There is a somewhat related anecdote. Two trains are 100 miles apart on a straight track, facing each other and travel at 25 miles per hour toward the other. At the same time, a fly flies at 100 miles an hour from one train to the other and, when it reaches the other train, turns around instantaneously and flies toward the other train, and so on. When the trains crash, what is the total distance the fly flew?
There is an easy way and a harder way to compute the answer. Someone posed the question to John von Neumann. After a moment, he answered, 200 miles. Correct. Now, Johnny, how did you figure it out? I summed the series. On Tue, Apr 1, 2014 at 9:40 AM, Jose Mario Quintana < [email protected]> wrote: > (Offline) > > The hour hand and the minute hand of a clock are superimposed at 0:0 > (12:00). When is the next time they will be superimposed again? One can > approach the solution time as follows: one hour later the minute hand is > where the hour hand was an hour ago; however, the hour hand has moved to > the 5 minutes (one o'clock) position; 5 minutes later the minute hand is > where the hour hand was 5 minutes ago; however, the hour hand has moved to > the 5 + 5 % 12 minutes position; and so on. Producing the next time the > hands will be superimposed is reduced to finding the sum of the series (the > limit of the partial sums). > > The situation has been compared to Zeno's Achilles and the tortoise > paradox. Furthermore, many argue that Zeno's Achilles and the tortoise > paradox cannot withstand "modern" mathematical analysis; nonetheless, a few > still contend that the issue is not really settled yet because the > existence of a limit for a sequence of partial sums does not necessarily > imply that a limit has been reached. There is a related pronouncement: > > "I protest against the use of infinity magnitude as something accomplished, > which is never permissible in mathematics. Infinity is merely a figure of > speech, the true meaning being a limit." > > C.F. Gauss > > (Incidentally, Gauss could have been a J addict: "I am never satisfied > until I have said as much as possible in a few words.") > > In any case, Zeno apparently was a resourceful man and had several > arguments against the possibility of motion; one in particular seems to > defy analysis. More on that subject and a solution for finding all the > superimposing hands times, expressed in hours and minutes, follow after the > countdown... > > ,.@|.@i. 31 > 30 > 29 > 28 > 27 > 26 > 25 > 24 > 23 > 22 > 21 > 20 > 19 > 18 > 17 > 16 > 15 > 14 > 13 > 12 > 11 > 10 > 9 > 8 > 7 > 6 > 5 > 4 > 3 > 2 > 1 > 0 > > > This is my quick and lazy solution to the puzzle: There are 11 times when > the hands superimpose because the hour hand completes a full revolution in > 12 hours (and the hands will be again superimposed). In addition, the > times must be evenly distributed because the hands move around at constant > rates. The rest is also easy (using J), for example, > > (<. ([ (,"0) 60 * -~) ])@(12&% * i.) 11 > 0 0 > 1 5.45455 > 2 10.9091 > 3 16.3636 > 4 21.8182 > 5 27.2727 > 6 32.7273 > 7 38.1818 > 8 43.6364 > 9 49.0909 > 10 54.5455 > > Golden points: find all the superimposing times for a modified clock where > the short hand completes a (PHI -1) revolution per hour (PHI is the golden > section: 1.61833988... ). > > Zeno's dichotomy paradox: "that which is in locomotion must arrive at the > half-way stage before it arrives at the goal "is similar to the Achilles > and the tortoise paradox but backwards; so, the approximating sum cannot > even start because there is no (chronologically speaking) first term. The > issue has been depicted vividly via a thought experiment; here is my > recollection, with a lot of color added in the process: > > Zeno's and his paradoxes of motion have been challenged experimentally and > he is engaged in a 500 amphoras of wine bet (the amount has been carefully > chosen by the annoyed challengers): a well-marked hungry female fly will be > released at point A, which is very close to the destination point B (where > a bait of rotten meat has been set). If the fly does not reach point B in > an already agreed reasonably short period of time Zeno wins; otherwise, he > is a looser, once and for all. > > Zeno wants to make absolutely sure the fly will never reach her > destination. He knows there are plenty of infallible fly zappers for hire > but they charge 1,000 amphoras of wine per kill (infallibility does not > come cheap); thus, defeating the purpose of hiring one. However, Zeno > knows best... He hires one zapper with strict instructions to zap the fly > at the exact instant when the fly reaches the midpoint (1%2 point); he > hires another zapper with strict instructions to zap the fly at the exact > instant when the fly reaches the 1%4 point, and so on. > > Zeno then rests smiling mischievously while sipping some wine knowing full > well that there is plenty more coming his way and his reputation will > remain intact (he could not even be accused of any wrong doing): the fly > cannot start her journey without being zapped AND no zapper would be able > to claim the kill! > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
