exupero's blog

Calendar recurrence

There are 14 possible Gregorian calendars, given that a Gregorian year can begin on any day of the week and has either 365 or 366 days. In this post I'm going to explore how often each of those calendars can be reused.

Leap years cycle through beginning on each day of the week, starting two days earlier in the week than the previous leap year. Thus, a 366-day calendar can only be reused once every 28 years:

January 1992
SMTWTFS
1
234
567891011
12131415161718
19202122232425
262728293031
January 1996
SMTWTFS
1
23456
78910111213
14151617181920
21222324252627
28293031
January 2000
SMTWTFS
1
2345678
9101112131415
16171819202122
23242526272829
3031
January 2004
SMTWTFS
1
23
45678910
11121314151617
18192021222324
25262728293031
January 2008
SMTWTFS
1
2345
6789101112
13141516171819
20212223242526
2728293031
January 2012
SMTWTFS
1
234567
891011121314
15161718192021
22232425262728
293031
January 2016
SMTWTFS
1
2
3456789
10111213141516
17181920212223
24252627282930
31
January 2020
SMTWTFS
1
234
567891011
12131415161718
19202122232425
262728293031

365-day calendars recur more frequently, though, because of the insertion of leap years, not quite so regularly. This year's calendar started on Saturday, as did 2011's calendar, and before that 2005's. In general, non-leap year calendars recur after either 6 or 11 years: after 6 years for years immediately following a leap year, after 11 years for other non-leap years. That means a non-leap calendar recurs after 6 years, then after 11 years, then after another 11 years, then after 6 years again, repeating in what, like leap recurrence, is a 28-year cycle.

January 2005
SMTWTFS
1
2345678
9101112131415
16171819202122
23242526272829
3031
January 2006
SMTWTFS
1
234567
891011121314
15161718192021
22232425262728
293031
January 2007
SMTWTFS
1
23456
78910111213
14151617181920
21222324252627
28293031
January 2008
SMTWTFS
1
2345
6789101112
13141516171819
20212223242526
2728293031
January 2009
SMTWTFS
1
23
45678910
11121314151617
18192021222324
25262728293031
January 2010
SMTWTFS
1
2
3456789
10111213141516
17181920212223
24252627282930
31
January 2011
SMTWTFS
1
2345678
9101112131415
16171819202122
23242526272829
3031
January 2012
SMTWTFS
1
234567
891011121314
15161718192021
22232425262728
293031
January 2013
SMTWTFS
1
2345
6789101112
13141516171819
20212223242526
2728293031
January 2014
SMTWTFS
1
234
567891011
12131415161718
19202122232425
262728293031
January 2015
SMTWTFS
1
23
45678910
11121314151617
18192021222324
25262728293031
January 2016
SMTWTFS
1
2
3456789
10111213141516
17181920212223
24252627282930
31
January 2017
SMTWTFS
1
234567
891011121314
15161718192021
22232425262728
293031
January 2018
SMTWTFS
1
23456
78910111213
14151617181920
21222324252627
28293031
January 2019
SMTWTFS
1
2345
6789101112
13141516171819
20212223242526
2728293031
January 2020
SMTWTFS
1
234
567891011
12131415161718
19202122232425
262728293031
January 2021
SMTWTFS
1
2
3456789
10111213141516
17181920212223
24252627282930
31
January 2022
SMTWTFS
1
2345678
9101112131415
16171819202122
23242526272829
3031