Last month I wrote:

I also came up with simple rules for calendars which gave pi in the average year length. Can anyone else find such a rule? Or rules that give e, Euler's constant, or other interesting numbers in the average year length?

It looks like nobody else is interested in working on it. So I'll give my answers. By a leap-year rule, I mean a rule that adds some whole number of days (possibly zero or negative) to some years based on the year number. If a year has a negative length, the necessary number of days are borrowed from the following year. If that causes the following year to also have a negative length, the necessary number of days are borrowed from the year after that, etc. If this is impossible, i.e. all future years put together don't have enough days to pay off the accumulating debt, the calendar breaks down. The calendar rule I mentioned last month was that a day is added for each odd number (including 1) that the year number is divisible by, and subtracted for each even number that the year number is divisible by. The average year length converges to 365 plus the natural log of 2. Calendar rule pi: Add four days for each number congruent to 1 mod 4 that the year number is divisible by (e.g. 1, 5, 9, 13), and subtract four days for each number congruent to 3 mod 4 that the year number is divisible by (e.g. 3, 7, 11, 15). The average year length converges to 365 + pi. Calendar rule pi^2: Add six days for each square that the year number is divisible by (e.g. 1, 4, 9, 16). The average year length converges to 365 + pi^2. Calendar rule e: Add one day for each factorial number that the year number is divisible by (e.g. 1, 1, 2, 6, 24). Note that 1, which all years are divisible by, counts as two factorial numbers (0! and 1!). The average year length converges to 365 + e. Calendar rule e^2: Add N^2 days for each factorial number that the year number is divisible by, where N is which number's factorial it is. The average year length converges to 365 + e^2. (This generalizes to e to any natural-number power.) Calendar rule 1/e: Add or subtract one day for each factorial number that the year number is divisible by. Add if it's the factorial of an odd number, subtract if it's the factorial of an even number. The average year length converges to 365 + 1/e. I'm still looking for good rules for phi, Euler's constant, and the square root of 2. Can anyone find any? Or for any other "interesting" numbers? (It's always possible to find a solution using greedy Egyptian fractions, but that's ugly.) Some rules don't converge. For instance what if you add a day for each distinct prime number that the year number is divisible by? The average length grows without limit, but very slowly. Turn it around. What if you *subtract* a day for each distinct prime number that the year number is divisible by? This calendar will eventually break down into a cascade of day-borrowing that can never be caught up. But not for a very, very, very long time. Can anyone figure out when? I do know that the first negative-length year won't occur until after the year 10^1050. But that year will have no trouble borrowing from the following year; the breakdown won't be until much later.