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. Also, over the years, consensus has determined which possible coverage periods are "premium" versus "non-premium." In this regard, premium periods have to been shown to include not only standard holidays, but the weekends following school vacation weeks in February and April. Taking this into account, there are between 39 and 41 non-premium coverage periods and 14 or 15 premium coverage periods in any given year, which means 4 or 5 non-premium coverage periods and 1 or 2 premium coverage periods per doc for an 8-person coverage group. There are 1537 coverage periods in the 28-year cycle (1133 non-premium and 404 premium periods). Since neither 1133 nor 404 are divisible by 8 (coverage docs), neither 28-year premium nor non-premium coverage cycles arithmetically align. The least common multiple of 404 and 8 is 808, corresponding to a 56-year repeating cycle (for premium coverage periods only. 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 8, either. The non-premium cycle, therefore, requires 3200 years to re-align, and the premium cycle requires 800 years (least common multiples of [16185, 8], [5772, 8]), which is the time it will take to get back to 2013, calendar/coverage-wise.

Since a 3200-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 premium vs. non-premium periods. Since 6 out of the 14 calendar types have 56 (divisible by 8) coverage periods, this leads to "clumping" of premium coverage for certain individuals for certain premium periods, and does not result in equitable premium coverage distribution.

The next simplest scheme would be to "uncouple" premium and non-premium coverage periods, having each rotate separately. For practical purposes, this works fine when neither number of premium periods (14 and 15) is integer divisible by the number of docs (to work perfectly, those quantities would need to be relatively prime), as portrayed in the center table on the linked page. However, if we go back down to 7 docs, it doesn't. Why? 8 of the 14 calendar types have 14 premium periods, and there are blocks of 4 consecutive years which each have that number of premium periods. Since 14 is divisible by 7, this would result in 1 doc having the same premium period 4 or even 5 years in a row.

A more generalizable solution, which is the one currently implemented, is illustrated in the right-hand table (and on the right on this page). In this scheme, which should be obvious from the table itself, the 1st premium period of the year rotates through the Coverage Group on an 8-year cycle, assuring equitable premium period 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 - 4 docs get 2 of them and 4 docs only get 1. Boo hoo. This could be solved by incorporating a "doc offset" for the next 28 year cycle (in 2044) with a different doc.
(2) Due to the uncoupling of premium and non-premium coverage, in a given year, 1 doc could have 5 non-premium and 2 premium coverage periods (total of 7) and another could have 4 non-premium and 1 premium coverage periods (total of 5). This is a hypothetical but not a real concern. Projecting the schedule out for the remainder of this century, the difference in highest and lowest assigned coverage periods is cumulatively only 3. 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.