ZENO'S COFFEEHOUSE Results #12
The Twelfth Coffeehouse Challenge
Here is the problem, as presented. along with the responses. Congratulations to the winners!
Space and Time Fly
Having become somewhat familiar with puzzles from the namesake of the Coffeehouse, Maggie realized that Zeno's logic often involved calculation of infinite sequences and the like, which are occasionally mind-boggling, to say the least. One night, late
at the Coffeehouse, she was challenged with the following problem, posed by another Zeno devotee:
Consider this thought-experiment. Imagine two train locomotives facing each other on a single track, two hundred miles away from each other. Both locomotives begin moving toward the other simultaneuously at fifty miles per hour, ending in a head-on cr
ash. (Assume that they accelerate instantaneously to that speed.) Sitting on the front of one of the locomotives at the beginning is a lonely fly. When the two begin to move, the fly will move quickly toward the other locomotive; when it reaches it, it wi
ll turn around and return to the first locomotive, then turn around again and fly back to the second; it will continue shuttling back and forth until the locomotives crash (when it will get nicely and thoroughly smashed between them). The fly flies a cons
tant seventy-five miles per hour. Question: before the spectacular finale, how far will the fly travel??
Maggie's quick mind began racing, calculating limits and all that. Can the Zeno's patrons help out? How far will the fly tavel, and how did you figure it out?
For starters, I told Jamie Dreier that I would include his charming
account of Von Neumann, which seemed particularly apt:
Well, the story is that someone once asked this question of John von
Neumann. After a moment, von Neumann gave the correct answer. The
puzzle-poser was disappointed. "Most people try to sum the infinite
series," he said. Von Neumann replied, "Is there a different method?"
-Jamie
Here's an example of a winning entry:
It was submitted by Jason
Johns on Tuesday, April 15, 1997 at 03:23:
state: GA
city: Moody AFB
country: USA
comments: The fly will travel 150 miles. This is due to the fact that the
trains will each travel 100 miles at 50 mph and crash in 2 hours,
therefore if the fly travels at 75 mph in two hours it will have traveled
150 miles.
From nobody@gumbo.valdosta.peachnet.edu Wed Apr 9 05:01 EDT 1997
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username: lauri.kultti@helsinki.fi
city: helsinki
country: FINLAND
comments: The trains crash after two hours - each having travelled 100 miles. In two hours the fly - at the speed of 75m/h - travels 150 miles. So, there is your answer. On the other side, you can understand the question a little differently (as one might
suspect is the case because the question itself is old familiar one). You ask how far will the fly travel. Well, at the beginning the fly is in the front of the other locomotive. At the end the fly is half way to the other locomotives starting point, tha
t is 100 miles from it's (fly's) original position. So, the fly,in matter of fact, has travelled only 100 miles.
The answer is 150 miles or 100 miles, depending on how you interpret the question.
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From nobody@gumbo.valdosta.peachnet.edu Wed Apr 9 09:07 EDT 1997
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From: (Mark Hartswood)
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username: mjh@dcs.ed.ac.uk
city: Edinburgh
country: Scotland
comments: The trains will collide after two hours (the time taken
for each to travell 100 miles). Thus the fly would have
been airborn for two hours - at seventy-five miles per hour
it will travel 150 miles in this time.
What infinities?
BTW, its a shame neither driver could understand the warning
messages given by the fly on it's desperate mercy dash between
the trains.
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From nobody@gumbo.valdosta.peachnet.edu Wed Apr 9 16:15 EDT 1997
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username: James_Dreier@brown.edu
state: RI
city: Providence
country: USA
comments: The fly will travel 150 miles before it is squished.
It will take 2 hours for the locomotives to reach each other.
In 2 hours, a fly traveling at a constant 75mph will, of
course, travel 150 miles.
(I hope you tell the [probably apocryphal] John von Neumann
story when you give the answer!)
