(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
