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Topic: Extension of Spider and Fly Problem (Read 1051 times) |
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Sir Col
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impudens simia et macrologus profundus fabulae
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I am sure that this classic problem is on the website in some guise, but this is an extension.
Original problem...
A spider sits in one corner of a room, S, measuring 4m by 3m by 2m. A fly is caught in a web in the opposite corner, F.
The original question asks for the shortest route from S to F.
By "opening out" the room to produce a net of the cuboid, the first time we see this, we are surprised to note that the shortest route takes an unexepected path. The two candidates being SF1= [sqrt]41 and SF2=[sqrt]45; SF1 being the shortest in this example.
Extension...
Given any sized cuboid room, is the shortest route always found by travelling to some point on the longer base wall?
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JocK
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Re: Extension of Spider and Fly Problem
« Reply #1 on: Dec 12th, 2004, 8:34am » |
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on Dec 10th, 2004, 4:31pm, Sir Col wrote:| Given any sized cuboid room, is the shortest route always found by travelling to some point on the longer base wall? |
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:: Yes, for the shortest route to the opposite corner the spider has to avoid crawling across the two walls (or floor and ceiling) with smallest area.
Intuitively, this is not surprising as crawling like this, the 'kink' in the double-straight-line path is minimal (i.e. it is the closest approximation to the straight-line path the spider can make) ::
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| « Last Edit: Dec 12th, 2004, 8:41am by JocK » |
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Sir Col
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Re: Extension of Spider and Fly Problem
« Reply #2 on: Dec 12th, 2004, 10:17am » |
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True, but if the floor measured a by b, with a>b and SF1 representing the segment of the "shortest" path that meets the edge of length a, then is SF1 always less than SF2?
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JocK
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Re: Extension of Spider and Fly Problem
« Reply #3 on: Dec 12th, 2004, 10:43am » |
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on Dec 12th, 2004, 10:17am, Sir Col wrote:| True, but if the floor measured a by b, with a>b and SF1 representing the segment of the "shortest" path that meets the edge of length a, then is SF1 always less than SF2? |
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Not sure if I understand you correctly. If a > b then SF1 is less than SF2. But if the third dimension c (the height of the room) exceeds a, neither is the shortest path. Again, the shortest path is to avoid the walls of smallest area (in this case the floor and ceiling each of area a x b).
But I think I read you incorrectly and you have something different in mind?
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Barukh
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Re: Extension of Spider and Fly Problem
« Reply #4 on: Dec 12th, 2004, 10:44am » |
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What you are actually asking: Is it true that (b+c)2 + a2 < (a+c)2 + b2 whenever a > b?
The answer is yes.
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| « Last Edit: Dec 12th, 2004, 11:14pm by Barukh » |
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Sir Col
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impudens simia et macrologus profundus fabulae
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Re: Extension of Spider and Fly Problem
« Reply #5 on: Dec 12th, 2004, 11:00am » |
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That is exactly what I am asking, but I wanted to avoid giving too much away; deriving the inequality you presented goes half way to solving the problem.
Of course you're right, Barukh, the answer is, yes. But perhaps we can leave the proof for someone else? Obviously you're free to answer it, but it's only an "easy" problem and is probably not going to challenge you very much.
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Barukh
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Re: Extension of Spider and Fly Problem
« Reply #6 on: Dec 12th, 2004, 11:17pm » |
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on Dec 12th, 2004, 11:00am, Sir Col wrote:| Of course you're right, Barukh, the answer is, yes. But perhaps we can leave the proof for someone else? Obviously you're free to answer it, but it's only an "easy" problem and is probably not going to challenge you very much. |
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Sir Col, I apologize, I misinterpreted your question.
My last post is hidden now (better later than never).
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| « Last Edit: Dec 12th, 2004, 11:17pm by Barukh » |
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