Ask Wednesday: The husband yet again on the calendar

Fourteen calendars in sestina

By Morris Dean

The last time we interviewed the husband who last month with his wife celebrated their 47th wedding anniversary, he explained why a particular date recurs on the same day of the week according to a pattern, using for example the date April 29, which occurred on Monday this year (see calendar above right). As you may recall from that interview, April 29 falls on Monday again in the years 2019, 2024, 2030, and 2041. Note the number of intervening years, or intervals:

2013 [6 yrs] 2019 [5 yrs] 2024 [6 yrs] 2030 [11 yrs] 2041

And note that the sum of the four intervals is 28 [6 + 5 + 6 + 11 = 28].
    Today, after explaining why we need only fourteen calendars altogether, the husband shows us his template of calendar-recurrence patterns.
    [Our timorous questions are in italics.]

You say we need but fourteen calendars?
Yes, that gives us all possible patterns.
The reason is that any given year
must start on one of only seven days:
seven calendars—plus seven for leaps.
They cover a 28-year cycle....

Why is it a 28-year cycle?
Our common Gregorian calendar
needs twenty-eight to get seven that leap
(one leap year every four is the pattern).
Two calendars for each of seven days
on which begins a leap or non-leap year.

Is't true that, for twenty-one of the years
of a given 28-year cycle
(when February has twenty-eight days),
seven of the said fourteen calendars
must be reused in line with a pattern
by which from year to year they jump, or leap?

Yes. This lovely template—

Years 01-28 (which correspond to 2001-28)
begin on the weekday shown by the column headings

                                        —shows how they leap
and when they're used over the course of years
according to determining pattern.
What you should attend to is the cycle
for reusing the fourteen calendars
shown by the columns under the weekdays.

For instance, take the column for Tuesday,
when 2013 began. We leapt
here from—or can reuse the calendar
of—year 2002, eleven years
before in the 28-year cycle.
Each column shows its own leaping pattern.

What about each determines its pattern?
To answer that, just focus on each day-
column: the template reveals the cycle.
Where falls each year prior to the next leap
year—four, three, two, or one before such year?
Answer says when next the same calendar.^*

Annual patterns of predictive leaps
are shown by day for whatever year:
the cycle for reuse of calendar.

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* You might appreciate having it spelled out. Compare Observations 1-5 below with the template, which we repeat for your convenience:

Years 01-28 (which correspond to 2001-28)
begin on the weekday shown by the column headings

1. For each leap year (four years before another leap), it will be 5 yrs. before another year starts on the same day, but of course you'll have to use the regular-year calendar for it. (Twenty-eight yrs. must pass before you can use the same leap-year calendar again.)
2. For each year three years before a leap year, it will be 6 yrs. before another year starts again on the same day of the week—that is, before you can use that year's calendar again.
3. For each year two years before a leap year, similarly, it will be 6 yrs. before another year starts again on the same day. However, that year will be a leap year, so you'll have to use the alternative, leap-year calendar.
4. For each year one year before a leap year, it will be 11 yrs before another year starts again on the same day and you can use the same calendar. (The jump of 11 yrs. is represented by a blank in the template.)
5. In the 28-year cycle, you use each of the seven regular-year calendars three times and each of the seven leap-year calendars once.

To use the template for earlier years through 1901 or later years through 2099, simply lay it on top of them so the templates abut. For example, the template years 01-28 correspond to the calendar years 1917-44, 1945-72, 1973-2000, 2001-28, 2029-56, 2057-84. (The years 1901-16 correspond to template years 13-28, and the years 1985-2099 correspond to 01-15.)
    Why are the years 1900 and 2100 not included? They (and other years) which are divisible by 100 but not divisible by 400 are not leap years—the extra day (February 29) is not needed in order to keep the Gregorian calendar in sync. In order to use the template for periods involving a 00 year that is not a leap year, the appropriate regular-year calendar must be used for the 00 year itself, and the template must be shifted to apply to subsequent years. I will explain the shift in a subsequent article if people are interested.

Note that you can't extend back in time to years prior to the adoption of the Gregorian calendar (1752 in Britain and the British Empire, including the eastern part of what is now the United States).

Quiz question for extra credit: Why after 2019 does April 29 occur on the same day of the week again in 2024? Why doesn't it have to wait until 2030, the way January 1 through February 28 have to do?
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Copyright © 2013 by Morris Dean

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