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Marie-Jo W.
guest
Dec-27-07, 01:04 PM (EST)

"Puzzle involving exponential"

 Dear Puzzle Friends,I am not a mathematician but just a math puzzle enthusiast... I placed a month ago an adaptation of an old math problem on my site, see text below: A high quality rubber band is fastened and hung from a horizontal pole with a cannonball at its end. Two facing ladybugs are crawling along this rubber band toward each other. From their respective starting positions (8 cm apart -- see image), each small beetle crawls toward the other at a speed of 1 cm per second. However, in the length of time each beetle crawls 1 cm, the cannonball, thanks to the force of gravity, stretches the rubber band an additional 8 cm. Will the poor ladybugs ever meet? And, if yes, when? If not, why?!(see the whole problem and solution at: https://www.archimedes-lab.org/monthly_puzzles_69.html )My answer was that the beetles will meet after:ln(n) + 0.5772156649 + 1/2n ≈ 4that is approximately after exp^3.422784336 ≈ 30.65464915 sec.But someone disagreed with me and posted a new solution:exp^4 - 1 = 53.598... sec.Who is right?I would be very happy if a math expert would help me to settle this problem. Thanks in advance for your help!Marie-Jo Waeber

Subject     Author     Message Date     ID
Puzzle involving exponential Marie-Jo W. Dec-27-07 TOP
RE: Puzzle involving exponential alexb Dec-28-07 1
RE: Puzzle involving exponential Marie-Jo Dec-30-07 3
RE: Puzzle involving exponential mpdlc Dec-30-07 2
RE: Puzzle involving exponential Marie-Jo Dec-30-07 4
RE: Puzzle involving exponential mpdlc Dec-30-07 5
RE: Puzzle involving exponential Marie-Jo Jan-02-08 6
RE: Puzzle involving exponential Pierre Charland Jan-19-08 7

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alexb
Charter Member
2159 posts
Dec-28-07, 09:23 AM (EST)

1. "RE: Puzzle involving exponential"
In response to message #0

 I understand from your site that there are two approaches that are judged different: discrete and continuous. When the latter is presented there is a remark that the discretization differentiates between two sequences of events: stretching/crawling and crawling/stretching, and that at the limit there is no difference. It also argued that the answer provided by a discretization depends on the chosen unit interval. I think that the two argumnts should settle the problem. Neither the gravity nor the bugs are not concerned with the fact that they became a part of a puzzle. They act independently of us puzzle fans, and the bugs meet when they meet: at the unique moment in time. Since the descrete solution depends on the two aforementioned factors (sequence and interval) it may only provide a numeric approximation to the real solution.

Marie-Jo
guest
Dec-30-07, 12:49 PM (EST)

3. "RE: Puzzle involving exponential"
In response to message #1

mpdlc
Member since Mar-12-07
Dec-30-07, 11:34 AM (EST)

2. "RE: Puzzle involving exponential"
In response to message #0

 Beside the difference that arises in discretization process, I believe the solution posted have some inconsistencies. And at a first look in absence of tons of ingenuity like Archimedes brain we must resort in calculus or to solve numerically.If interested you can look the attach filempdlc
Marie-Jo
guest
Dec-30-07, 01:08 PM (EST)

4. "RE: Puzzle involving exponential"
In response to message #2

 >Beside the difference that arises in discretization process, >I believe the solution posted have some inconsistencies. And >at a first look in absence of tons of ingenuity like >Archimedes brain we must resort in calculus or to solve >numerically. >>If interested you can look the attach fileDear mpdlc,Thanks for your help... I had a look at your attached file and find it very complete. Can I post your answer on my site? I wonder, however, if there isn't a simpler way to help the visitors to understand the problem with clear step-by-step examples or references.My site is first of all intended for general interested people (puzzle math fans), I just want to put a 'spark' in the visitors' mind by giving them indications/information where to find the math tools to solve particular problems. The visitor should afterwards be able to understand and solve the problem by himself. Best wishes,Marie-Jo

mpdlc
Member since Mar-12-07
Dec-30-07, 01:40 PM (EST)

5. "RE: Puzzle involving exponential"
In response to message #4

 Of course you can posted, the only problem I see is that my typing cannot rival with the beautiful taste of the you Italian website.I tried initially to solve by discrete steps using Finite Differentials but I, a retired Naval Architect, must refresh a subject which I almost have not used in my career, especially since to find the equivalent of integration factor is FDE as straightforward as in ODE.I am traveling for new year, meanwhile in the airplane I maybe a get shortcut. If interested in use harmonic series in your puzzle conner, I will try to find some which I email to your site so you can reword and embellish them. mpdlc

Marie-Jo
guest
Jan-02-08, 08:27 AM (EST)

6. "RE: Puzzle involving exponential"
In response to message #5

 Thanks! I will appreciate it.Best wishes,Marie-Jo

Pierre Charland
Member since Dec-22-05
Jan-19-08, 11:18 PM (EST)

7. "RE: Puzzle involving exponential"
In response to message #0

 I have some references concerning a variation of this problem (one worm walking the length of a stretching rubber band).-- Martin Gardner, Time Travel, (The Rubber Rope, ch.9 #1 p.111)-- Martin Gardner, aha! Gotcha, (The Rubber Rope, p.145)-- Graham & Knuth & Pasternick, Concrete Mathematics 1st ed, (6.3 p.260; 9.49 p.479)AlphaChapMtl

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