Robin: “Holy, Leap Year math,” Batman, “did you know a Leap Day is more likely to occur on a Monday or Wednesday?
Batman: “Why yes,” Boy Wonder, “The Gregorian Calendar repeats a cycle every 400 years. Therefore Feb. 29 can occur 15 times on a Monday or Wednesday, 14 Times on a Friday or Saturday and 13 times on a Sunday, Tuesday, or Thursday, in that cycle”.
Robin: “I make that 97 leap years. 400 divided by 97, let me see, is 4.12 – How can it be?”
Batman: “It’s 4.1237 to be more precise. Remember, a leap year is skipped every 100 years, except every 400 years. 4.1237 days are the actual number of days needed to keep the calendar in synchronization with the Equinox. By this reasoning, the average  year would be 365.2425 days (365 + 1/4 − 1/100 + 1/400). Not entirely accurate as the actual number is 365.242375. Do your math and the discrepancy is 0.000125 days. That is almost the amount that would accumulate to an inaccuracy of one day in 8000 years.”
Robin: “Gosh,”
Batman, “That’s Brilliant!”
Batman: “Not quite,”
Boy Wonder, “That’s Gregorian!” 

Batman & Robin

This little dialog between Batman and Robin (in my fictional conversation) illustrates some Leap Year Math. Yesterday was a Leap Day in this year 2012. As it happens, a Leap Day is more likely to occur on a Monday and a Wednesday and did land on a Wednesday this year. I didn’t just make this up and it is a true fact, but Leap Year Math reveals some interesting calendar calculations. I thought I would  break it down in more detail for those that are more curious in the following explanation. If you are satisfied with my Batman & Robin dialog then you need read no further, but if you wish to know more in detail, then read on…

As everyone knows, an ordinary year has 365 days and a leap year has 366 days. The time the Earth takes to go around the sun is approximately 365.25 (365. 242375 to be more exact) days. Since that 0.25 days are left off the calendar, an extra day must be added every 4 years to keep the calendar accurate with the motion of the earth around the sun. It is not quite right, however, and every 100 years, it is off by 0.7625 (25 – 24.2375) days too many, so leap year is skipped. This still leaves a discrepancy 0.2375, which accumulates over centuries, and is added back in every 400 years with a leap year (4 x 0.2375 = 0.95), which is fairly close. Keeping the calendar in synch with the motion of the earth and only off by 0.05 days (1.2 hours) over 400 years is not bad (0.000125 days per year). It’s not exactly an elegant way of doing things either, but it works.

So, the rules are:

1.) Every 4th year and extra day is added
2.) Every 100 years there is no leap year
3.) Every 400 years is another leap year

So 1600, 2000 and 2400 are leap years but 1700, 1800, 1900, 2100, 2200 and 2300 are not. Apart from that, every year divisible by 4 (2012, 2016, 2020, etc.) is a leap year. Make sense? So, what will be done with that 0.05 days every 400 years. Well, the Gregorian Calendar, which instituted modifications over the Julian calendar, to correct some discrepancy and fine tune things, was only devised and decreed by Pope Julius Gregory XIII in 1582, and corrected the 10 day difference the Julian calendar had accumulated with the equinox. The logic & history can be summarized as follows:

  • 10 days were dropped in October 1582.
  • New rules were set to determine the date of Easter.
  • Rules for calculating Leap Years changed to include that a year is a Leap Year if:
    1. The year is evenly divisible by 4;
    2. If the year can be evenly divided by 100, it is NOT a leap year, unless;
    3. The year is also evenly divisible by 400. Then it is a leap year.

So, what about that 0.05 days over 400 years. Well, after 8000 years the calendar will only be less than 1 day off (minus one day actually). There is no rule to account for this (as yet) that I know of. The Earth’s orbit and the Earth’s rotation is changing over time, so it’s complicated to calculate. Leap seconds are used to account for the Earths rotation slowing (longer mean solar day) and because the second as defined by NIST standards is different from the mean solar second. Anyway, just a bit of Leap Year Math for the curious…

“Thirty days hath September,
April, June, and November;
All the rest have thirty-one:
Except February: it has twenty-eight we find,
unless it’s leap year, then it has twenty-nine.”

Hope everyone enjoyed the extra day this year. Now onto March, and beware the Ides – But that’s another calendar story for another day…