The Doomsday rule, Doomsday algorithm or Doomsday method is an algorithm of determination of the day of the week for a given date. It provides a perpetual calendar because the Gregorian calendar moves in cycles of 400 years. The algorithm for mental calculation was devised by John Conway in 1973,John Horton Conway, Richard Guy, John Horton Conway, Elwyn Berlekamp : "Winning Ways: For Your Mathematical Plays, Volume. 2: Games in Particular", pages 795–797, Academic Press, London, 1982, . drawing inspiration from Lewis Carroll's perpetual calendar algorithm.Lewis Carroll, "To Find the Day of the Week for Any Given Date", Nature, March 31, 1887. Martin Gardner, The Universe in a Handkerchief: Lewis Carroll's Mathematical Recreations, Games, Puzzles, and Word Plays, pages 24–26, Springer-Verlag, 1996. It takes advantage of each year having a certain day of the week upon which certain easy-to-remember dates, called the doomsdays, fall; for example, the last day of February, April 4 (4/4), June 6 (6/6), August 8 (8/8), October 10 (10/10), and December 12 (12/12) all occur on the same day of the week in the year.
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 (modulo 7) between that date and the date in question to arrive at the day of the week. The technique applies to both the Gregorian calendar and the Julian calendar, although their doomsdays are usually different days of the week.
The algorithm is simple enough that it can be computed mentally. Conway could usually give the correct answer in under two seconds. To improve his speed, he practiced his calendrical calculations on his computer, which was programmed to quiz him with random dates every time he logged on.
Doomsdays for contemporary years Doomsday for the current year in the Gregorian calendar () is . Simple methods for finding the doomsday of a year exist.
+ Anchor days for years from 1796 through 2105The table is filled in horizontally, skipping one column for each leap year. This table cycles every 28 years, except in the Gregorian calendar on years that are a multiple of 100 (such as 1800, 1900, and 2100 which are not leap years) that are not also a multiple of 400 (like 2000 which is still a leap year). The full cycle is 28 years (1,461 weeks) in the Julian calendar and 400 years (20,871 weeks) in the Gregorian calendar. Sun Mon Tue Wed Thu Fri Sat 1796 1797 1798 1799 1800 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2033 2034 2035 2036 2037 2038 2039 2040 2041 2042 2043 2044 2045 2046 2047 2048 2049 2050 2051 2052 2053 2054 2055 2056 2057 2058 2059 2060 2061 2062 2063 2064 2065 2066 2067 2068 2069 2070 2071 2072 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 2085 2086 2087 2088 2089 2090 2091 2092 2093 2094 2095 2096 2097 2098 2099 2100 2101 2102 2103 2104 2105
Finding the day of the week from a year's doomsday One can find the day of the week of a given calendar date by using a nearby doomsday as a reference point. To help with this, the following is a list of easy-to-remember dates for each month that always land on the doomsday.
The last day of February is always a doomsday. For January, January 3 is a doomsday during common years and January 4 a doomsday during leap years, which can be remembered as "the 3rd during 3 years in 4, and the 4th in the 4th year". For March, one can remember either Pi Day or "March 0", the latter referring to the day before March 1, i.e. the last day of February.
For the months April through December, the even numbered months are covered by the double dates 4/4, 6/6, 8/8, 10/10, and 12/12, all of which fall on the doomsday. The odd numbered months can be remembered with the mnemonic "I work from 9 to 5 at the 7–11", i.e., 9/5, 7/11, and also 5/9 and 11/7, are all doomsdays (this is true for both the day/month and month/day conventions).
John Conway wrote that: "Summary: 'Last' in Jan and Feb, otherwise nth in even months, in odd ones". He clarified that: "The sign is + for long odd months (31 days), and − for short ones (30 days)".
Several well-known dates, such as Independence Day in United States, Boxing Day, Halloween and Valentine's Day in common years, also fall on doomsdays every year.
Month Memorable date Month/day Day/month MnemonicComplete list of days January January 3 (common years), January 4 (leap years) 1/3 OR 1/4 (1/31 OR 1/32) 3/1 OR 4/1 (31/1 OR 32/1) the 3rd 3 years in 4 and the 4th in the 4th (or: last day of January, pretending leap years have a January 32) 3, 10, 17, 24, 31 OR February February 28 (common years), February 29 (leap years) 2/0 OR 2/1 (2/28 OR 2/29) 0/2 OR 1/2 (28/2 OR 29/2) last day of January, pretending leap years have a January 32 (or: last day of February) 0, 7, 14, 21, 28 OR March "March 0," March 14 3/0 AND 3/14 0/3 AND 14/3 last day of February, Pi Day 0, 7, 14, 21, 28 April April 4 4/4 4/4, 6/6, 8/8, 10/10, 12/12 4, 11, 18, 25 May May 9 5/9 9/5 9-to-5 at 7-112, 9, 16, 23, 30 June June 6 6/6 4/4, 6/6, 8/8, 10/10, 12/12 6, 13, 20, 27 July July 11 7/11 11/7 9-to-5 at 7-11 4, 11, 18, 25 August August 8 8/8 4/4, 6/6, 8/8, 10/10, 12/12 1, 8, 15, 22, 29 September September 5 9/5 5/9 9-to-5 at 7-11 5, 12, 19, 26 October October 10 10/10 4/4, 6/6, 8/8, 10/10, 12/12 3, 10, 17, 24, 31 November November 7 11/7 7/11 9-to-5 at 7-11 0, 7, 14, 21, 28 December December 12 12/12 4/4, 6/6, 8/8, 10/10, 12/12 5, 12, 19, 26
Since the doomsday for a particular year is directly related to weekdays of dates in the period from March through February of the next year, common years and leap years have to be distinguished for January and February of the same year.
