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Moving Beyond a Proverb

The English have a proverb: A broken clock is right twice a day. That is comforting, and you would even know when it is right!

So, let's muddy the proverbial waters somewhat. Suppose you have a set of broken analog clocks, but the hands continue to move in a mathematically-describable pattern:

  1. Clock 1 gains one minute every hour
  2. Clock 2 loses one minute every hour
  3. Clock 3 runs backwards, but at the correct pace of a correct clock
Assume it is now exactly 15 minutes after six, and that none of the clocks have displays designating AM vs PM.

Question 1: For Clocks 1-3, what is the probability that each will show the correct time?

Question 2: When all three clocks show the same time, what time is it?

Question 3: Suppose Clock 4 is really confused, in that it gains time according to the Fibonacci sequence. That is, it gains 1 minute the first hour, 1 minutes the second hour, 2 minutes the next hour, 3 minutes the next hour, 5 minutes the next hour, etc... If it is now correctly set at exactly 15 minutes after six, when will the Fibonacci clock show the right time again?

Note: Like many other people, your clock may run correctly, but it is set a few minutes fast to be sure that you arrive somewhere on time. However, the situation is self-defeating, in that one can easily adjust knowing how many fast the clock is. Thus, someone has created "the Procrastinator's Clock. It's guaranteed to be up to 15 minutes fast. However, it also speeds up and slows down in an unpredictable manner so you can't be sure how fast it really is. Furthermore, the clock is guaranteed to not be slow...." This is the clock I need!


Hint: Generate an equation to show the time...you may have to use some variation of clock (or modular) arithmetic.

Another option is to model the clocks using a spreadsheet...


Solution Commentary: Sorry, no specific answers. The first question is quite straightforward, but the second question is a real poser! Since I made up the problems,. you can assume I probably made up the answers as well, and thus they would not be very "timely."