Goines On: The bumps

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Over several days, Goines further cogitated the “bump problem” that arose when a 00 non-leap year came along and interrupted the standard 28-year cycle, with its pattern of 6₁-11-6₂-5 sub-cycles. He believed he could specify a pattern for the 400-year stretches as well, each one comprising three 00-years that aren’t leap years.
    He had already traced the 6₁-11-6₂-5 sub-cycles back to 1909, within 8 years of the 1900 non-leap year, and up to 2099, where the 6₁-11-6₂ portion of 28-year cycle butted up against the 2100 non-leap year. He now needed to examine that bump at 2100, and the bumps at 2200 & 2300 as well.
    By specifying how the 28-year cycles were interrupted at those bumps, he would have defined the 400-year pattern from 2021 through 2420, and it would repeat for 2421–2820, 2821–3220, and so on (for as long as Pope Gregory XIII’s calendar ruled).
    Goines’ analysis of the 2100 bump yielded (for the years 2077–2116) the 40-year pattern consisting of

1. a 6₁-11-6₂ run (23 years),
2. the “bump years”: |*|_|_|_|L|_| (6 years, with the * indicating where 2100 falls), and then
3. a 6₂-5 run (11 years).

    His analysis of the 2200 bump yielded (for the years 2173–2212) a 40-year pattern identical to that of 2100, except for 2200’s falling in a different spot:

1. a 6₁-11-6₂ run (23 years),
2. the bump years: |L|_|_|_|$|_| (6 years, with the $ indicating where 2200 falls), and then
3. a 6₂-5 run (11 years).

    But the 2300 bump yielded a surprise: just 12 bump years (for the years 2297–2308, with the # indicating where 2300 falls): 

       |_|_|_|#|_|_|_|L|_|_|_|L|

immediately followed (starting in the year 2309) by a standard 28-year cycle – “standard” including the defining fact that 2309 begins on the same day of the week (Friday) as 2021, 2049, 2077, etc.

    Goines summed 40 + 40 + 12 for a total of 92 years, which didn’t seem to be an accident, because that number plus the sum of the surrounding eleven standard 28-year cycles is 400 – 92 + (11 x 28) = 400. The overall 400-year pattern would repeat over and over (so long as the Gregorian Calendar ruled). 

    Goines smiled at the thought that he now had the key to forever in Gregory Land. He constructed a table of the two 400-year periods, to be read top-to-bottom, left-to-right (with the first year of each period’s three bumped cycles shown in bold with underscores):

​  |——————————————————||——————————————————||———————> 

2021 2117 2213 2309 2421 2517 2613 2709 2821

2049 2145 2241 2337 2449 2545 2641 2737 2849

2077 2173 2269 2365 2477 2573 2669 2765 2877

          2297 2393           2697 2793

    Goines smiled at himself, really, for he knew he was joking. The only “forever” involved here was the short time remaining during which he might perform his parlor trick...
    …Unless someone were interested enough in the trick to want to master it and continue the performance. But was there any such person?
    Goines sighed. It didn’t really matter. Besides, some clever 12-year-old could write an app to do it. Goines wondered now whether he should outline the procedure, to guide the kid. He thought he could still do that, still draw on his early experiences designing and coding file-construction programs for IBM….

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