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From nobody@gumbo.valdosta.peachnet.edu Wed Apr 9 16:15 EDT 1997
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obody@gumbo.valdosta.peachnet.edu Wed Apr 9 16:29 EDT 1997
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username: antrit@mbox.vol.it
city: PALERMO
country: ITALY
comments: The two locomotives take 2 hours before crashing.
A fly flying a costant 75 mph runs 150 miles in the meanwhile.
It's OK?
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From nobody@gumbo.valdosta.peachnet.edu Fri Apr 11 11:21 EDT 1997
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state: fl
city: tampa
country: USA
comments: i have another question:
which train will the fly be on when the other train impacts it?
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From nobody@gumbo.valdosta.peachnet.edu Sun Apr 13 19:06 EDT 1997
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username: jd@w3link.com
state: NC
city: Hickory
country: USA
comments: The fly will fly 150 miles. (Sum n=0->inf 120/5^^n)
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From nobody@gumbo.valdosta.peachnet.edu Tue Apr 15 03:23 EDT 1997
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state: GA
city: Moody AFB
country: USA
comments: The fly will travel 150 miles. This is due to the fact that the
trains will each travel 100 miles at 50 mph and crash in 2 hours,
therefore if the fly travels at 75 mph in two hours it will have traveled
150 miles.
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From nobody@gumbo.valdosta.peachnet.edu Tue Apr 15 21:52 EDT 1997
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username: eagleard@comcen.com.au
state: Vic
city: Melbourne
country: Australia
comments: Answer is 150 miles.
The trains travel 100 miles each before crashing. As they
travel at 50 miles per hour, two hours will elapse before
the crash. The fly travels at a constant 75 mph; hence the
fly will travel (75 mph* 2h) = 150 miles.
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From nobody@gumbo.valdosta.peachnet.edu Fri Apr 18 00:33 EDT 1997
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username: ludwig@susqu.edu
state: Pa
city: Selinsgrove
url: http://www.susqu.edu/students/l/ludwig
country: USA
comments: From v=d/t, the trains will travel for a total
time of t=d/v = 200mi/(50mi/hr) = 4 hours.
The fly flies at a speed of 75 mi/hr.
>From v=d/t d=vt = (75mi/hr)*(4 hr) = 300 miles
Therefore, the fly will travel 300 miles before it
is crushed. No need for an infinite series.
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From nobody@gumbo.valdosta.peachnet.edu Sun Apr 20 07:18 EDT 1997
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From: (Casey O'Callaghan)
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username: casey@injersey.com
state: NJ
city: Toms River
url: http://nj5.injersey.com/~casey
country: USA
comments: Assuming instantaneous direction and velocity changes, the fly will travel 150 miles before being obliterated. This is because the trains will travel for two hours before crashing; since the fly is going 75mph, and will travel for two hours, he
will travel 150miles.
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From nobody@gumbo.valdosta.peachnet.edu Sun Apr 20 13:07 EDT 1997
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username: alanrgreenland@hotmail.com
state: IN
city: Kokomo
country: USA
comments: With a closing speed of 100 mph (2 x 50 mph each), the
trains will collide in two hours (since they begin 200
miles apart). The fly, at a constant speed of 75 mph, will
cover 150 miles in that time. [The harder question is: how
many times will the fly have to turn around?!]
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From nobody@gumbo.valdosta.peachnet.edu Mon Apr 21 23:12 EDT 1997
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username: mfpatton@mindspring.com
state: AL
city: Montevallo
url: www.mindspring.com/~mfpatton/
country: USA
comments: The trains each go their half of the 200 miles in 2 hours.
Then they collide. So the fly flies 150 miles, as the crow flies.
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From nobody@gumbo.valdosta.peachnet.edu Tue Apr 22 01:21 EDT 1997
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username: katzkatz@netvision.net.il
country: Israel
comments: The fly will travel 150 miles. I takes 2 hours for the
locomotives to crash and that is what the fly can do at the
speed of 75 miles per hour.
