to C. He's already used 1/4 tank and keeps 1/4 tank
0.875 0.25 0.25 0.25 0.875 0.875 0.875 All three return home
1 0 0 0 1 1 1 All three arrive on fumes




Time Fuel A Fuel B Fuel C Loc. A Loc. B Loc. C Comments
0 1 1 1 0 0 0 A, B, and C depart clockwise
0.0625 0.875 0.875 0.875 0.0625 0.0625 0.0625 C transfers 1/8 tank to A and B
0.0625 1 1 0.625 0.0625 0.0625 0.0625 C turns back
0.125 0.875 0.875 0.5 0.125 0.125 0 C refuels
0.125 0.875 0.875 1 0.125 0.125 0 C leaves
0.25 0.625 0.625 0.75 0.25 0.25 0.125 B refuels A
0.25 1 0.25 0.75 0.25 0.25 0.125 B turns back
0.3125 0.875 0.125 0.625 0.3125 0.1875 0.1875 C refuels B with a quarter tank
0.3125 0.875 0.375 0.375 0.3125 0.1875 0.1875 C turns back
0.5 0.5 0 0 0.5 0 0 A and B reach island on fumes, refuel
0.5 0.5 1 1 0.5 0 0 A and B depart counter-clockwise
0.6875 0.125 0.625 0.625 0.6875 0.8125 0.8125 C transfers 1/4 tank to B
0.6875 0.125 0.875 0.375 0.6875 0.8125 0.8125 C turns back
0.75 0 0.75 0.25 0.75 0.75 0.875 B refuels A with 3/8 tank
0.75 0.375 0.375 0.25 0.75 0.75 0.875 B turns back
0.875 0.125 0.125 0 0.875 0.875 1 C refuels
0.875 0.125 0.125 1 0.875 0.875 1 C turns back
0.9375 0 0 0.875 0.9375 0.9375 0.9375 C refuels A and B with 7/24 tank each
0.9375 0.2917 0.2917 0.2917 0.9375 0.9375 0.9375 C turns back
1 0.1667 0.1667 0.1667 1 1 1 All arrive with 1/6 of a tank


Pretty similar method. Your way allowed plane B to take a little tea break at the island. However, mine has all three arriving back with a little extra fuel.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Wizard
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December 3rd, 2017 at 2:31:22 PM permalink
Quote: RS

6

A and B leave simultaneously. At 1/6 around the world, each plane should have 2/3 a tank remaining. B gives A 1/3 a tank, letting B have enough fuel to return to his home island safely. This will let A get 4/6 around the world without additional help.

Going the other way around, C, D, E and F leave at the same time. C gives D 1/3 a tank at 2/6 around the world and E gives F 1/3 a tank at the same spot, C and E flip a U-Turn (like B did, above) and get home. D and F each have a full tank. D goes to A at A's 4/6 around the world and gives him 1/3 a tank. A and D return to F, where F gives them each 1/3 a tank. The 3 now each have 1/3 a tank, return to the island.

E and F would obviously be leaving later than C and D, such that they can all meet at the appropriate times.




I didn't verify every line but will take your word that it works. However, as OD and me have shown, it can be accomplished with three planes only.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
RS
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December 3rd, 2017 at 2:35:56 PM permalink
Quote: Wizard

Quote: RS

6

A and B leave simultaneously. At 1/6 around the world, each plane should have 2/3 a tank remaining. B gives A 1/3 a tank, letting B have enough fuel to return to his home island safely. This will let A get 4/6 around the world without additional help.

Going the other way around, C, D, E and F leave at the same time. C gives D 1/3 a tank at 2/6 around the world and E gives F 1/3 a tank at the same spot, C and E flip a U-Turn (like B did, above) and get home. D and F each have a full tank. D goes to A at A's 4/6 around the world and gives him 1/3 a tank. A and D return to F, where F gives them each 1/3 a tank. The 3 now each have 1/3 a tank, return to the island.

E and F would obviously be leaving later than C and D, such that they can all meet at the appropriate times.




I didn't verify every line but will take your word that it works. However, as OD and me have shown, it can be accomplished with three planes only.


Yeah, I sorta forgot you could use the same plane for more than one flight. =(
OnceDear
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December 3rd, 2017 at 2:39:58 PM permalink
Quote: Wizard

Your way allowed plane B to take a little tea break

It's all a matter of priorities. Reserve fuel or tea break. No contest!
Psalm 25:16 Turn to me and be gracious to me, for I am lonely and afflicted. Proverbs 18:2 A fool finds no satisfaction in trying to understand, for he would rather express his own opinion.
Ayecarumba
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December 3rd, 2017 at 11:49:08 PM permalink
Quote: Wizard

Quote: Ayecarumba

One. The problem doesn't state that the plane can't land to refuel.



Planes may not land to refuel. In other words, no caches of gas allowed.




Since the planes can't land to refuel, you will need more than 3 to finish the circuit.
Simplicity is the ultimate sophistication - Leonardo da Vinci
RS
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December 4th, 2017 at 12:50:44 AM permalink
Ayecarumba is right! If a plane can't land to refuel, then I'm the winner. All hail King RS!!!
OnceDear
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December 4th, 2017 at 5:10:17 AM permalink
Quote: Ayecarumba

Quote: Wizard

Quote: Ayecarumba

One. The problem doesn't state that the plane can't land to refuel.



Planes may not land to refuel. In other words, no caches of gas allowed.




Since the planes can't land to refuel, you will need more than 3 to finish the circuit.

of course, they may land on the island to refuel. Wizard's quote above was mis-phrased and not part of the original question.
Psalm 25:16 Turn to me and be gracious to me, for I am lonely and afflicted. Proverbs 18:2 A fool finds no satisfaction in trying to understand, for he would rather express his own opinion.
Wizard
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December 4th, 2017 at 6:30:24 AM permalink
Quote: RS

Ayecarumba is right! If a plane can't land to refuel, then I'm the winner. All hail King RS!!!



That was in response to someone else whose solution involved them landing in places besides the island to refuel. Planes may only land on the island. I should have been more clear.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Ayecarumba
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December 4th, 2017 at 12:12:17 PM permalink
There is also the logistics problem of having a plane double back, refuel, then head out the opposite way to meet the others before they run out of fuel. Does the problem also suspend the relationship between speed and fuel consumption?
Simplicity is the ultimate sophistication - Leonardo da Vinci
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