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### Patterns in the Perpetual Calendar

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Date: 03/17/2000 at 03:09:16
From: Ella
Subject: Patterns in the perpetual calendar

I'm trying to find an answer to my question:

Why do some years (like 1717, 1923, 1934, 1945, 1951, 9162, 1973 etc)
that have the same calendar, have this calendar repeated every 6, then
11, then 11 again, and then again 6, 11, 11, 6 years?

And why are leap years calendar repeated every 28 years?

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Date: 03/17/2000 at 09:12:03
From: Doctor Rob
Subject: Re: Patterns in the perpetual calendar

Thanks for writing to Ask Dr. Math, Ella.

The reason for the 28-year cycle is that 28 is the least common
multiple of 4 and 7. 7 is the number of days in a week, and 4 is the
gap between leap years.

If a certain year had 1 January as Sunday, and it was a leap year,
then the following would be the day of the week on which 1 January
fell during the 28-year cycle:

Year No.  Day    Year No.  Day    Year No.  Day    Year No.  Day
1 L    Sun       8      Tue      15      Thu      22      Sat
2      Tue       9 L    Wed      16      Fri      23      Sun
3      Wed      10      Fri      17 L    Sat      24      Mon
4      Thu      11      Sat      18      Mon      25 L    Tue
5 L    Fri      12      Sun      19      Tue      26      Thu
6      Sun      13 L    Mon      20      Wed      27      Fri
7      Mon      14      Wed      21 L    Thu      28      Sat

This happens because 365 = 52*7 + 1, and 366 = 52*7 + 2, so that over
a normal year, the day of the week goes up by 1, and over a leap year,
it goes up by 2.

You see that the gaps between Sundays are 5, 6, 11, 6, for Mondays
they are 6, 5, 6, 11, for Tuesdays they are 6, 11, 6, 5, and so on.
The reason 6 appears is that one leap year occurs between the first
and second occurrence, 5*1 + 1*2 = 7, and 5 + 1 = 6. The reason 5
appears is that two leap years intervene, one being the year of the
first occurrence, 3*1 + 2*2 = 7, and 3 + 2 = 5. The reason 11 appears
is that three leap years intervene, one of them causes that day of the
week to be skipped, 8*1 + 3*2 = 2*7, and 8 + 3 = 11.

If we drop the leap year calendars, which do repeat on a cycle of 28,
then the normal year gaps between identical calendars follow the 11,
11, 6, 11, 11, 6, ... pattern, and note that 11 + 11 + 6 = 28.

Each day of the week will appear four times during this cycle, once on
a leap year and three times on normal years. There will be 7 leap
years in the cycle and 21 normal years.

Of course this 28-year cycle can be disrupted in a year like 1900
which is a year divisible by 100 but not by 400, and so is *not* a
leap year, but the pattern holds from 1901 through 2100, the next
exceptional year.

If you take exceptional years into account, there is a 2800-year
cycle, since 2800 is the least common multiple of 4, 7, 100, and 400.

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```

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Date: 03/17/2000 at 12:44:06
From: Doctor TWE
Subject: Re: Patterns in the perpetual calendar

Hi Ella - thanks for writing to Dr. Math.

You are on the right track by considering leap years. Non-leap years
have 365 days, which is 52 weeks plus 1 day. If there were no leap
years, every year would start off one weekday later than the year
before it (and every other date on that year would be one weekday
later than the year before as well). 1/1/2001 falls on a Monday, so
1/1/2002 will fall on a Tuesday, 1/1/2003 will fall on a Wednesday,
and so on. If it weren't for leap years, we'd have a nice, simple
pattern as follows:

Year   Weekday of 1/1
----   --------------
2001   MONDAY
2002   Tuesday
2003   Wednesday
2004   Thursday
2005   Friday   (assuming that 2004 was NOT a leap year)
2006   Saturday
2007   Sunday
2008   MONDAY
2009   Tuesday  (assuming that 2008 was NOT a leap year)
:       :

If this were the case, isomorphic years (those years whose calendars
are identical) would occur every 7 years.