Example To find which day of the week Christmas Day of 2027 is, proceed as follows: in the year 2027, doomsday is on Sunday. Since December 12 is a doomsday, December 25, being thirteen days afterwards (two weeks less a day), will fall on a Saturday. Christmas Day is always the day of the week before doomsday. In addition, July 4 (U.S. Independence Day) is always on the same day of the week as a doomsday, as are Halloween (October 31), Pi Day (March 14), and December 26 (Boxing Day).
Mnemonic weekday names Since this algorithm involves treating days of the week like numbers modulo 7, John Conway suggested thinking of the days of the week as "Noneday" or "Sansday" (for Sunday), "Oneday", "Twosday", "Treblesday", "Foursday", "Fiveday", and "Six-a-day" to recall the number-weekday relation without needing to count them out in one's head. Day of week Indexnumber Mnemonic Sunday 0 Noneday orSansday Monday 1 Oneday Tuesday 2 Twosday Wednesday 3 Treblesday Thursday 4 Foursday Friday 5 Fiveday Saturday 6 Six-a-day There are some languages, such as Slavic languages, Chinese, Estonian, Greek, Portuguese and Hebrew, that base some of the names of the week days in their positional order. The Slavic, Chinese, and Estonian agree with the table above; the other languages mentioned count from Sunday as day one.
Finding a year's doomsday First take the anchor day for the century. For the purposes of the doomsday rule, a century starts with '00 and ends with '99. The following table shows the anchor day of centuries 1600–1699, 1700–1799, 1800–1899, 1900–1999, 2000–2099, 2100–2199 and 2200–2299.
Century Anchor day Mnemonic Index numberWeekday mnemonic 1600–1699 Tuesday — 2Twoday 1700–1799 Sunday — 0Noneday 1800–1899 Friday — 5Fiveday 1900–1999 Wednesday We-in-dis-day(most living people were born in that century, see List of countries by median age) 3Treblesday 2000–2099 Tuesday Y-Tue-K or Twos-day(Y2K was at the head of this century) 2Twosday 2100–2199 Sunday Twenty-one-day is Sunday(2100 is the start of the next century) 0Noneday 2200–2299 Friday — 5Fiveday For the Gregorian calendar: Mathematical formula Tuesday = anchor. Algorithmic Let if then anchor = Tuesday if then anchor = Sunday if then anchor = Friday if then anchor = Wednesday. For the Julian calendar: Sunday = anchor. Note: .
Next, find the year's anchor day. To accomplish that according to Conway:John Horton Conway,
Divide the year's last two digits (call this ) by 12 and let be the floor of the quotient. Let be the remainder of the same quotient. Divide that remainder by 4 and let be the floor of the quotient. Let be the sum of the three numbers (). (It is again possible here to divide by seven and take the remainder. This number is equivalent, as it must be, to plus the floor of divided by four.) Count forward the specified number of days ( or the remainder of ) from the anchor day to get the year's one.
For the twentieth-century year 1966, for example:
As described in bullet 4, above, this is equivalent to:
So doomsday in 1966 fell on Monday.
Similarly, doomsday in 2005 is on a Monday:
Why it works
The doomsday's anchor day calculation is effectively calculating the number of days between any given date in the base year and the same date in the current year, then taking the remainder modulo 7. When both dates come after the leap day (if any), the difference is just (rounded down). But 365 equals 52 × 7 + 1, so after taking the remainder we get just
This gives a simpler formula if one is comfortable dividing large values of by both 4 and 7. For example, we can compute
which gives the same answer as in the example above.
Where 12 comes in is that the pattern of almost repeats every 12 years. After 12 years, we get . If we replace by , we are throwing this extra day away; but adding back in compensates for this error, giving the final formula.
For calculating the Gregorian anchor day of a century: three “common centuries” (each having 24 leap years) are followed by a “leap century” (having 25 leap years). A common century moves the doomsday forward by
days (equivalent to two days back). A leap century moves the doomsday forward by 6 days (equivalent to one day back).
So c centuries move the doomsday forward by , but this is equivalent to . Four centuries move the doomsday forward by ; so four centuries form a cycle that leaves the doomsday unchanged (and hence the “mod 4” in the century formula).
The "odd + 11" method
A simpler method for finding the year's anchor day was discovered in 2010 by Chamberlain Fong and Michael K. Walters.Chamberlain Fong, Michael K. Walters: "Methods for Accelerating Conway's Doomsday Algorithm (part 2)", 7th International Congress on Industrial and Applied Mathematics (2011). Called the "odd + 11" method, it is equivalent to computing
.