Perpetual calendar

• Offices and retail establishments often display devices containing a set of elements to form all possible numbers from 1 through 31, as well as the names/abbreviations for the months and the days of the week, so as to show the current date for the convenience of people who might be signing and dating documents such as checks. Establishments that serve alcoholic beverages may use a variant that shows the current month and day, but subtracting the legal age of alcohol consumption in years, indicating the latest legal birth date for alcohol purchases. A very simple device consists of two cubes in a holder. One cube carries the numbers zero to five. The other bears the numbers 0, 1, 2, 6 (or 9 if inverted), 7 and 8. This is perpetual because only one and two may appear twice in a date and they are on both cubes.
• Certain calendar reforms have been labeled perpetual calendars because their dates are fixed on the same weekdays every year. Examples are The World Calendar, the International Fixed Calendar and the Pax Calendar. Technically, these are not perpetual calendars but perennial calendars. Their purpose, in part, is to eliminate the need for perpetual calendar tables, algorithms and computation devices.
• In watchmaking, "perpetual calendar" describes a calendar mechanism that correctly displays the date on the watch 'perpetually', taking into account the different lengths of the months as well as leap years. The internal mechanism will move the dial to the next day.[6]

These meanings are beyond the scope of the remainder of this article.

Perpetual calendars use algorithms to compute the day of the week for any given year, month, and day of month. Even though the individual operations in the formulas can be very efficiently implemented in software, they are too complicated for most people to perform all of the arithmetic mentally.[7] Perpetual calendar designers hide the complexity in tables to simplify their use.

A perpetual calendar employs a table for finding which of fourteen yearly calendars to use. A table for the Gregorian calendar expresses its 400-year grand cycle: 303 common years and 97 leap years total to 146,097 days, or exactly 20,871 weeks. This cycle breaks down into one 100-year period with 25 leap years, making 36,525 days, or one day less than 5,218 full weeks; and three 100-year periods with 24 leap years each, making 36,524 days, or two days less than 5,218 full weeks.

Within each 100-year block, the cyclic nature of the Gregorian calendar proceeds in exactly 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 day of the week after the leap year began. Every four years, the starting weekday advances five days, so over a 28-year period it advances 35, returning to the same place in both the leap year progression and the starting weekday. This cycle completes three times in 84 years, leaving 16 years in the fourth, incomplete cycle of the century.

A major complicating factor in constructing a perpetual calendar algorithm is the peculiar and variable length of February, which was at one time the last month of the year, leaving the first 11 months March through January with a five-month repeating pattern: 31, 30, 31, 30, 31, ..., so that the offset from March of the starting day of the week for any month could be easily determined. Zeller's congruence, a well-known algorithm for finding the day of week for any date, explicitly defines January and February as the "13th" and "14th" months of the previous year in order to take advantage of this regularity, but the month-dependent calculation is still very complicated for mental arithmetic:

${\displaystyle \left\lfloor {\frac {(m+1)26}{10}}\right\rfloor \mod 7,}$

Instead, a table-based perpetual calendar provides a simple look-up mechanism to find offset for the day of week for the first day of each month. To simplify the table, in a leap year January and February must either be treated as a separate year or have extra entries in the month table:

• Antikythera Mechanism
• Determination of the day of the week
• Doomsday rule
• Long Now Foundation
• Year 10,000 problem

Wikimedia Commons has media related to Perpetual calendars.

• New Perpetual Calendar for any year
• Perpetual Calendar in JavaScript
• Markl, Xavier (15 June 2016). "Monochrome-Watches A technical perspective calendar watches".