But leap years complicate the pattern. Because leap years have an
extra day (52 weeks plus 2 days), each year following a leap year
begins two weekdays later than the previous year - we "leap" over a
day. (This is where the term "leap year" comes from.)

With leap years, the repeating cycle for isomorphic years shrinks to 6
years (typically), because one day is "leaped over" during the cycle.
For example, in 2004, Friday is leaped over, so the next year
isomorphic to 2001 (starting on a Monday) moves up from 2008 (if there
were no leap years) to 2007 (with leap years) - see the table below.

Year   Weekday of 1/1
----   --------------
2001   MONDAY
2002   Tuesday
2003   Wednesday
2004   Thursday   (Friday leaped over)
2005   Saturday
2006   Sunday
2007   MONDAY
2008   Tuesday    (Wednesday leaped over)
2009   Thursday
:       :

Sometimes, however, the calendar we're looking at (say for example,
the one that starts on a Monday) is the one that *gets* leaped over.
For example:

Year   Weekday of 1/1
----   --------------
2007   MONDAY
2008   Tuesday   (Wednesday leaped over)
2009   Thursday
2010   Friday
2011   Saturday
2012   Sunday    (MONDAY leaped over)
2013   Tuesday
2014   Wednesday
2015   Thursday
2016   Friday    (Saturday leaped over)
2017   Sunday
2018   MONDAY
:       :

The year 2007 starts on a Monday, so the next isomorphic year should
be 2013 (2007+6), but since the previous year (2012) is a leap year,
Monday gets leaped over, and we have to go another 6 years from 2012 -
the year which causes Monday to be skipped. Thus, we go from 2007 to
2018 before getting back to a Monday - a total of 11 years (two 6's
minus the Monday that was leaped over).

Sometimes, too, the year that is "supposed" to be isomorphic falls on
a leap year. A leap year can't be isomorphic to a non-leap year
because they don't have the same number of days. For example, 2018
starts on a Monday. The next isomorphic year is "supposed" to be 2024,
but that is a leap year. Tuesday gets leaped over in 2024 AND Sunday
gets leaped over in 2028 before the next isomorphic year to 2018.
Thus, there are 11 years separating these isomorphic years instead of
6.

Year   Weekday of 1/1
----   --------------
2018   MONDAY
2019   Tuesday
2020   Wednesday  (Thursday leaped over)
2021   Friday
2022   Saturday
2023   Sunday
2024   MONDAY     (Tuesday leaped over)
2025   Wednesday
2026   Thursday
2027   Friday
2028   Saturday   (Sunday leaped over)
2029   MONDAY
:       :

These leap year complications form a 28-year cycle in which isomorphic
non-leap years occur in intervals of 6 years - 11 years - 11 years,
etc. Why is the cycle 28 years? Because there are 7 weekdays and leap
years occur every 4 years. The smallest multiple of both 7 and 4 is
28.

The days leaped over on successive leap years follow the pattern
Monday-Saturday-Thursday-Tuesday-Sunday-Friday-Wednesday-
Monday-Saturday etc.
The year 2000 leaps over a Sunday. Based on this information, I think
you can figure out why isomorphic leap years occur every 28 years.

There is one more complication, however. Some years evenly divisible
by 4 are NOT leap years, and therefore don't leap over a day. This
causes further "irregularities" in the pattern. Years ending in 00
(for example, 1700, 1800, 1900, 2100, etc.) are not leap years UNLESS
the century (the first two digits) is divisible by 4. Here's a
partial chart:

Year   Leap year?
----   ----------
:     :
1500   No
1600   Yes
1700   No
1800   No
1900   No
2000   Yes
2100   No
2200   No
2300   No
2400   Yes
2500   No
:     :

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
http://mathforum.org/dr.math/
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