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From nobody@gumbo.valdosta.peachnet.edu Tue Apr 22 04:36 EDT 1997
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username: nande@pacific.net.sg
country: singapore
comments: four hours to collision ->300 miles
fly will be knackered
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From nobody@gumbo.valdosta.peachnet.edu Tue Apr 22 07:54 EDT 1997
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username: deacon@merlin.net.au
state: SA
city: Adelaide
country: Australia
comments: Distance: an unmeasurable distance unless space can be
quantized ( to the nearest Planck length? ) and if so, does
space have a wave equivalent form?
Time: forever ( for the fly )
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From nobody@gumbo.valdosta.peachnet.edu Tue Apr 22 18:25 EDT 1997
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username: wag@heald.edu
state: CA
city: San Francisco
country: USA
comments: 100 miles (less the infinitesimal amount of distance trains travel as the fly is "turning around quickly".
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From nobody@gumbo.valdosta.peachnet.edu Thu Apr 24 15:49 EDT 1997
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username: branch@humnet.ucla.edu
state: CA
city: Los Angeles
country: USA
comments: Enough information is given about the trains to entail that they will crash in two hours; since the fly's speed is a constant 75 m.p.h., it will travel 150 miles.
There's a story, probably apocryphal, that when Johnny van Neumann was presented with this problem, he stared off into space for a few seconds, and then gave the correct answer. His interlocutor said, "Good, you saw the trick, and didn't try to sum the s
eries," to which von Neumann replied, "Trick? What trick? I did sum the series."
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From: (Linda Sayle)
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username: Linda@doughty.source.co.uk
city: London
country: England
comments: It will be 2 hours before the trains crash. The fly will fly
150 miles in two hours. (you say he flies at 75 miles per
hour)The number of journeys he makes between the trains is
irrelevant.
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From nobody@gumbo.valdosta.peachnet.edu Fri May 2 10:45 EDT 1997
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username: mfepgdp@fs1.art.man.ac.uk
city: Manchester
url: http://www.man.ac.uk/~mfepsdp/potts.htm
country: England
comments: Ans : 150 miles
The fly keeps moving at 75mph until the trains crash.
The trains start 200 miles apart and are moving together
at a combined speed of (50+50) = 100 mph.
So the trains travel for (200/100) = 2 hrs
Hence the fly keeps travelling for 2 hrs.
So the distance covered by the fly = (2*75) = 150 miles
Poor fly :-<
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From nobody@gumbo.valdosta.peachnet.edu Sun May 4 13:11 EDT 1997
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username: lance@freelance.com
city: Jersey City
url: http://www.freelance.com
country: USA
comments: 150 mi., relative to an assumed Galilean frame of reference.
The locomotives are 200 mi. apart at the beginning. Distance
Between them is diminishing at 100/hr. Therefore, total
flight time for the fly will be 2 hrs X 75mph = 150.
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From nobody@gumbo.valdosta.peachnet.edu Mon May 5 13:00 EDT 1997
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username: johnlong@udel.edu
state: DE
city: wilmington
country: USA
comments: If the question is how far away from the starting point the
fly will be when it get smooshed, then regarless of how many
times he changes direction the answer will be 100 miles. The
point where the trains will meet and where the fly will
perish will be exactly halfway between where the trains start
from because they are moving at the same speed.
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From nobody@gumbo.valdosta.peachnet.edu Mon May 5 13:19 EDT 1997
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Email: johnlong@udel.edu
Name: John Long
Rating: Way Cool
Text: I love Zeno's Coffeehouse! I'm sure I'll
visit again. In the bio about Zeno, the
word "impossible" is misspelled.
HomePage: none
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From nobody@gumbo.valdosta.peachnet.edu Tue May 6 00:51 EDT 1997
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username: 943872@swin.edu.au
state: VIC
city: Melbourne
country: Australia
comments: 75*2 = 150 miles
The trains will hit each other after exactly two hours. At 75 mp/h,
the fly will travel 150 miles in two hours.
Alternative answer: 100 miles, because it stops travelling at the
collision point, which is 100 miles from where it started.
