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Patterns in the Perpetual Calendar

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?

Please help.

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   

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 

     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 

The days leaped over on successive leap years follow the pattern 
     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   
Associated Topics:
Middle School About Math

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