If the end of the world was supposed to come on the first day of a new century, what would be the chances that it would happen on a Sunday?

  • $\begingroup$ Hint: The answer is not 14.28%. $\endgroup$ – John Bupit May 17 '14 at 18:59
  • $\begingroup$ I know. I'm typing up an answer as we speak. $\endgroup$ – Joe Z. May 17 '14 at 19:00
  • $\begingroup$ Okay, done. What do you think? $\endgroup$ – Joe Z. May 17 '14 at 19:19
  • $\begingroup$ @JohnBupit is it 14,29%? When you round it properly. (1/7=0,14285) $\endgroup$ – martijnn2008 May 17 '14 at 20:16

The answer is slightly trickier than you might think at first - it has to do with a quirk in the calendar we currently use, the Gregorian Calendar.

The Gregorian Calendar was instated by Pope Gregory (hence the name) in 1582 to account for a slight disparity in the Julian calendar that made the days drift about 18.75 hours every century. The Julian Calendar had one leap day every 4 years, which accounted for the gap between 365 days and a year, but overshot it by just a little too much to be unnoticeable. By the time Jesus had been gone for over 1,500 years, the dates on which they were celebrating Easter (and the dates on which the seasons were occurring) were drifting much too far away from their original dates for the church's comfort.

So, the Catholic Church under Pope Gregory decided to remove the leap years for three of the four century years, specifically the ones that weren't divisible by 400. So while the year 1600 had a February 29 as usual, the years 1700, 1800, and 1900 didn't have an extra day at all. This corrected the disparity by an average of 18 hours every century, to the Catholic Church's satisfaction. (The remaining 0.75 hours will only start to be noticeable in about 20,000 years' time.)

Now, what does this have to do with the question at hand? Well, the Gregorian calendar runs a 400-year cycle with its leap years in this way, and every 400 years there are exactly 97 occurrences of February 29. This is a total of (365 * 400 + 97) = 146097 days, which happens to be exactly divisible by 7. So every 400 years, the days of the week complete one cycle as well.

This means that ultimately, the beginning of a century can only have one of up to four possible days of the week, because after every fourth century, the cycle restarts.

With this in mind, let's check the dates for each of the first days of the century from 2101 to 2401 (as centuries begin with the -01 year; the -00 year is actually the last year of the previous century):

  • January 1, 2101 falls on a Saturday.
  • January 1, 2201 falls on a Thursday.
  • January 1, 2301 falls on a Tuesday.
  • January 1, 2401 falls on a Monday.

And this cycle repeats for every set of 400 years onwards.

So tough luck, the first day of the century never falls on a Sunday. The probability is zero.

But wait, what if we do count the -00 year as the beginning of the century? Then, the days of the week are as follows:

  • January 1, 2100 falls on a Friday.
  • January 1, 2200 falls on a Wednesday.
  • January 1, 2300 falls on a Monday.
  • January 1, 2400 falls on a Saturday.

As it happens, Sunday doesn't appear here either, and in fact, Sunday is the only day of the week that doesn't appear in either list, and so is the only day that cannot be the beginning of a century, regardless of whether you consider -00 or -01 to be the first year.

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    $\begingroup$ And if you believe centuries start in years starting with 01, so the first day of the 21st century was Jan 1, 2001, the probability is still zero. As 2400 is a leap year, Jan 1, 2401 falls on Monday. $\endgroup$ – Ross Millikan May 17 '14 at 22:32
  • $\begingroup$ @RossMillikan: I wasn't aware it was a matter of belief. The Gregorian calendar does not include a year 0, so how could our decades and centuries start in that year? $\endgroup$ – Michael Myers May 18 '14 at 2:50
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    $\begingroup$ Because not enough people know that fact, and we sort of just assume that the rollover of the digit is what begins the "first" year of that decade. I can revise my problem statement to match. $\endgroup$ – Joe Z. May 18 '14 at 3:08
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    $\begingroup$ @MichaelMyers: I don't know of an official definition of the span of a century. I do know that popular culture is split, and much more often starts a century on 00. Though I know there was no year 0, I think the pattern of the leading digits is more important that keeping the first century with a duration of 100 years, so I agree with starting centuries on 00. But I recognize it is a case of defining things the way you like. $\endgroup$ – Ross Millikan May 18 '14 at 4:00
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    $\begingroup$ The answer should be edited, I think, to show a major part of the puzzle's elegance--that none of the years ending in 00 or 01 start on Sunday. $\endgroup$ – supercat Jul 2 '14 at 20:15

0 percent. This is because the first day of a century can never fall on a Sunday.

There are 365 days in a year (which is not a leap year) - which is exactly 52 weeks plus one day. Therefore, from a common year to the following year, New Year's Day advances by one day of the week.

From a leap year (which has 366 days) to the year following, New Year's Day advances by two days of the week.

  1. So, advancing from one century to another, we advance a 100 days plus a few extra days for leap years. Every fourth year, except the year divisible by 100 is a leap year making a total of 24 ($\frac{100}{4} - 1$) leap years in a century. Therefore, we advance a total of 124 days of the week in a century. 124 divided by 7 leaves a remainder of 5 - so we're only moving 5 days of the week ahead.

  2. After 2 centuries have passed, we advance $5 + 5 = 10$ days - or 3 days from the start, because $10 =3 \mod 7$.

  3. After 3 centuries, we've advanced 15 days - or 1 day from the start.

  4. On the 4th century, however, we have to take into account the fact that years divisible by 400 have a leap day. So, instead of advancing 20 days, we advance 21 days - or 0 days. We're back at the day we started 4 centuries ago.

We can now check the 4 days that a century can begin with by examining the days of 2100 to 2400:

  • The century 2100 starts on a Friday.

  • 2200 subsequently starts 5 days later, a Wednesday.

  • 2300 then starts on 3 days ahead of Friday, i.e. a Monday,

  • and 2400 starts on Saturday, i.e. 1 day from Friday.

  • We return to a Friday in the year 2500, and the cycle continues.

Because of this Friday-Wednesday-Monday-Saturday cycle, we cannot have a century starting on Sunday (and a Tuesday and a Thursday) - which is a good thing because we get an extra holiday!

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    $\begingroup$ Your "extra holiday" argument doesn't hold any water given the way most governments structure it - if January 1 is on a weekend, people get January 2 or January 3 off instead. $\endgroup$ – Joe Z. Jun 4 '14 at 1:40

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