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The Hands on a Clock

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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
informed us that the answer was not 24.  What do you think the

Thank you!

Mike Petersen
Mike Rock
Dan Langley
Drew Hackett
Brett Johnson
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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

-Doctor Syd,  The Geometry Forum

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