(Kindly select your frame of reference _now_.)
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From nobody@gumbo.valdosta.peachnet.edu Wed May 7 17:11 EDT 1997
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username: jha@stac.com
state: CA
city: San Diego
country: USA
comments: Train A is assumed to be at point 0, traveling with positive motion, right.
Train B is assumed to be at point 200, traveling with negative motion, left.
Acceleration is assumed to be instantaneous.
Velocity is assumed to constant.
Velocity of A is 50 m/h. Velocity of B is -50 m/h.
Distance equation yields, A(x)=50t + 0, B(x)=-50t + 200.
Examination reveals that the trains will collide in two hours.
A(x) = B(x) (collision)
50t = -50t +200
t = 2
Seeing that the fly travels at 75 m/h, at a constant velocity,
it will cover 150 miles in two hours.
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From nobody@gumbo.valdosta.peachnet.edu Thu May 8 12:56 EDT 1997
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username: jimsang@bu.edu
state: ma
city: New Bedford
url: none
country: USA
comments: The only thing you have to know is that rate times time equals distance. The trains will crash after 2 hours because they crash halfway at 100 miles going 50 miles an hour so the total time the fly is also flying is two hours as well. Going 75
miles an hour times two hours yields a travelled distance of 150 miles.
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From nobody@gumbo.valdosta.peachnet.edu Thu May 8 16:18 EDT 1997
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username: threno@julian.uwo.ca
state: ON
city: London
country: CANADA
comments: Too easy! The trains will collide in 2 hours. Since the fly travels at 75 mph, it will have traveled a total of 150 miles before the "spectacular finale".
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From nobody@gumbo.valdosta.peachnet.edu Fri May 9 11:10 EDT 1997
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username: jrp@netspace.org
state: RI
city: Providence
url: http://www.netspace.org
country: USA
comments: The trains will meet after 2 hours, since it will take each
one 2 hours to get to the midpoint. In 2 hours, a fly
moving at 75 miles per hour will have traveled 150 miles.
No need for limits here - simple mechanics of motion.
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From nobody@gumbo.valdosta.peachnet.edu Sat May 17 21:37 EDT 1997
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username: rrfrye@aol.com
state: CA
city: Newark
country: USA
comments: The fly will travel 150 miles.
At 50 mph the trains will crash in 2 hours. The fly travels
constantly, at 75 mph, for those 2 hours.
The description of the flies flight from train to train is
an interesting distraction.
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From nobody@gumbo.valdosta.peachnet.edu Tue May 20 17:45 EDT 1997
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username: joeriben@zedat.fu-berlin.de
city: Berlin
country: Germany
comments: That's easy: the fly will travel 150 miles. It takes 2 hours
until the trains will crash. Within this 2 hours, the fly
moves constantly with a speed of 75mph.
-> 2h * 75mph = 150 miles. (hey - you mean the time the fly
is travelling is 8/5 + 8/25 + 8/125 + 8/625 + ... ? - but
it's not the fly who creates time!)
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From nobody@gumbo.valdosta.peachnet.edu Thu May 22 05:26 EDT 1997
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From: (A. Lopez Ariste)
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username: arturo@mesopa.obspm.fr
city: Paris
url: http://uhuara.obspm.fr/~arturo
country: France
comments: 150 miles! The two locomotives take 2 hours before they
crash. The fly spends 2 hours flying at a constant velocity
of 75 mph, then the answer.
This problem was solved in less than 10 seconds by J. von
Newmann, and when he was asked if he already knew the
"trick" to solve directly , he answered quite surprised:
"There is something faster than to calculate the series?!"
