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Century

cinelli
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Century

#481440

Postby cinelli » February 19th, 2022, 10:29 am

Let us agree that the first day of the century appears on, for example, 1 January 2001. When is the next time that the first day of the century appears on a Sunday?

Cinelli

pje16
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Re: Century

#481442

Postby pje16 » February 19th, 2022, 10:37 am

not bothered
none of us will be around to see it :lol:

mc2fool
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Re: Century

#481500

Postby mc2fool » February 19th, 2022, 1:50 pm

God won't allow a Holy day of rest to be conflated with a day of Bacchian hangover recovery ... but what's he got against Wednesdays? :D

SteelCamel
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Re: Century

#481526

Postby SteelCamel » February 19th, 2022, 3:42 pm

Since in the Gregorian calendar 400 years is a whole number of weeks (146097 days, exactly 20871 weeks) the cycle of weekdays repeats every 400 years. So the answer is 1 January 2101, 2201, 2301 or 2401, or there is no answer - since after that it will just repeat.
1 Jan 2101 - Saturday
1 Jan 2201 - Thursday
1 Jan 2301 - Tuesday
1 Jan 2401 - Monday
So in the Gregorian calendar no century ever starts on Sunday (or Wednesday, or Friday).

Of course the Gregorian calendar doesn't track the motion of the Earth perfectly, and changes in the Earth's orbit will only make it less accurate over time. Assuming humans are still around in a few thousand years, we'll need to revise the calendar. And if we're still using weeks, and the traditional weekday names, it's quite possible that this revised calendar will have a century beginning on Sunday. So I can't tell you when the next one is, but given enough time it probably will happen.

Incidentally, the last time it happened was Sunday 1 January 1301, as England was on the Julian calendar at that point - though for most purposes 25th March not 1 January was considered the start of the year, which makes working out when centuries start even more complex! Also 1 January 601, though I'm not sure what calendar was actually observed in the 5th century. If we'd kept using the Julian calendar, 1 January 2001 would have been a Sunday, and the next one in the future would be 1 January 2701.

UncleEbenezer
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Re: Century

#481567

Postby UncleEbenezer » February 19th, 2022, 7:53 pm

1 Jan 2001 was a Monday.

365/7 has remainder 1, so that moves one day a year. Or two on leap years. In a normal century there are 76 normal years and 24 leap years (the 00 being an exception), so it moves by ((76 + 2*24) mod 7) = 5 days. So,

1 Jan 2101 is a Saturday
1 Jan 2201 is a Thursday
1 Jan 2301 is a Tuesday.

But centuries that are multiples of 400 (including the one in our lifetimes) don't have a missing leap year, so there's an extra day and it moves by six days. Hence the would-be Sunday on 1 Jan 2401 is another Monday, and as Steelcamel already posted, we're into a cycle that repeats every 400 years.

The answer is, sometime after they change the calendar. But not so radically as to change out of all recognition.

I'm reminded of a familiar story:
W S Gilbert wrote:For some ridiculous reason, to which, however, I’ve no desire to be disloyal,
Some person in authority, I don’t know who, very likely the Astronomer Royal,
Has decided that, although for such a beastly month as February,
twenty-eight days as a rule are plenty,
One year in every four his days shall be reckoned as nine and twenty.
Through some singular coincidence – I shouldn’t be surprised if it were owing to the
agency of an ill-natured fairy –
You are the victim of this clumsy arrangement, having been born in leap-year,
on the twenty-ninth of February;
And so, by a simple arithmetical process, you’ll easily discover,
That though you’ve lived twenty-one years, yet, if we go by birthdays,
you’re only five and a little bit over!

9873210
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Re: Century

#481607

Postby 9873210 » February 20th, 2022, 7:42 am

UncleEbenezer wrote:The answer is, sometime after they change the calendar. But not so radically as to change out of all recognition.

So, when might that be?

The error in the length of the mean Gregorian year suggests this would be needed in about 7000 years.

However, since Babylonian times the mean solar day has been increasing by about 1.8ms/century. If this continues* that error will sum to one day in about 5000 years, two days in 7200 years and three days in 9000 years and will be larger than the Gregorian error. Assuming that we continue to add leap seconds as needed to keep mean solar noon at about 1200 hours. Also, assuming we decide to fix the error by tweaking the calendar rather than tweaking Earth's rotation and/or orbit.

* It is possible that melting of polar ice caps and other effects due to climate change may lengthen the day faster than the historical average.

UncleEbenezer
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Re: Century

#481620

Postby UncleEbenezer » February 20th, 2022, 10:01 am

9873210 wrote:
UncleEbenezer wrote:The answer is, sometime after they change the calendar. But not so radically as to change out of all recognition.

So, when might that be?

How should I know? I ain't called Nostradamus! The movements of heavenly bodies may be an exact science, but the translation of them into human calendars is arbitrary.

Perhaps we should flag up an in memoriam here. John Conway (RIP) - of many a mathematical game, and known personally to some of us, not least Gengulphus (RIP) - had a party trick of instantly calculating the day of the week of any date in history you could throw at him. Well, here's the formula for any time since (was it?) 1752.

cinelli
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Re: Century

#481969

Postby cinelli » February 21st, 2022, 9:19 pm

SteelCamel and UncleEbenezer have the right answer, which is never. If you are using a linux computer, the command

$ cal 1752

displays the 1752 calendar. That was the year which had only 355 days.

Cinelli

UncleEbenezer
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Re: Century

#483247

Postby UncleEbenezer » February 28th, 2022, 2:22 am

cinelli wrote:SteelCamel and UncleEbenezer have the right answer, which is never. If you are using a linux computer, the command

$ cal 1752

displays the 1752 calendar. That was the year which had only 355 days.

Cinelli

Methinks mc2fool was actually first to answer, albeit cryptically.

ObPedant: cal is actually much older than Linux (going back I think to the 1970s), and much more widely-available.

SteelCamel
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Re: Century

#483473

Postby SteelCamel » March 1st, 2022, 7:30 am

UncleEbenezer wrote:ObPedant: cal is actually much older than Linux (going back I think to the 1970s), and much more widely-available.


The most widely used version originates in BSD, and yes it's on most Unix-like systems. There's also ncal which has a few extra options - that was what I used to get the hypothetical future dates in the Julian calendar.

An interesting piece of trivia - the Gregorian reform didn't reset the calendar back to the alignment with the seasons created by Julius Caesar. It reset it to 325AD, by which time it was already a few days out. The original design for the Julian calendar had the spring equinox on the 25th March, while it's usually about the 21st on the Gregorian calendar.


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