How often in years do calendars repeat with the same day-date combinations?

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Calendars repeat in a regular cycle, at least within a century, because each year has 1 day more than exactly 52 weeks, and leap years add another extra day . This combination results in a sequence of repeated calendars in a 28-year cycle. For NON-LEAP YEARS, a given arrangement of days will repeat in 6 years, then 11, then 11 years, then begin a new cycle. Crossing a century changes this because only every 4th century year (e.g. 2000 but not 2100, 2200, or 2300) is a leap year. LEAP YEAR calendars repeat every 28 years.

When Calendars repeat
There is a very simple pattern for determining "when" calendars repeat with the same day/date combinations. Define any year as one of four things: a "leap year", the "1st year after a leap year", the "2nd year after a leap year", or the "3rd year after a leap year". Add 28 to a "leap year" to get the next year that it will repeat. Add 6 to the "1st year after a leap year" to get the next year that it will repeat; add 11 to the "2nd year after a leap year" to get the next year that it will repeat. Also add 11 to the "3rd year after a leap year" to get the next year that it repeats. Ex: 2010 is the 2nd yr after a leap year and will repeat in 2010 + 11 = 2021. 2011 repeats in 2022; 2012 repeats in 2040; 2013 repeats in 2019; 2014 repeats in 2025; 2015 repeats in 2026; 2016 repeats in 2044; 2017 repeats in 2023; and so on.


How often calendars repeat
  • Any leap year calendar will repeat in exactly the same way every 28 years.
  • Any "1st year after a leap year" will repeat in a 6-11-11 cycle, ie, it will repeat in 6 years, then come up again in 11 years, come up again in 11 more years, then repeat the 6-11-11 year cycle.
  • Both "2nd year after a leap year" and "3rd year after a leap year" will repeat in an 11-11-6 cycle.


Notice the patterns involve 6, 11, and 28. The cycles of 6-11-11 and 11-11-6 add to 28.

It can be 5, 6, or 11 years before an individual date-day combination comes up again. Individual days will repeat on a 6, 5, 6, 11 year cycle. For example, January 1, 2000 (a leap year) was a Saturday. The years that January 1 fell on or will fall on Saturday are: 1955, 1966, 1972, 1977, 1983, 1994, 2000, 2005, 2011, 2022, 2028, 2033, 2039, and 2050.

Possible Calendars
There are 14 possible combinations:
A year can begin on one of 7 weekdays, and the year can be a regular year or a leap year.
Note that the cycle doesn't repeat on a 14 year basis. Because the year can start on one of 7 days, and a leap year comes every 4 years, the cycle is more complicated, but any given year can have its calendar taken from one of the 14 possible calendars.
In a regular (non-leap) year, the year ends on the same day of the week as it starts (2005 started on Saturday, and will end on Saturday) That's 52 weeks and one day. The next year starts on the next day of the week, and should end on that same weekday. (2006 starts on a Sunday, and ends on a Sunday.)
A leap year will end on the weekday immediately after the weekday on which it starts, and is 52 weeks and 2 days long. (2004 started on a Thursday, and ended on a Friday).
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