Dear Straight Dope:

I read recently online that "the first day of a century can never fall on a Sunday," which seems incredible and probably wrong as the maths seems to imply that millennium years (1000, 2000 etc.) are leap years. What is the truth?

SDStaff Karen Lingel replies:

It’s not all that surprising: the calendar follows a predictable pattern, and so do the days of the week. Since first days of the century occur so rarely (oh, about every 100 years or so), it’s entirely possible that the alignment of these two fixed repeating patterns might not allow the New Year’s Day of a year evenly divisible by 100 – which we’ll call the first day of the century, the protests of pedants notwithstanding – to fall on a Sunday. But let’s do the calculations and see.

OK: There are 365 days in a common year (a year without a leap day). Dividing 7 into 365, we get 52 plus a remainder of 1 – a common year 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 (DOTW). So if New Year’s Day of a common year falls on a Monday, the next year’s will be a Tuesday.

For a 366-day leap year, the remainder is 2, which means that going from a leap year to the year following, New Year’s Day advances by two DOTW – e.g., from a Monday to a Wednesday.

So: what happens to New Year’s Day over a period of several centuries? Well, in a single century New Year’s Day advances by 100 DOTW (one per year) plus an extra day for each leap year. These come every four years, of course, *except* that a year evenly divisible by 100 *isn’t* a leap year. Therefore in each century there are 24 extra leap days (100 divided by 4, minus 1), meaning that from one century to the next, New Year’s advances by a gross total of 124 DOTW. But each complete week’s worth of days doesn’t do anything to advance New Year’s – seven days of DOTW advancement just gets you back to the day you started on – so we divide 7 into 124 and get a remainder of 5. (For you nonmath types: we call this procedure “taking modulo 7 of 124.”) Thus from one century to the next the net DOTW advancement is five days.

As a second century goes by, the DOTW advances another five days, for a total of ten days from the beginning of the first century. Taking modulo 7 of 10, we get three days of net DOTW advancement over the 200 years.

A third century advances New Year’s another 5 days, for a total of 15 DOTW; modulo 7 of this = 1 day of net DOTW advancement.

For the fourth century, we have to take into account the third leap year rule: years divisible by 400 *do* have a leap day. So for the fourth

century, we advance the DOTW another five days plus an additional leap day, plus the 15 days accumulated over the first three centuries, for a gross total of 21 days advanced. Take the modulo 7 of that, and you get a net total of 0: that is, the fifth century starts on the same day of the week that the first century did. Then the pattern starts over again.

There are no agreed-upon leap year rules with timescales longer than 400 years, so what we have is one fixed pattern that repeats itself exactly every 400 years. Since there are only four starts-of-the-century in each four centuries, New Year’s can fall on only four of the seven days of the week. The winners depend solely on where the days of the week happened to line up with the cycle when the crazy leap year system was adopted. As it works out, our system is set up so that out not only Sundays get left out, but Tuesdays and Thursdays as well. The first New Year’s of each 400-year cycle falls on a Friday, the next century starts on a Wednesday, the next on a Monday, and the next one – the century beginning with a year that’s divisible by 400 (like, e.g., 2000) – starts on a Saturday.

(Note to pedants: If you define your centuries as beginning with the year ending in 01, then the cycle goes Saturday-Thursday-Tuesday-Monday. So being pedantic still won’t get you a century that starts on a Sunday.)

Now keep in mind that a fair amount of tinkering with the calendar occurs now and again. Our current (Modern Gregorian) calendar was instituted on Friday, 15 October 1582, and took a couple hundred years to be generally adopted. Prior to this, the year 1100 had started on a Sunday, but timekeeping was so generally screwed up back then that when the Gregorian system was adopted, a bunch of days had to be removed to get the dates to synch up correctly with the seasons. (In the U.S., the missing days were September 3 through 13, 1752.)

There is already talk (premature in my opinion) that to get the current calendar to remain accurate, we’ll need to eliminate one extra leap day every 4,000 years. If this happens, the 400-year pattern will shift by a day, and the first day of the century will loop through a Thursday-Tuesday-Sunday-Friday cycle. January 1, 4300, would be a Sunday.

I’m guessing, though, that if humans aren’t extinct by Jan 1, 4300, they’ll be using stardates or something.

Send questions to Cecil via cecil@straightdope.com.

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