- find the time interval it takes for the first overlap to occur
- inverse it to get overlaps per hour
- multiply by 24 hours to get the answer.
So, lets find the time interval of the first overlap.
at T hours and t minutes the location of the minute hand (m) will be (in terms of a circle of 60 degrees):
m = t
and hour hand:
Equating the two we get:
So, an overlap occurs after 60/11 minutes in the first hour
and therefore after
1+(60/11)/60 hours = 12/11 hours
therefore overlaps per hour = 11/12
and in 24 hours = 24*11/12 = 22 overlaps.