Monday Math Madness #26: Been Around the World
15 Feb 2009 Quan Quach 22 comments 922 views
The Problem Statement
Ever since Quan was a little boy, his lifelong dream has been to fly around the world in an airplane.
- Quan lives on an island wherein there are 100 airplanes, all created equally with identical characteristics.
- Each airplane has a fuel tank that contains enough fuel to fly exactly half way around the world.
- All of the airplanes travel at the same speed, and use their gas at the same rate.
- Airplanes can exchange fuel with other airplanes while in flight.
- The island is the only source of fuel.
- For the purposes of this problem, assume that there is no time lost refueling on either the air or ground.
- All airplanes must make it back to the island safely.
- Edit #1: Assume the size of the island is really small compared to the size of the world (e.g. a point).
- Edit #2: The path around the world must be the greatest circle, e.g. the path of the equator is considered a greatest circle around the earth.
- Edit #3: Planes cannot land anywhere except the island. The flight around the world must be continuous.
- Edit #4: Planes can land, take off, and change direction instantaneously (there is no penalty for these actions).
Question: What is the lowest number of airplanes required to allow one airplane to travel all the way around the world?
The Prize
The winner will receive a 10 dollar gift certificate to Amazon.com!
The Rules
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Email your answers to mondaymathmadness at gmail dot com.
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Inelgible for any one person to win more than once per year. But you should still submit your answer!
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Answer must be explained. You must show your work! We will accept answers in the form of a MATLAB script as well. We will be the final judge on whether an answer was properly explained or not.
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The deadline to submit answers is February 23rd 2009, Monday 11:59 PM Pacific Standard Time
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The winners will be chosen randomly from all the submittals using a random number generator.
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The winner will be announced at 9:00 AM PST February 27th, 2009.
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Comments for this post should only be used to clarify the problem. Please do not discuss ANY potential solutions.
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Please spread the word about our contest by stumbling this webpage!
22 Responses to “Monday Math Madness #26: Been Around the World”
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How big is the island?
Assume the island is really small compared to the size of the world (e.g. a point).
Where is the island located, and how much around the Earth does he want to fly? That is, does he want to make a full circle around the middle of the world, say at the equator, if that’s where the island is, or, is it enough to go full-way around a higher latitude?
Renata,
Assume that they want to fly around the earth in the greatest circle.
Quan
[...] has MMM #26 posted. This one is tricky. We’ve received a number of different answers and even the [...]
Are planes allowed to land at any arbitrary point on the Earth, such that they stop consuming fuel until they take off again?
Also, can planes take off and land and change direction at any time at no cost?
Ron,
Good question. For the purposes of this problem, planes cannot land anywhere except the island. The flight around the world must be continuous.
Planes can land, take off, and change direction with no penalty.
Quan
“Each airplane has a fuel tank that contains enough fuel to fly exactly half way around the world.” –Are we to further assume that each fuel tank will hold no more than this amount of fuel?
Correct Nate, otherwise, why not just fill up the first plane with all the fuel you need!
Can a plane go partial way then turn around, go back to the island, refuel (instantaneously, but not necessarily), and take off again or would that count as a new plane taking off?
if the fuel is enough to fly half the way and back^^, one plane is enough to fly the circle round the world. if not, i dont have an idea jet.
Hi Quan,
Just wondering if you could help me out with some questions that I have. Also, does anyone know if the 2007b version has GUI capabilities?
Thanks
I can see an answer that requires only 4 planes. But how can I proove that 3 planes ins impossible?
[...] don’t miss this week’s Monday Math Madness #26: Been Around the World. You have until Monday night to send in your answer to this challenging puzzle from Quan Quach at [...]
The fist plane could go 1/2 way around the world, at which point it would take 4 planes flying from 3/8ths of the way to refill the first plane for the rest of its trip and be able to get back to the 3/8s point.
You would also have to have 4 planes come to refuel the last 4 planes so that they could get back to the 1/4th way point.
Now you have 8 planes at the 1/4th of the way point that need to be refueled.
They can be refueld by 4 planes for the 1/8th of the way point and they all return back to 1/8th point. Now there are 12 planes that need to be refueled. They can be refueled by 6 planes and everyone returns home safely.
So I am going to guess it takes “19″ planes to go around the world and get everyone home safely. I dont know if my math is right on this, But I like how it looks anyway!
I dont know if my answer from before was right or wrong, but I liked the problem, because it made me think about the Nasa Moon Rockets.
It is not enpough to have enough fuel to take the rocket to the moon. You have to have enough fuel to lift the rocket fuel as well, and then you have to heve enough rocket fuel to lift that rocket fuel, and then fuel to lift that fuel, and when you finally get there…What do the big bosses want you to do? Grab a whole bunch of rocks to bring back with you.
What if I grab too many rocks?….Well.,The take home fuel runs out and everybody dies.
Thats just swell!
Answer is 57 planes
Explanation:
a) 1 plane can fly to 1/2 round the world
b) 2 planes: Second plane flies to 1/12 the distance around the world, transfers 1/3 its fuel to the first plane and returns. At this point the second plane has burnt 1/3 of its fuel, transferred 1/3 of fuel to the first plane and has 1/3 of fuel left to get back to the island. First plane with additional 1/3 tank of fuel (ie full tank) can travel (1/2+1/12) =0.583 around the world
c) 3 planes: Third plane flies to 1/20 the distance around the world, transfers 2/5 of its fuel to the second plane and 1/5 to the first plane each. The third plane returns now. Ie after flying 1/20 of distance around the world, the second and first plane have full tank of fuel, and can repeat step b). Therefore the total distance traveled by first plane is (1/2+1/12+1/20) = 0.6333 distance around the world.
Mathmatically this can be expressed as
Total distance traveled by first plane = ½ + SIGMA (for n=1) 0
(for n=2 to n) 1/4/(2n-1)
Where n= number of planes
Summing up the series gives n=57planes to travel distance 1.000822 around the world (PS: Use excel as my school math on arithmetic series summation failed here)
Just sent my solution. I hope it still counts in light of the answers above.
Crap missed the deadline.
Appologies for pasting the reply in the forum itself.
Do we get to know the correct answer?
@anka
Yup, we post solutions and winner roughly 2 weeks from the Monday of the contest
http://blinkdagger.com/monday-math-madness/monday-math-madness-26-winner-around-the-world
The solution is posted here:
http://blinkdagger.com/monday-math-madness/monday-math-madness-26-winner-around-the-world