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

1:00 am feb 12...

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.

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

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

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.

Not if it's a digital clock...

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.

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).

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."

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:

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!

It's an astronomical clock in Prague

Good find! Since you found that in Wikipedia, it's probable that the OP found it there too. Then the question is: Was that image an inspiration for the challenge, or was it simply the first image of a 24-hour clock that he or she found? I suspect the latter...

"Orloj" in Czech.

Interesting. Is "Orloj"a translation of "Clock", or of "Orrery"?

The clock in the original illustration looks a little like both.

The Spanish word for "Clock" is Reloj"; rather similar...

In your overview I see 22 different times, but the next day we will have the same times, however, the question asks for 1000 different times... Could we miss a leap second or something like that?

Is mentioning the leap year a tip or is it just a distraction?

Just using those 22 times seems... too simple to me.

...So it does not matter whether it is a leap year or not

45 remainder 10 means that it happens in the morning of the 46th day - so it will be on Feb 15 10:54:32.(72).

GA

Your 32 seconds is correct, I don't know where I got 27 from.

Correct! Thanks.

Tell 'em what he's won! (Answer: good answer points!)

Both hands move at fixed speeds. In the hour it takes the minute hand to complete a revolution, the hour hand has moved 1/12 of a circuit. In a 12 hours period, the interval between overlaps is 11/12, or 65.45 minutes. Double this for a 12 hour day, arrive at 22 times per day.

1,000/22 = 45.45 days. The 45th day of the year is always February 14, bringing us to February 15.

There are 1,440 minutes in a day. That brings us to the 10 a.m. hour, where the 1,000th overlap occurs at 10:54 a.m. on February 15.

NOTES
- The question asks for the 1,000th different time. Meaning the overlap that occurs right before midnight, and right after, only counts as one overlap.
- Leap years and seconds, and astronomical and digital clocks, are not factored in. Those were red herrings.

= ≠ ≈.

Exact and approximate are not the same, although they are often confused.

Although you got (so far) 9 "good answers", I think you are wrong by one day. Having 22 overlaps per day means 22*45=990 overlaps at the end of the day 45. And the 1000th overlap occurs at the 10th overlap on day 46, which is Feb 15 (so it will be on Feb 15 10:54:32.(72), as I wrote in #6).

Rixter already acknowledged your correct answer in post #8.

But, I'm the only one so far to have given your answer a GA vote.

When I wrote #6 he had 2 GA votes - since then he got 7 more (although his answer is wrong). This indicates that some people here didn't read beyond #2, probably considering that since it is the correct answer it has no meaning to read other comments. Their loss...

"1,000/22 = 45.45 days. The 45th day of the year is always February 14."

But the 46th day of the year which is where 45.45 occurs is the 15th.

Suppose you'd been asked for the 10th overlap:-

10/22 = 0.45 days

Would you have declared this to be on the 31st of December?

Feb 15th 10:54 and 27 seconds in the morning.

To be fully analytical, the nearest-second times are only approximate.

The hands coincide every 1-1/11 hours, or every hour plus 60/11 minutes.

Thus 22 times per 24-hour day is correct, as is 45 days + 10 extra times on the 46th day (Feb. 15). This is still true on a 24-hour clock.

The 1000th repeat is at Feb 15, 10:54+6/11minutes = 10:54:32+8/11 ≈ 10:54:32.73.

I think Alex Roman's .(72). indicates point seven two recurring.

Yes.

Yes, and to the nearest 1/100 second, 0.72 repeated rounds to 0.73.

Or if we count the first overlap at 00:00:01 on New Year, the 1000th overlap will be at 15th Feb, 9:49+couple of seconds!

"This is still true on a 24-hour clock."

No!

The crossing times in digital hours for a 12-hour clock are: n*12/11, where "n" is the crossing number. This gives 1.090909... hours per crossing. 11 crossings gives exactly 12 hours, and that 11th crossing is the zeroth crossing for the next 12 hours.

The crossing times in digital hours for a 24-hour clock are: n*24/23; again "n" is the crossing number. This gives 1.043478... hours per crossing. 23 crossings gives exactly 24 hours, and that 23rd crossing is the zeroth crossing for the next 24 hours.

So there is one more crossing per day on a 24-hour clock than on a 12-hour clock.

My Excel spreadsheet uses a different rounding algorithm than Rixter's calculations show, so some of the seconds values differ by one digit. Otherwise it agrees with his values.

I think it depends on which type of 24-hour clock. For the 24-part dial, you are right and I stand corrected. For the dual-sweep 12-part dial, I believe I'm still correct.

I have used 24-hour time on shipboard, but never with a 24-part clock dial.

Agreed. To me, what you describe as "the dual-sweep 12-part dial" is simply a 12-hour clock with an extra set of numbers.

1.05am 10th Feb 2022

Great Scott:

One could always go and find out...

Sorry, I fail to see what that image has to do with the challenge, or any post replying to it. It is indeed strange, with the last departure ≈30 years after the "present time"...