I don’t have any specific puzzle in this particular topic but I thought I can cover the bases so that any problem based on this concept can be answered.
There is a ready-made solution from Wikipedia which has formulas to figure the angle out,
Degree_H = (60H + M)/2
Degree_M = 6M
Angle = | Degree_H – Degree_M |
or
Angle = | (60H – 11M) / 2 |
eg: For Time 6:43,
Degree_H = (360+43) / 2 = 201.5
Degree_M = 6* 43 = 258
Angle = | 201.5 – 258 | = 56.5
The analytic method is simple for some times (Hours only like HH:MM = x:00 or times with perfect 60:min ratio like 6:15, 6:30)
Note: i) For every minute movement of the minute hand, the hour hand moves too. So angle between 6:30 is not 0 degrees.
ii) The problem is complicated if the minute mentioned is some odd number lik 6:11 or 6:43 and so on.
Personally, I just remember the formula and Note (i)