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Weeks start with Monday and end on Sunday. Each week's year is the Gregorian year in which the Thursday falls. The first week of the year, hence, always contains 4 January. ISO week year numbering therefore usually deviates by 1 from the Gregorian for some days close to 1 January.
Applying the Doomsday algorithm involves three steps: determination of the anchor day for the century, calculation of the anchor day for the year from the one for the century, and selection of the closest date out of those that always fall on the doomsday, e.g., 4/4 and 6/6, and count of the number of days between that date and the date in ...
The basic approach of nearly all of the methods to calculate the day of the week begins by starting from an "anchor date": a known pair (such as 1 January 1800 as a Wednesday), determining the number of days between the known day and the day that you are trying to determine, and using arithmetic modulo 7 to find a new numerical day of the week.
Similarly, the last ISO week of a year may have up to three days that are actually in the Gregorian calendar year that is starting; if three, they are Friday, Saturday, and Sunday. The Thursday of each ISO week is always in the Gregorian calendar year denoted by the ISO week-numbering year. Examples: Monday 29 December 2008 is written "2009-W01-1"
Leap years have two letters, so for January and February calculate the day of the week for January 1 and for March to December calculate the day of the week for October 1. Leap years are all years that divide exactly by four, with the following exceptions: Gregorian calendar – all years divisible by 100, except those that divide exactly by 400.
Zeller's congruence is an algorithm devised by Christian Zeller in the 19th century to calculate the day of the week for any Julian or Gregorian calendar date. It can be considered to be based on the conversion between Julian day and the calendar date.
In a Gregorian mean year, there are 365.2425 days, and thus exactly 52 + 71 ⁄ 400 or 52.1775 weeks (unlike the Julian year of 365.25 days or 52 + 5 ⁄ 28 ≈ 52.1786 weeks, which cannot be represented by a finite decimal expansion). There are exactly 20,871 weeks in 400 Gregorian years, so 6 January 1625 was a Monday just as was 6 January 2025.
Within each 100-year block, the cyclic nature of the Gregorian calendar proceeds in the same fashion as its Julian predecessor: A common year begins and ends on the same day of the week, so the following year will begin on the next successive day of the week. A leap year has one more day, so the year following a leap year begins on the second ...