Nathan Grigg

Leap Years

On Sunday, I was reading a page on Wikipedia, when this quote caught my attention:

The Gregorian leap cycle, which has 97 leap days spread across 400 years, contains a whole number of weeks (20871).

This is somewhat surprising to me, since I assume that this was by accident and not by design, which means there was only a 1-in-7 chance of it happening.

One result of this is that the calendar for 2000 is the same as the calendar for 2400, which makes a perpetual calendar such as this one quite a bit easier to specify.

Another consequence is that not every calendar is equally likely, which I captured in this silly Mastodon post:

Did you know that October 8 is 3.5% less likely to fall on a Sunday than a Saturday? Enjoy your rare day!

I have few followers, but I was pretty sure my old internet pal Dr. Drang would take the bait (he did).

If I were more patient, I could have waited until next February 29, which falls on a Thursday. Since leap days are more rare than non-leap days, the disparity is greater, and my 3.5% could have been 14%. Wednesday is the more common day for February 29, which again reminds us that we are currently on one of the less common paths through the perpetual calendar.

But calendars are fun to think about, so I didn’t stop there.

A year is about 365.24219 days long. The Julian calendar has a leap year every 4 years, for an average year length of 365.25 days. The error, of course, adds up relatively quickly, as we eventually noticed.

The Gregorian calendar skips leap years on years divisible by 100, unless they are also divisible by 400, leaving the 97 leap years per 400 years quoted above. This makes a year on average 365 + 97/400 = 365.2425 days long, which is closer! It takes 3,225 years before you drift a day, which I guess is good enough?

At this point, I stumbled upon the Revised Julian calendar, which skips leap years on years divisible by 100, unless they are also either 200 or 600 mod 900. This makes a year 365 + 218/900 = 365.24222 days long, which is even better. Now it takes over 30,000 years before you drift a day. The rule is much more confusing, though, although it has the benefit (by design) that it matches the Gregorian calendar exactly for the years 1600-2799. This lets you claim you are following a more accurate calendar without really making a fuss. Also, 900 years of the Revised Julian calendar is not a whole number of weeks, so the Revised Julian perpetual calendar would actually have a 6,300-year cycle.

Finally, I spent some time thinking about what I would have done to handle the leap year problem if I ran the world. The answer is so obvious that it makes you doubt the wisdom of our ancestors. The Julian calendar drifts by 1 day every 128 years. We could have had a calendar (could I call it the Griggorian?) that had leap years every year divisible by 4 except those also divisible by 128. This gets you a year with an average length of 365 + 31/128 = 365.2421875 days, which would mean one day of drift every 400,000 years.

I admit that computing divisibility by 128 is harder (for a human) than 400, but otherwise the rule is clearly simpler. The other downside is that 128 years of this calendar is not a whole number of weeks, which means that the perpetual calendar would have a 896-year cycle. But as long as I’m in charge, we might as well solve that by starting every year on a Sunday.