Math

[expat]

Well, stating *how* to do it and *doing* it are two separate things, no?

[Bangkoktraveler]

What if it is an old watch that has only one hand?

[TheCorinthian]

Answer: After 12 o'clock, the minute hand races ahead of the hour hand. By the time the minute hand has gone all the way round the clock and is back at 12, one hour later (i.e., at 1 o'clock), the hour hand has moved to indicate 1. Five minutes later, the minute hand reaches 1 and is almost on top of the hour hand, but not quite, since by then the hour hand has moved ahead a tiny amount more. So the next time after 12 that the minute hand is directly over the hour hand is a bit after 1:05. Similarly, the next time it happens is a bit after 2:10. Then a bit after 3:15, and so on. The eleventh time this happens, a bit after 11:55, has to be 12 o'clock again, since we know what the clock looks like at that time. So the two hands are superimposed exactly 12 times in each 12 hour period.

 

To answer the second part of the puzzle, you have to figure out those little bits of timer you have to keep adding on. Well, after 12 o'clock there are eleven occasions when the two hands match up, and since the clock hands move at constant speeds, those 11 events are spread equally apart around the clock face, so they are 1/11th of an hour apart. That's 5.454545 minutes apart, so the little bit you keep adding is in fact 0.454545 minutes. The precise times of the superpositions are, in hours, 1 + 1/11, 2 + 2/11, 3+ 3/11, all the way up to 11 + 11/11, which is 12 o'clock again.

[expat]

...snip...since the clock hands move at constant speeds, those 11 events are spread equally apart around the clock face...

 

That's exactly the correct thinking to get it easily.

 

In other words, they meet up every 11th of an hour, or every 5 minutes, 27 and 3/11ths seconds.

[Redbaron]

11 times in 12 hours, yes

[TheCorinthian]

11 times in 12 hours, yes

 

 

Nope. 12.

[TroyinEwa/Perv]

So whomever answered "24 times" was right.....give the man his prize, Johnny.

[Redbaron]

I would have thought since they overlap less than once an hour it couldn't be 24 times... ie they overlap just over every hour and 5 minutes.. I got these times (roughly)

 

12:00

1:05

2:11

3:16

4:22

5:27

6:33

7:38

8:44

9:49

10:55

12:00

1:05

2:11

3:16

4:22

5:27

6:33

7:38

8:44

9:49

10:55

 

which is 22 times. They don't overlap during the hour following 11:00 (am or pm), that's why it's 22.. Unless you count 1200 twice (as in noon and midnight) it is 11x in a 12 hour period.

[TroyinEwa/Perv]

Just going by the teacher's answer. Teacher gives the grades.

[Sporty]

Would it not be the case at 11:59 + 99/100 they move together, to Noon or Midnight ? Therefore 24 times, twice in the smallest fraction of a second apart?

 

Or 23 times, because Midnight (00:00) is the new day ? Or does the move off of Midnight to 00:00:01 also happen, smallest fraction ? Back to 24?

 

What about the smallest fraction before and after Noon ? Would that be 24, 25 or 26?