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For the next week it took over my life. Listen, I'm no mathematician, and I don't have time for this sort of nonsense—I've got early deadlines to meet and bare cupboards to fill. Listen, I said to myself all week long, as the files of figures invaded my dreams and the sheafs of notes overflowed my desk (by the end of the third day I couldn't even find the notes for the articles I was supposed to be writing, I think I threw them out on the backsides of a a particular unproductive line of reasoning)—listen, I said: you don't have time for this.
But a puzzle can get to be like a fever, and it will run its course.
Let us examine that index and see whether we can determine a pattern to the sequence of the 14 possible calendars. Indeed we can.
(a) Using "9, 4, 5, 6" as our opening phrase, we discover that the years 1776-1815 bracket one pattern, which we will call A; 1816-1843 bracket a variant pattern, which we will call B; 1844-1871 repeat B; 1872-1911 bracket another variant, which we will call C; but then 1912-1939, 1940-1967, 1968-1994, and 1995-(2022) all repeat pattern B.
(b) Let us analyze these patterns:
(1) Pattern B consists of a sequence of 28 digits and seems to occur all the time except when the sequence runs through a centennial year (e.g. 1800 or 1900, years that disrupt the pattern because they are not leap years).
(2) Patterns A and C are both patterns of 40 sequential digits; both happen to run through a centennial year, and both sequences follow pattern B up to that centennial, whereupon the B pattern is disrupted by a "missed leap" and we note two different strategies (A and C), such that within the limit of 40 digits the sequence returns to the "opening phrase" ("9, 4, 5, 6 … "), thus beginning a new B pattern.
(c) We can thus hypothesize that the calendar follows through in sequences of two types:
(1) B type patterns of 28 years, and
(2) A and C of 40 years.
The pattern will always consist of a succession of Btypes except as it crosses "nonleap" centennials, when an interruption of A-C type will occur. We note that centennials divisible by 400 (e.g. 1600, 2000) are leap years, and that the sequences running through them will be B types.