Goines On: Those elusive leap years

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In the few days remaining before their flight to California, Goines wondered whether the solution to the problem of calculating the day of the week for dates in past or future years might be a lot simpler than he had been imagining. What if he just used his mnemonic for the current year (“Mi Ki Ko Ranch” rattles Miquel) and then “shifted” to the year of the date in question?
    The calendar for next year varies from this year’s calendar by starting either 1 day or 2 days later in January, depending on whether this year is, or is not, a leap year (2021 is not). Leap years have 366 days, or 35 weeks plus 2 days. Last year’s calendar starts a day or two earlier in the week for the same reasons. Goines wasn’t going to solve this before they left, so he tried to stop thinking about it and finish packing.
    One of the things Goines packed was his scribblings of calendar charts, but on the flight to MSP for their connection to the flight to SFO he soon tired of the stress of thinking about calendars. He needed a vacation from that. He decided to try to nap the rest of the way to California.
    During the week at Coos Bay, he kept his scribblings stacked on the table at the end of the futon in their cabin, but he looked at them only once, without progress. And the few times on walks that his thoughts turned to calendars, the thoughts went around in wearying circles, revealing no inner light. He tried to concentrate instead on the light coming off the bay, out of the clouds.

During the first few days back home, Goines had better luck with calendars. He focused on the fact that shifts of multiple years into the past or future would accumulate the single-year shifts of either 1 or 2 days. (He set aside for later Pope Gregory XIII’s little glitch that years ending in 00 and not integrally divisible by 400 aren’t leap years, so they would add a shift of only 1 day, not 2.)
    Over days of experimentation – drawing more charts on paper and trying to articulate the discoveries he made – Goines finally thought he had formulated the procedure he could follow to determine the day of the week for any date covered by the Gregorian calendar. All he had to do was calculate the shift from the same date this year to the year in question.
    He set about testing it. For example, say August 24, 1743. Okay, what day of the week does August 24 fall on this year (2021)? On Tuesday. So, the question is: how many day-shifts back would there be for the 278 years from 2021 to 1743?
    Okay, how many leap years are involved? It would seem, from dividing 4 into 278, that there are 69 leap years. But 278 ÷ 4 = 69.5, not just 69. What about the two days represented by that 0.5? Do they include a leap year, or not? Do we or don’t we add 1?
    This point gave Goines a lot of trouble. He considered the range 2021 back in time to 2015, involving a shift of 6 years; did they include 2 leap years, or just 1? Since 2016 and 2020 (in that 6-year range) are both leap years, the 6 years involved 2 leap years, so there were 8 day shifts between the two years, and 8 ÷ 7 = 1, with a remainder of 1; therefore, August 24, 2015, fell one day of the week earlier than in 2021: on Monday.
    So, for the 1743 example (shifting back 278 years), do we round 69.5 down to 69, or up to 70? 2021 is one year after a leap year, and 1743 was one year before a leap year (same as in the case of 2015 – 2021). Better round 69.5 up to 70.

    And then there was the matter of Pope Gregory XIII’s glitch – applying to 1800 and 1900 in this example. Subtract 1 for each of them: 70 – 2 = 68. So, to the 278-year difference between 1743 & 2021 Goines added 1 more for each of those 68 leap years, for a total of 346 day shifts.
    And 346 ÷ 7 = 49, with a remainder of 3. Result: August 24 fell three days earlier in the week in 1743 than it did in 2021 – on Saturday, not Tuesday.

    Goines smiled at the thought that he could leave it to whoever asked him for August 24’s day of the week in 1743 to go check the Saturday result for themselves. You could google for any year’s calendar, and Google seemed to be prepared to tell you – Google seemed to have them all!

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