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.