24 times...so simple..its equals to the hours in a day

22 times. (times given are approx) 1:05a 2:10a 3:15a 4:20a 
5:25a 6:30a 7:35a 8:40a 9:45a 10:50a 12:midnight etc. Both 
of the "eleven o'clock" overlaps never occur; they turn 
into the "midnight" and "noon" overlaps. The hour hand 
is "running away" from the minute hand.

The answer is twenty-three.

Most people quickly realize that the answer has to be 
twenty-four, give or take. The issue is nailing down 
that "give or take" part. Recognize, first of all, that 
there is nothing capricious about the overlaps. Both hands 
move at fixed speeds.

Therefore, the time interval between overlaps is a 
constant. This constant interval is a little more than an 
hour. At midnight, the hour and minute hands are exactly

superimposed. It takes an hour for the minute hand to make 
a complete circuit In that same time, the hour hand has 
moved 1/12 of a circuit to the numeral 1. It then takes 
another five minutes for the minute hand to catch up to 
where the hour hand was, in which time the hour hand has 
crept a bit farther.... Before getting sucked into a Zeno's 
Paradox, let's settle for the moment by saying that the 
interval is a little more than sixty-five minutes. We also 
know that the exact interval has to divide evenly into 
twenty-four hours, since the day ends as it started, with 
both hands up and overlapping. In fact, it has to divide 
evenly into twelve hours. The way the hands move in the 
P.M. is an exact replay of the way they move in the A.M. 

Focus on the twelve-hour period from midnight to noon. The 
hands cannot overlap twelve times in that period, for if 
they did, it would mean that the interval between overlaps 
was 12/12, or exactly one hour — and we know it's a bit 
more than sixty-five minutes. No, there must be eleven 
overlaps in a twelve-hour period. That means the interval 
between overlaps is 12/11 hour, which comes to 65.45 
minutes. This must be the precise interval that we balked 
at calculating a moment ago. Doubling eleven gives twenty-
two overlaps in a twenty four- hour period. Twenty-two is 
the answer — unless you want to split hairs. Should you 
count the overlap at the midnight that begins the day, and 
also at the midnight that ends the day, the answer is 
twenty-three.

It can be reduced to a fencepost problem... 

if you have two hands x (faster hand) and y (slower hand),
then the number of fenceposts is the number of circuits x
needs to make for y to make one complete circuit. The number
of overlaps of x and y is the number of spaces between the
fenceposts.

The answer is 44 times FOR TRUE OVERLAPS.
As an explanation, I will approach the answer in following 
steps. 

Notations:
Hour hand:=H minute hand:=M
Assumption: We consider only H and M (and not the hand for 
seconds counter) hence overlap of 2 hands.

1) A very tempting answer (which is also given above) would 
have been to consider time points such as 12:00 noon, 1:05 
am/ pm etc. and so on leading to final value of 24 but 
consider the following situation. Suppose that the overlap 
happened at 12:00 noon then, since H covers 30 degree angle 
in an hour whereas M covers 360 degree, how can the next 
overlap happen at 1:05 because then starting from 12:00 H 
has moved 30 degree in 1 hour to reach exactly the point 
for 1 o clock marker while in the same time M will move 
only 360 degree to reach 12 o clock marker agian to make 
the time as 1:00 PM and by the time M reaches the point for 
1 o clock marker, H will have moved a bit not to make it an 
overlap. 
Hence it will nullify the answer which is being discussed.

2) Now let us move towards the correct answer. Note that 
the angular speeds of the tips of H and M are 30 degree/ 
Hour and 360 degree / hour respectively.

3) Let us start from a point where both have already 
overlapped because if no such point is available then the 
question becomes redundant. H will indicate the tip of hour 
hand and M will do the same for the minute one.

4) Now had this been a problem of both H and M starting 
from the same point, moving on a stratight line with 
different speeds, they would have never met again. 

5) But here since they are moving on a circle, it is 
equivalent to the aforementioned example of straight line 
with the end points of the line identified to make them 
move round and round in a circle.

6) Let the starting point of overlap be A. After 1 hour M 
is 30 degree behind H (Recall the example in <1> ) and if 
it is going to overlap with H in next T hours then it has 
to cover 30 degree more than H does in this time (T hours). 
Since M cover (360-30=)330 degree more in 1 hour, it will 
take (30/330=) 1/11 hour for the next overlap.

7) so starting from the point A of first overlap it has 
taken (1+ 1/11=) 12/11 hour for another overlap to happen.
So ein 12/11 hour we have had 2 overlaps and hence in 24 
hours we will have (2 * (11/12) * 24)= 44 overlaps (unitary 
method).

8) Note that the point A of first overlap is an arbitraty 
choice.

The correct answer is 23. 
Sorry! there has been a mistake of double counting at the 
end of the above answer posted by me in hurry 
(Clicked "post answer" and could not take back). 
The calculation of 12/11 hour required for the second 
overlap is correct. But then in this time there is 1 more 
overlap (not 2)and hence in 24 hours time there will be 22 
more overlaps and taking the first one into account the 
number will be 23.

all of ya r wrong the correct answer is 22 get it throu 
ya's head OK

22 is crct
but y not 11:55 ???