End of the World on a Sunday

The first day of a century can never fall on a Sunday; nor on a Wednesday or a Friday.

The answer 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 the probability of the first day of the century falling on a Sunday is zero.

So 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 you can see, 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.