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# When the minute hand of a clock points exactly at a full minute

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When the minute hand of a clock points exactly at a full minute, the hour hand is exactly three minutes away. Give all possible times in a 12-hour period that satisfy condition.

Oct 29, 2020

#1
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Here is my attempt at your question:

I have calculated the following times based on separation of the hour hand and the minute hand by exactly 3 minutes, or 3 x 6 degrees =18-degree angle.
Time starts at noon hour till midnight. Time is in hours, minutes and fraction of a minute(x 60 gives you seconds):

1 : 2.181818182
1 : 8.727272727
2 : 7.636363636
2 : 14.18181818
3 : 13.09090909
3 : 19.63636364
4 : 18.54545455
4 : 25.09090909
5 : 24
5 : 30.54545455
6 : 29.45454545
6 : 36
7 : 34.90909091
7 : 41.45454545
8 : 40.36363636
8 : 46.90909091
9 : 45.81818182
9 : 52.36363636
10 : 51.27272727
10 : 57.81818182
11 : 56.72727273

Oct 29, 2020
#2
+10583
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When the minute hand of a clock points exactly at a full minute, the hour hand is exactly three minutes away. Give all possible times in a 12-hour period that satisfy condition.

Hello Guest!

$$Z_{hour}=\frac{t}{12min}\\ t=12min\cdot Z_{hour}\\ t=1min\cdot Z_{minut}\\ t\in \{60\cdot 1\ minute\}$$

t [min] =              0   12    24    36    48    60          648  660  672  684  696 720

$$Z_{hour[min]}$$          0     2      4     6     8       5             50   52     54   56    58   60

$$Z_{minut}$$               0    12    24   36    48      0             0    12     24   36    48   60

The hour hand and the minute hand are never exactly three minutes apart when the minute hand points exactly to the minute on the dial.

!

Oct 30, 2020
edited by asinus  Oct 30, 2020
edited by asinus  Oct 30, 2020