Anne and Mike have an argument about the hands of the clock:

Anne: "The minute hand takes one hour to make a complete circle. Therefore, it will pass the hour hand once every hour!"

Mike: "But by the time the minute hand has made a full circle, the hour hand will have moved 1/12th of the circle ahead. And by the time the minute hand gets to that spot, the hour hand will have moved forward yet again. This will continue indefinitely, so it is obvious that contrary to what it may seem like, the minute hand will never pass the hour hand."

In reality, how often does the minute hand pass the hour hand?

Mike's arguement is the same as Zeno's paradox of Achilles and the tortoise. There are better solutions that can be found elsewhere on the web, but suffice it to say that Mike assumes an inifite sum of numbers is always infinity, which is not true.

One way of finding the solution is to take the number of passings divided by the total time. In the span of 12 hours (12:00 to 12:00), the hour and minute hand pass 11 times (once during 12, 1, 2,..., 8, 9, and 10 - they do not meet during the 11th hour, because when they do meet it's 12:00 again). So you have 11 passings in 12 hours, which is 12/11 hours per passing, slightly more than once every 65 minutes, 27.2727 seconds.

Posted by Ender on 2002-05-23 04:27:41