login
Minute with hour hand overlap problem on analog clock. Fractions of seconds.
4

%I #23 Mar 15 2015 10:36:09

%S 0,3,6,9,1,4,7,10,2,5,8

%N Minute with hour hand overlap problem on analog clock. Fractions of seconds.

%C At which a.m. times h:m:s (with fractions of seconds) does the minute hand overlap the hour hand on an analog clock? This is problem 43 of the referenced Loyd/Gardner book, which also gives the solution (pp. 41-42, solution pp. 180-181 in the German version).

%D Sam Loyd, Mathematische Raetsel und Spiele, ausgewaehlt und herausgegeben von Martin Gardner, Dumont, Koeln, 1978, 3. Auflage 1997.

%D Sam Loyd, Mathematical puzzles, selected and edited by Martin Gardner, Dover, 1959.

%F a(n)/11 gives the fraction of the second for the (a.m.) hour h = n = 0,1,2,...,10 when the minute hand overlaps the hour hand on an analog clock, provided the minute is A178181(n), and the full second is A183032(n). See the eleven times given in EXAMPLE.

%F a(n) = floor((300*n)/11) (mod 12), n=0..10. (See the Mathematica code given by _Robert G. Wilson v_, and also the solution in Loyd's book with (27 + 3/11)s = 300/11 s.)

%e The eleven overlap times are:

%e 00:00:00 plus 0/11 s,

%e 01:05:27 plus 3/11 s,

%e 02:10:54 plus 6/11 s,

%e 03:16:21 plus 9/11 s,

%e 04:21:49 plus 1/11 s,

%e 05:27:16 plus 4/11 s,

%e 06:32:43 plus 7/11 s,

%e 07:38:10 plus 10/11 s,

%e 08:43:38 plus 2/11 s,

%e 09:49:05 plus 5/11 s,

%e 10:54:32 plus 8/11 s.

%e The next time would be 12:00:00.

%t Table[ Mod[ Floor[300/11 n], 12], {n, 0, 10}]

%K nonn,fini,full,easy

%O 0,2

%A _Wolfdieter Lang_, Dec 20 2010