# Clock time puzzle

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The hour and minute hands are at equal distance from the 6 hour, what time will it be exactly? There are 12 answers to this puzzle.  For every hour, there is a position of the minute hand such that the hour and minute hands are equal distance from the 6.

on 9th November 2015. on 10th November 2015.

EXPLANATION-
You need to solve (n+(180θ)/360)×30=θ, where n is the number of hours past 6 o’clock. The solution is
θ=18013(2n+1)deg

For different values of θ you would get different times.
For example, for n=6 you get θ=180 which is 12 o’clock.
For n=0 you get θ=180/13=13.8, which, using 30 degree=5 min, is  6h27m41.5s

Since the clock is symmetric than  we can calculate the time as if the distance should be the same from 12 o’clock.

The general approach to solving problems like this is to take into consideration the rate of change of the angle per minute.The hour hand makes a complete rotation of 360 degrees in 12 hours. In other words, 360 degrees in 720 minutes,which means 0.5 degrees per minute.The minute hand rotates 360 degrees per hour, meaning 6 degrees per minute.
And so the angle formula for the hour hand will be:0.5* the hour according to the hour hand * 60 (minutes in an hour) + 0.5* the number of minutes according to the minute hand.The formula for the minute hand is: 6 * the number of minutes according to the minute hand.So in our question if we mark the hour hand with H and the minutes hand with M we can write the question as:0.5*H*60 + 0.5*M = 360 -6M >> 6.5M = 360 -30HSo the solutions are:1) 1:50:462) 2:46:093) 3:41:324) 4:36:555) 5:32:186) 6:27:417) 7:23:048) 8:18:279) 9:13:5010) 10:09:1311) 11:04:3612) 12:00:00