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
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