Magnanimous Month

Copyright ©2011 by Paul Niquette All rights reserved.

 
athematics will have limited utility in analyzing dates on the Gregorian Calendar. Irregular durations for the months may not be the only impediment to generalized algorithms.  There is also the matter of leap year, with its extra day added quadrennially.  No intervals are divisible by seven except the duration of February -- hey, in non-leap years.  Accordingly, the days of the week precess drunkenly through the months.

If we treat four consecutive years as a unit (quad-year, let's call it), then one quad-year comprises 4 x 365 + 1 = 1,461 days, which is not divisible by seven.  By putting seven quad-years end-to-end, the resulting 10,227 days is divisible by seven and covers all possible calendars.  Seven consecutive quad-years will repeat themselves exactly every 28 years. 

As a finite domain, the puzzle can be solved by determinism.  Your spreadsheet is a handy tool for carrying out an otherwise tedious process, and here are the results -- 28 Magnanimous Months, alongside the number of months in between...
 
July, 2011
March, 2013
August, 2014
May, 2015
January, 2016
July, 2016
December, 2017
March, 2019
May, 2020
January, 2021
October, 2021
July, 2022
December, 2023
March, 2024
August, 2025
May, 2026
January, 2027
October, 2027
December, 2028
March, 2030
August, 2031
October, 2032
July, 2033
December, 2034
August, 2036
May, 2037
January, 2038
October, 2038
19
16
8
7
5
16
14
13
7
8
8
16
2
16
8
7
8
13
14
16
13
8
16
19
8
7
8
8

This pattern will repeat itself ad infinitum.  By inspection, then, here is the solution to the puzzle...

The times between successive Magnanimous Months range between 2 months and 19 months.

Epilog

Twelve years went by after the publication of Magnanimous Month.  Your puzzle-master received the following message:

This coming February cannot come in your life again. 
This year's February has...

4 Sundays
4 Mondays
4 Tuesdays
4 Wednesdays
4 Thursdays
4 Fridays
4 Saturdays
This happens once ever 823 years.  This is called miraclein.
The message goes on to promise miracles for immediate forwardings to others. 

Solvers recognize at once that
every February has exactly the same four-of- each weekday distribution except in leap-years.  Indeed, every month of every year has at least those same quantities plus bonuses of 0, 1, 2, or 3 days-of-the-week.

By the way, the word 'miraclein' cannot be found in 55 dictionaries.


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