Monday, January 3, 2011

How many times do the hours and minutes hand overlap in 24 hours

There is a simple mathematics to this puzzle. All you have to do is 
  • 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:
h=5T+(t/60)*5

Equating the two we get:
t=60/11T
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.