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From nobody@gumbo.valdosta.peachnet.edu Thu May 22 09:00 EDT 1997
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username: darkwriter
state: Mb
city: Winnipeg
country: Canada
comments: 150 miles, as the trains will not collide for 2 hours, the fly will fly for 2 hours at 75 miles per hour
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From nobody@gumbo.valdosta.peachnet.edu Thu May 22 09:00 EDT 1997
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username: darkwriter
state: Mb
city: Winnipeg
country: Canada
comments: 150 miles, as the trains will not collide for 2 hours, the fly will fly for 2 hours at 75 miles per hour
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From nobody@gumbo.valdosta.peachnet.edu Thu May 22 18:30 EDT 1997
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username: ionescu@accent.net
state: PQ
city: Dorval
country: CANADA
comments: The fly flies two hours (200/100=2) then 150 miles.
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From nobody@gumbo.valdosta.peachnet.edu Fri May 23 17:49 EDT 1997
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username: shack@esinet.net
state: VA
city: Charlottesville
country: USA
comments: The locomotives are closing at 100 miles per hour (each travels at 50 miles per hour). They have 200 miles to cover, that means they will crash in 2 hours. The fly travels at 75 mph, so in those 2 hours it will travel 150 miles.
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From nobody@gumbo.valdosta.peachnet.edu Sun May 25 15:10 EDT 1997
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username: mcklusky@ix.netcom.com
state: Ky
city: Lexington
country: USA
comments: Because both trains start out simultaniously, at the same
speed they should meet at the 1/2 way point.
This distance would be 100 miles from either starting point.
Traveling at 50mph it would take each train 2 hours to
reach this point and colide...
The fly, beginning his journey at the same time
the trains do... would also be traveling for 2 hours at
a constant rate of 75mph.
Provided that when the fly (moving at 75mph) does not
disintegrate when meeting a train moving in the opposite
dirction at (50mph) it should cover 150miles.
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From nobody@gumbo.valdosta.peachnet.edu Tue May 27 19:33 EDT 1997
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From: (Erich Plondke)
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username: ejplondk@grits.valdosta.edu
state: GA
city: Valdosta
url: http://www.freenet.tlh.fl.us/~plonk
country: USA
comments: The trains move at 50 Mi/Hr.
The trains meet after 2 hours.
The fly moves at 75 Miles/Hour.
The fly travels 150 miles.
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From nobody@gumbo.valdosta.peachnet.edu Fri May 30 00:19 EDT 1997
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Below is the result of your feedback form. It was submitted by Terence Bradshaw () on Friday, May 30, 1997 at 00:19:12
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username: tbradshaw@dclink.com
state: DC
city: Washington
url: http://www.dclink.com/poohbear
country: USA
comments: At first I made the mistake of actually trying to figure everything, but then I remembered that this is a Zenoic problem. Thus, the answer is that the fly will fly an infinite distance. I say this because the distance and time here are constan
tly decreasing. This wouldn't be a problem if the fly stayed put on one of the trains. Then he'd simply travel a hundred miles before going splat. But since he decides to be strange anf fly between the two back and forth he will constantly be travellin
g a shorter distance. Eventually the reductions in distance become infinitesmal, as does the time taken, so the fly is caught in infinity. Of course, back in the real world, the fly is dead.
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From nobody@gumbo.valdosta.peachnet.edu Mon Jun 2 13:44 EDT 1997
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Date: Mon, 2 Jun 1997 13:44:41 -0400 (EDT)
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To: rbarnett@grits.valdosta.peachnet.edu
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Below is the result of your feedback form. It was submitted by Ramon A. Vicioso () on Monday, June 2, 1997 at 13:44:41
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username: vicioso@msn.com
state: Fl
city: Melbourne Beach
country: USA
comments: 100 miles
Since the "spectacular finale" occurs midway between the two train starting points and the fly is a participant in the "spectacular finale" the fly travels 100 miles.
(The answer would be different if we asked the question, how far would the fly fly?)
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From nobody@gumbo.valdosta.peachnet.edu Thu Jun 5 20:53 EDT 1997
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To: rbarnett@grits.valdosta.peachnet.edu
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Below is the result of your feedback form. It was submitted by Tom () on Thursday, June 5, 1997 at 20:53:44
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state: Tx
city: Houston
country: USA
comments: 100 miles
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