The Hands on a Clock

Date: 2/1/96 at 10:25:11
From: Anonymous
Subject: time (about the hands on the clock)

	A few students and myself were browsing the Web, when we 
came upon a question dealing with the hands (minute, hour) of a 
clock.  The question is: How many times, in a 24 hour period, do 
the hour and minute hands meet?  We've tried and we have found a 
possible answer. We think the answer is 25 times.  The question 
informed us that the answer was not 24.  What do you think the 
answer is?

Thank you!

Mike Petersen
Mike Rock
Dan Langley
Drew Hackett
Brett Johnson

Date: 2/1/96 at 17:16:13
From: Doctor Syd
Subject: Re: time (about the hands on the clock)

Dear Mike, Mike, Dan, Drew, and Brett:

Hello!  This is an interesting question!  I'm curious about how 
you came up with an answer of 25.  I think the answer is a bit 
less.  Let's look at a 12-hour period of time.  Then, using 
similar reasoning, you can, on your own, figure out what the 
answer is for the 24 hour period of time.  

Okay, so let's consider the 12 hour period between 12:01 pm and 
12:01 a.m.  At 12:01 p.m., the minute hand and the hour hand 
aren't at an intersection point, right? (They were just a minute 
before, at noon, but not now!)

Consider the next period of time from 12:01 to 1:00.  All that 
happens here is that the minute had moves all the way around, 
once, to the 12 spot, and the hour hand moves from being very very 
close to the 12 to being on the 1.  Since the minute hand started 
in front of the hour hand, the minute hand moves faster than the 
hour hand, and the minute hand ends before the hour hand (at 
1:00), we know that in this period of time there were no 
intersections.  

Now, consider the time period from 1:00 to 2:00.  The hour hand 
moves from 1 to 2, and the minute hand makes one complete 
revolution, right?  So, we have one intersection time in this 
hour.  Similarly, there is one intersection time from 2:00 to 
3:00; 3:00 to 4:00, and so on until we get from 10:00 to 11:00.  
Then, to complete our cycle, we have to count the number of 
intersections from 11:00 to 12:01, which is again going to be just 
1 time (at 12:00).  So, let's count up these intersections...it 
looks like we have 11 total intersections, right?  Does this make 
sense to you?

Do you buy the argument?  You may be wondering why I thought of 
the starting time as 12:01 and not just 12:00, and there is a good 
reason for that!  I  was getting confused about when I was 
counting the intersection at 12:00 and wanted to count it twice.  
The situation is more intuitively clear when we start at a time 
that is NOT an intersection point, in my opinion.

So, maybe using similar reasoning, you can figure out the answer 
for the 24 hour period.  Let us know if you need more help or 
would like to check your answer!  Thanks for writing!

-Doctor Syd,  The Geometry Forum