How many times a day a clock’s hands overlap?

Seem likes many have been beating around the bush with lengthy answers. I think this analytical solution will answer it once and for all:

Let t be the time (in minute) from 00:00 when the two hands first overlap. The angle each hand makes to number 12 is:

minute hand: 360/60t = 6t (degree)

hour hand: 360/12*60 = 0.5t (degree)

Two hands overlap if 6t mod 360 = 0.5t

equivalently: 0.5t + 360k = 6t (for non-negative integer k=0,1,2,...)

OR t = 360k/5.5

So within a day you need t < 24*60 = 1440 => max k = 22

We can easily find the exact moment where two hand matches.

First moment after 0:00 is for k=1 => t = 65.4545 or 1h5 minute and 27.272727.. seconds

and so on.