Your VCR had a calendar failure on January 1st? Need to reset it to a year which it can handle?

Got a nice wall calendar? Want to save it and use it again? How long will you have to wait?

On the next page is a calculator to answer these questions.

Year is the particular year you're interested in, e.g., 2001 CE. Start year and End year specify the range of years you're interested in.

For example, if you enter 2001, 1990 and 2200 then (after you click on 'Submit') the output is:

1st January 2001 CE is a Monday.  2001 is not a leap year.

The following calendar years in the range 1990 through 2200
are identical to 2001:
   1990                       2001  (11: 4018)           2007  ( 6: 2191)
   2018  (11: 4018)           2029  (11: 4018)           2035  ( 6: 2191)
   2046  (11: 4018)           2057  (11: 4018)           2063  ( 6: 2191)
   2074  (11: 4018)           2085  (11: 4018)           2091  ( 6: 2191)
   2103  (12: 4382)           2114  (11: 4018)           2125  (11: 4018)
   2131  ( 6: 2191)           2142  (11: 4018)           2153  (11: 4018)
   2159  ( 6: 2191)           2170  (11: 4018)           2181  (11: 4018)
   2187  ( 6: 2191)           2198  (11: 4018)

The numbers in parentheses (a:b) give the number of years since the previous year and the number of days from one 1st January to the 1st January of the next year of the same type.

As another example, if you enter 1904, 1960 and 2300 then (after you click on 'Submit') the output is:

1st January 1904 CE is a Friday.  1904 is a leap year.

The following calendar years in the range 1960 through 2300
are identical to 1904:
   1960                       1988  (28:10227)           2016  (28:10227)
   2044  (28:10227)           2072  (28:10227)           2112  (40:14609)
   2140  (28:10227)           2168  (28:10227)           2196  (28:10227)
   2208  (12: 4382)           2236  (28:10227)           2264  (28:10227)
   2292  (28:10227)

Since 1st January of any given year occurs on one of seven weekdays, and a year is either a leap year or not, there are fourteen possible calendar years.  As the results above show, the calendar years do not proceed in a regular fashion.  This is because a year consists either of 52 weeks plus 1 day or (in a leap year) 52 weeks plus 2 days, so the day of the week of the 1st January of the next year is either one or two days later in the week.  And when a year with 1st January on the same day of the week comes around again it may or may not be a leap year.

The possible numbers of days elapsed from the 1st of January to the 1st of January of the next year of the same type is always one of: 2,191 (6 years), 4,018 (11 years), 4,382 (12 years), 10,227 (28 years) and 14,609 (40 years). Thus to use that nice wall calendar again you'll have to wait at least six years and at most forty years.

Each of these numbers of days is a multiple of 7 because the 1sts January of years in a sequence of years of the same type always occur on the same day of the week. Thus the possible numbers of weeks from one 1st January to the next are: 313 (prime), 574 (2*7*41), 626 (2*313), 1,461 (3*487) and 2,087 (prime). It's curious that 1,461 happens to be the number of days in four successive Gregorian years, and here it emerges again as a possible number of weeks between successive 1sts January in years of the same type. A coincidence? No. On 2001-01-19 CE Karl Palmen wrote to the CALNDR-L mailing list:

First I note that 6 and 11 years can only occur for common years,
because they are not a multiple of 4.  28 and 40 years occur only
for leap years; for common years, they'd split into smaller periods.
12 (I think) may occur for either common or leap years and must contain
a dropped leap-year.

The 28-years is simply the Julian cycle which is 7*1461 days. Since the
Gregorian calendar follows the Julian pattern, except for some century
years it's bound to occur for leap years.

The periods add up as follows (for number of days/weeks as well as years)
12 = 6+6
28 = 6+11+11
40 = 6+6+6+11+11 = 12+28

It may be worth counting the leap days and adding to the number of years

Years Leap Total Total/7
 6     1     7     1
11     3    14     2
12     2    14     2
28     7    35     5
40     9    49     7

The total/7 is a leap-week count.

To use the online calculator click here.

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On this subject see also "Cycles in the Gregorian Calendar"
in Lance Latham's Standard C Date/Time Library
(R&D Books, Miller-Freeman, 1998), pp.249-260.