exupero's blog

# Month recurrence

Last February I wrote about how often a year's calendar can be reused. Non-leap year calendars can be reused at 6- or 11-year intervals, while leap year calendars can only be reused once every 28 years. After writing that, I noticed a month in a 2016 calendar (a leap year) that aligned with the same month in 2022 (a non-leap year), which I didn't expect but makes perfect sense, since months other than February recur more frequently than years as a whole.

There are 14 possible annual calendars, one beginning on each day of the week in non-leap years and one beginning on each day of the week in leap years. But only February changes length in leap years, so the 14 different annual calendars are a result of the 14 different calendars for February. All other months have only seven possible arrangements.

Annual, non-leap year calendars recur after 6 years, then after 11 years, then after 11 years again, and beyond that the pattern repeats. Let's denote that cycle of intervals as 6-11-11. That pattern is driven by the recurrence of non-leap year Februaries, which follows the same pattern. (Leap year Februaries recur at 28-year intervals.)

Months other than February have a longer cycle of 6-11-6-5, which results from whether only a single leap year fell between recurrences (producing a 6-year interval), whether two leap years fell between (producing a 5-year interval), or whether leap year fell such that a recurrence was skipped (producing an 11-year interval).

These recurrence intervals are what we see most of the time. There are exceptions, which I'll cover in the next post.