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CR4 Challenge Question: Big clock in a New Year (Jan. 2022)

Posted December 31, 2021 12:00 AM
Pathfinder Tags: challenge question

The starting time is 00:00:01 on New Year's Day.

On what day and time will the clock's hour and minute hands overlap for the 1,000th different time?

(And keep in mind if it's a Leap Year or not!)

Image credit: Gadot/CC BY-SA 4.0

Author's note: the correct answer is Feb. 15 at 10:54 a.m. A previously reply mistakenly identified the day as Feb. 14.

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#1

Re: CR4 Challenge Question: Big clock in a New Year (Jan. 2022)

12/31/2021 2:36 AM

1:00 am feb 12...

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#2

Re: CR4 Challenge Question: Big clock in a New Year (Jan. 2022)

12/31/2021 8:42 AM

Well, the hands overlap 22 times a day. 1000/22 = 45 remainder 10. That puts it on Feb 14, the 10th overlap after 12:00:01.

12 hours = 43200 seconds. Overlaps occur 11 times in 12 hours or every 3927.272727... seconds. The 10th overlap would occur at second 39272.727272... or 10:54:32.72727272... in the morning of Feb 14.

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#3
In reply to #2

Re: CR4 Challenge Question: Big clock in a New Year (Jan. 2022)

12/31/2021 2:04 PM

Times hour and minute hands align each day:

1 01:05:27

2 02:10:54

3 03:16:21

4 04:21:49

5 05:27:16

6 06:32:43

7 07:38:10

8 08:43:38

9 09:49:05

10 10:54:32

11 12:00:00

12 13:05:27

13 14:10:54

14 15:16:21

15 16:21:49

16 17:27:16

17 18:32:43

18 19:38:10

19 20:43:38

20 21:49:05

21 22:54:32

22 00:00:00

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#9
In reply to #3

Re: CR4 Challenge Question: Big clock in a New Year (Jan. 2022)

01/01/2022 3:17 PM

The clock hands must cross every hour, that's why a broken clock is right at least 2 times a day...

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#10
In reply to #9

Re: CR4 Challenge Question: Big clock in a New Year (Jan. 2022)

01/01/2022 3:32 PM

NO! Since the hour hand has advanced 1 hour (roughly 30°) The minute hand must rotate a little over 390° before it reaches the hour hand position the next time. The "little over" is because the hour hand rotated further while the minute hand was rotating that 30+°.

So it is a little over 5 minutes more than an hour between crossings.

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#11
In reply to #10

Re: CR4 Challenge Question: Big clock in a New Year (Jan. 2022)

01/01/2022 5:48 PM

Not if it's a digital clock...

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#12
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Re: CR4 Challenge Question: Big clock in a New Year (Jan. 2022)

01/01/2022 9:50 PM

Now you've opened a whole new can of worms! All answers so far (including mine) have been assuming a standard mechanical 12-hour clock with physical hands.

The image you just showed is either a very non-standard clock, or it is defective. Why is it missing some minute marks? Perhaps they appear one at a time to show seconds... The hour hand appears to point at 10, or possibly a bit before 10, yet the minute hand is clearly pointing at 9 minutes past the hour.

This made me go back and look at the original post. It says a large clock, and the illustration does indeed appear to show a large, complex clock. The numbering on the clock is clearly not Arabic numerals, but it is pretty clear that it is a 24 hour clock. The hour hand must be the one with a sun on it, and I presume the black circle represents the moon. there are other pointers and their shadows that make it difficult to tell which is which.

I haven't yet fully analyzed it, but I suspect a 24-hour clock would have 23 occurrences of coincidence per day, rather than the 22 of a 12-hour clock.

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#16
In reply to #12

Re: CR4 Challenge Question: Big clock in a New Year (Jan. 2022)

01/02/2022 10:49 AM

If it's a 24 hour clock with a minute hand, you are correct, it should have 23 overlaps per 24 hour period. 1000/23 gives 43 days + remainder 11. Overlap 1000 should be on day 44, or Feb 13

The daily overlaps should occur at:

1 01:02:36

2 02:05:13

3 03:07:49

4 04:10:26

5 05:13:02

6 06:15:39

7 07:18:15

8 08:20:52

9 09:23:28

10 10:26:05

11 11:28:41

12 12:31:18

13 13:33:54

14 14:36:31

15 15:39:07

16 16:41:44

17 17:44:20

18 18:46:57

19 19:49:33

20 20:52:10

21 21:54:46

22 22:57:23

23 00:00:00

Overlap 1000 should be on day 44, or Feb 13, overlap 11 (11:28:41).

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#20
In reply to #16

Re: CR4 Challenge Question: Big clock in a New Year (Jan. 2022)

01/02/2022 9:50 PM

An interesting point, but are there actually any 24-hour clocks that divide the circle into 24 parts? All the ones have seen have 1-12 around the dial, with 13-24 shown in parallel for the second sweep. This keeps the minute positions normal; e.g., 15 minutes always "east."

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#21
In reply to #20

Re: CR4 Challenge Question: Big clock in a New Year (Jan. 2022)

01/02/2022 10:36 PM

Did you look at the image of a clock in the original Post, enlarged in my #12?

A simple search brought up several, including this one:

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#17
In reply to #12

Re: CR4 Challenge Question: Big clock in a New Year (Jan. 2022)

01/02/2022 1:03 PM

I can't help wondering whether the clock illustrated in the OP is intended to be a clue, a distraction, or was simply a convenient illustration of a large clock. Also, why is the word "Large" included at all? Does a large clock differ from a small or medium clock in the timing of hand crossings?

The symbols on the frontmost dial of the illustrated clock are clearly the signs of the zodiac. The less prominent Roman numerals agree with our current 12-hour time numbers, with one of the 12s at the top. There are (in black) a few Arabic numerals, with 1 at the left and 12 at the right. The more prominent symbols/numbers on the outside are a mystery to me. Why are they offset by 8 hours, and what is the origin of those symbols?

The pacific time zone is offset by 8 hours from GMT, but I'd be quite surprised if that photo represents a clock made for use in the Pacific Time Zone!

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#22
In reply to #17

Re: CR4 Challenge Question: Big clock in a New Year (Jan. 2022)

01/03/2022 7:13 AM