Keith F. Lynch:
Eventually, they have a crisis. For the first time, February will have a negative length, and there's no provision for dealing with that. Borrow a day from January or March? The puzzle is, in what year does this first happen?
The February days-added numbers are < https://oeis.org/A048272 >. Checking the 10000-term link that is provided there for numbers < -28 tells us that this first happens in the year 2880.
What then will be the first year in which a whole year has a negative length?
A program calculating 10^8 terms of A048272 has 14 unique terms < -365: -576, -512, -504, -480, -450, -448, -432, -420, -405, -400, -396, -392, -384, -378, first occurring, respectively, in years 86486400, 69189120, 57657600, 43243200, 69854400, 34594560, 28828800, 39916800, 85377600, 25945920, 70963200, 71850240, 17297280, 31046400. The smallest of these is 17297280. Gareth McCaughan:
so that the average year length is 365 + 1/4 log 2 sum {k} k(k-1)/2^k which I think equals 365 + log 2
The sum of my 10^8 terms of A048272 is 69314510 (log 2 ~ .69314718).