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Find time when hour and minute hands superimpose

  • Difficulty Level : Medium
  • Last Updated : 03 Dec, 2017

This puzzle is an extension of the famous interview problem, find angle between hour and minute hand at any given time. The questions asks us to find the time when they both superimpose.

The hour hand moves 360o in 12 hours and thus, 0.5 degrees in 1 min. The minute hands moves 360 degrees in 60 min, hence, 6 degrees in 1 minute. After H hours and M minutes,
Angle(Hour hand) = 0.5*(60*H + M)
and
Angle(Min Hand) = 6*M
For them to superimpose, both the above angles should be equal. Hence,
0.5*(60*H+M) = 6*M
(60*H+M) = 12*M
60*H = 11*M
M = 5.45*H

Now H varies from 0 to 11, we can correspondingly calculate the value of M for each H.
This gives times of: 0:00, 1:05.45, 2:10.90, 3:16.36, 4:21.81, 5:27.27. 6:32.72, 7:38.18, 8:43.63, 9:49.09, 10:54.54, and 12:00. (0.45 minutes are exactly 27.27 seconds)

References:
https://en.wikipedia.org/wiki/Clock_angle_problem

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