Yesterday we proved that the clock's two hands meet and point in the same direction 11 times between 6:00AM and 6:00PM (link below). Can you figure out the exact time(s) of their "meetings". Example: 12:00:00 noon.

civonamzuk
May 9, 2015

#4**+15 **

Exellent job, Alan! If you don't mind, I'll place my answer right beneath yours..

06:32:43.636363 AM

07:38:10.909090 AM

08:43:38.181818 AM

09:49:05.454545 AM

10:54:32.727272 AM

12:00:00 NOON

01:05:27.272727 PM

02:10:54.545454 PM

03:16:21.818181 PM

04:21:49.090909 PM

05:27:16.363636 PM

civonamzuk
May 9, 2015

#2**+10 **

@CPhill :/ It took me awhile to calculate all those numbers. But I did it. I'm sure you're gonna come up with the right numbers, soon. I helped you with 1 already. Btw, there's a shortcut in figuring out this one, but I realised it at the very end.

civonamzuk
May 9, 2015

#3**+10 **

These are the times I came up with by equating the two functions, littlehand and bighand, that I used to generate the graph in yesterday's problem. They are in hour-minute-second format on a 24-hour clock.

How do these times compare with yours civonamzuk?

.

Alan
May 9, 2015

#4**+15 **

Best Answer

Exellent job, Alan! If you don't mind, I'll place my answer right beneath yours..

06:32:43.636363 AM

07:38:10.909090 AM

08:43:38.181818 AM

09:49:05.454545 AM

10:54:32.727272 AM

12:00:00 NOON

01:05:27.272727 PM

02:10:54.545454 PM

03:16:21.818181 PM

04:21:49.090909 PM

05:27:16.363636 PM

civonamzuk
May 9, 2015

#5**+10 **

Some of my answers are slightly inaccurate. I rounded ** down **all the "seconds" numbers, whereas I should have rounded them to the

Look at the time between 7 and 8 am for example. Rounding your result to the nearest second gives 11 seconds, whereas I put 10 seconds.

I doubt if you could tell the difference when actually looking at the clock though!!

.

Alan
May 9, 2015

#6**+10 **

That's insignificant, Alan! You and I spent a lot of time solving this problem not knowing that there's a shortcut...

here's that shortcut:

12 hours / 11 =1.0909090909090909 = 1h, 5min & 27.272727sec

or

720min / 11 =65.4545454545454545 = 1h, 5min & ~27sec

or

360^{o} / 11 =32.727272727272727^{o} (converted into time units) = 1h, 5min, 27sec

Basically, 1h/5min/27sec is an interval ( as your yesterday's graph shows) that repeats 11 times.

Once again, you did a good job!

civonamzuk
May 9, 2015

#8**+10 **

I did this one (after a bit of "fumbling') in a slightly different manner...but...the results are really close to yours!!

I noted that the minute hand moves through 6° in a minute.....and I noted that the hour hand moves through (1/2)° every minute....so....

At 6:30PM, for instance.....the hour hand is 15° ahead of the minute hand. So it will take the minute hand....

(6 - 1/2)T = 15 .....T = about 2.727272727272727 min = 2 min 43.636363636363638 seconds to "catch up" to the hour hand ..i.e. the hands are "even' at about 6:32:43AM

So....solving the following "formulas" for the times between the hours when the hands are in the same position, we have

7 - 8 (5.5)T = 45 ...T = 8.1818181818181818 minutes afte 7:30AM = about 7:38:11

8 - 9 (5.5)T = 75 ....T = 13.6363636363636364 mintes after 8:30AM = about 8:43:38

And from here, I noted that each successive time would be (about) 1hr 5 min 27 seconds later than the previous one giving the remaining two times of .... 9:49:05AM and 10:54:32AM

And the hands don't cross again until noon.

The situation existing after 1PM is similar......

At 1PM, the hour hand is 30º "ahead" of the minute hand, so I calculated the time to "catch up" as:

(5.5)T = 30 ....T = 5.4545454545454545 minutes after 1PM = 1:05:27 PM

And I realized that the same 1hr 5min 27 second interval would exist ....so the remaining times are

2:10:54 , 3:16:21, 4:21:48 and 5:27:15 [PM, of course ]

My last two times are off slightly because of rounding.....but...close enough !!!

Mine isn't as "elegant"....but, maybe one day, I'll be as good as you guys....!!!

"I'll get you my two fine gentleman, and your little dogs, too !!! "

CPhill
May 9, 2015

#9**+1 **

Very neat shortcut civonamzuk! I like elegant solutions but I didn't spot this one.

(Actually, it didn't take me very long to produce my results. I wrote a very short computer program to do it, and let the computer do the work!).

.

Alan
May 10, 2015