Who is up first for the year?

Under the calendar system we use in Western civilization today, there are 14 distinct yearly calendars - one beginning on each weekday for a non-leap year and one beginning on each weekday for a leap year. Because of the Gregorian leap year system in use since 1582, (off by 1/2 second per century), these calendars rotate in a 400-year perpetual cycle. That is, 1613, 2013, 2413, etc., all have the same calendar and each is followed by the same calendar in 1614, 2014, 2414, etc. If you don't cross a century mark (--00), the cycles are 28 years.

So how does this relate to coverage?
Well, it turns out that the number of coverage periods in a year is not the same for each of the 14 calendar types. That number varies from 53 to 56 and depends upon how many "fixed" holidays (Jan 1, Jul 4, Nov 11, Dec 25) fall on Fri/Sat/Sun/Mon vs. Tue/Wed/Thu. There are between 39 and 41 non-premium coverage periods and 14 or 15 premium coverage periods in any given year, which means 5 or 6 non-premium coverage periods and 2 or 3 premium coverage periods per doc. There are 1537 coverage periods in the 28-year cycle (1133 non-premium and 404 premium periods). This is an update from the previous version of this page. I was erroneously considering Veteran's Day to always fall on Monday, when it always falls on November 11 regardless of the day of the week. Since neither 1133 nor 404 are divisible by 7 (coverage docs), neither 28-year premium nor non-premium coverage cycles are syzgial. Looking at the next cycle up, 400 years, there are 21957 coverage periods, (16185 non-premium and 5772 premium periods). Neither of these is divisible by 7, either. Both cycles, therefore, require 2800 years to re-align (least common multiples of [16185, 7], [5772, 7]), which is the time it will take to get back to 2013, calendar/coverage-wise.

Since a 2800-year cycle is probably too long to wait for mathematically-guaranteed equal coverage distribution, another scheme must be implemented. Several possibilities are illustrated on this page. The left-hand table depicts a straight rotation scheme blind to holiday vs. non-holiday periods. Since 6 out of the 14 calendar types have 56 (divisible by 7) coverage periods, this leads to "clumping" of holiday coverage for certain individuals for certain holidays, and does not result in equitable holiday coverage distribution.

The next simplest scheme would be to "uncouple" holiday and non-holiday coverage periods, having each rotate separately. The problem with this scheme is that 8 of the 14 calendar types have 14 holiday periods, and there are blocks of 4 consecutive years which each have that number of holiday periods. Since 14 is divisible by 7, this would result in 1 doc having the same holiday 4 or even 5 years in a row, as portrayed in the center table on the linked page.

A more equitable solution, which is the one currently implemented, is illustrated in the right-hand table. In this scheme, which should be obvious from the table itself, the 1st holiday of the year rotates through the Coverage Group on a 7-year cycle, assuring equitable holiday distribution. The only issues with this scheme are:
(1) In a 28 year cycle, there are only 12 years with a (premium) weekend between Christmas and New Years (Ch+). Obviously, those "extra" weekends cannot be equally apportioned - 5 docs get 2 of them and 2 docs only get 1. This could be solved by starting the next 28 year cycle (in 2041) with a different doc.
(2) Due to the uncoupling of premium and non-premium coverage, in a given year, 1 doc could have 6 non-premium and 3 premium coverage periods (total of 9) and another could have 5 non-premium and 2 premium coverage periods (total of 7). This is a hypothetical but not a real concern. Projecting the schedule out, until the year 2048, the difference in highest and lowest assigned coverage periods is cumulatively only 2. Check it out.
(3) Since the coverages are uncoupled, a doc could have a premium coverage period immediately preceded or succeeded by a non-premium coverage period. This is easily remedied via some minor arbitrary manual adjustments.