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,[1] [2] drawing inspiration from Lewis Carroll's perpetual calendar algorithm.[3] [4] [5] 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, 4/4, 6/6, 8/8, 10/10, and 12/12 all occur on the same day of the week in any 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.[6]
Doomsday's anchor day for the current year in the Gregorian calendar is . For some other contemporary years:
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | |
|---|---|---|---|---|---|---|---|
| 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 |
The 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.
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).[7]
Several well-known dates, such as Independence Day in United States, Boxing Day, and Valentine's Day in common years, also fall on doomsdays every year. The chart below includes only the mnemonics covered in the sources listed.
| Month | Memorable date | Month/Day | Mnemonic[8] [9] | Complete list of days | |
|---|---|---|---|---|---|
| January | January 3 (common years), January 4 (leap years) | 1/3 OR 1/4 (1/31 OR 1/32) | the 3rd 3 years in 4 and the 4th in the 4th (or: last day of January, pretending leap years have a January 32nd) | 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) | last day of January, pretending leap years have a January 32nd (or: last day of February) | 0, 7, 14, 21, 28 OR | |
| March | "March 0," March 14 | 3/0 AND 3/14 | 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-to-5 at 7-11 | 2, 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 | 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 | 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 | 9-to-5 at 7-11 | 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.
| Month | M | Doomsday | |||
|---|---|---|---|---|---|
| Jan | 1 | 3/4 | C/L | C/D | |
| Feb | 2 | 0/1 | |||
| Mar | 3 | 7/0 | M + 4 | C day | |
| May | 5 | 9 | |||
| Jul | 7 | 11 | |||
| Sep | 9 | 5 | M − 4 | ||
| Nov | 11 | 7 | |||
| Jan | 13 | 9/2 | B day | ||
| Apr | 4 | 4 | M | C day | |
| Jun | 6 | 6 | |||
| Aug | 8 | 8 | |||
| Oct | 10 | 10 | |||
| Dec | 12 | 12 | |||
| Feb | 14 | 13/−1 | M − 1 | B day | |
To find which day of the week Christmas Day of 2021 is, proceed as follows: in the year 2021, doomsday is on Sunday. Since December 12 is a doomsday, December 25, being thirteen days afterwards (two weeks less a day), fell 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).
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" in order to recall the number-weekday relation without needing to count them out in one's head.[10]
| day of week | Index number | Mnemonic | |
|---|---|---|---|
| Sunday | 0 | Noneday or Sansday | |
| Monday | 1 | Oneday | |
| Tuesday | 2 | Twosday | |
| Wednesday | 3 | Treblesday | |
| Thursday | 4 | Foursday | |
| Friday | 5 | Fiveday | |
| Saturday | 6 | Six-a-day |
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 (day of week) | |
|---|---|---|---|---|
| 1600–1699 | Tuesday | — | 2 (Twoday) | |
| 1700–1799 | Sunday | — | 0 (Noneday) | |
| 1800–1899 | Friday | — | 5 (Fiveday) | |
| 1900–1999 | Wednesday | We-in-dis-day (most living people were born in that century) | 3 (Treblesday) | |
| 2000–2099 | Tuesday | Y-Tue-K or Twos-day (Y2K was at the head of this century) | 2 (Twosday) | |
| 2100–2199 | Sunday | Twenty-one-day is Sunday (2100 is the start of the next century) | 0 (Noneday) | |
| 2200–2299 | Friday | — | 5 (Fiveday) |
Mathematical formula
Tuesday = anchor.
Algorithmic
Let
if then anchor = Tuesday
if then anchor = Sunday
if then anchor = Friday
if then anchor = WednesdayFor the Julian calendar:
Sunday = anchor.Note:
c=l\lfloor{year\over100}r\rfloor
Next, find the year's anchor day. To accomplish that according to Conway:[11]
| \begin{matrix}\left({\left\lfloor{ | y |
| 12 |
For the twentieth-century year 1966, for example:
| \begin{matrix}\left({\left\lfloor{ | 66 |
| 12 |
As described in bullet 4, above, this is equivalent to:
\begin{matrix}\left({66+\left\lfloor{
| 66 | |
| 4 |
So doomsday in 1966 fell on Monday.
Similarly, doomsday in 2005 is on a Monday:
| \left({\left\lfloor{ | 5 |
| 12 |
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
\left(y+\left\lfloor
| y | |
| 4 |
\right\rfloor\right)\bmod7.
This gives a simpler formula if one is comfortable dividing large values of by both 4 and 7. For example, we can compute
\left(66+\left\lfloor
| 66 | |
| 4 |
\right\rfloor\right)\bmod7=(66+16)\bmod7=82\bmod7=5
which gives the same answer as in the example above.
Where 12 comes in is that the pattern of
l(y+l\lfloor\tfrac{y}{4}r\rfloorr)\bmod7
l(12+\tfrac{12}{4}r)\bmod7=15\bmod7=1
l\lfloor\tfrac{y}{12}r\rfloor
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
(100+24)\bmod7=2+3=5
So c centuries move the doomsday forward by
\left(5c+l\lfloor{c\over4}r\rfloor\right)\bmod7
(5(c\bmod4))\bmod7
-2-2-2-1=-7, -7\equiv0 \pmod{7}
A simpler method for finding the year's anchor day was discovered in 2010 by Chamberlain Fong and Michael K. Walters,[12] and described in their paper submitted to the 7th International Congress on Industrial and Applied Mathematics (2011). Called the "odd + 11" method, it is equivalent to computing
\left(y+\left\lfloor
| y | |
| 4 |
\right\rfloor\right)\bmod7
It is well suited to mental calculation, because it requires no division by 4 (or 12), and the procedure is easy to remember because of its repeated use of the "odd + 11" rule. Furthermore, addition by 11 is very easy to perform mentally in base-10 arithmetic.
Extending this to get the anchor day, the procedure is often described as accumulating a running total in six steps, as follows:
Applying this method to the year 2005, for example, the steps as outlined would be:
The explicit formula for the odd+11 method is:
7-\left[
| y+11(y\bmod2) | |
| 2 |
+11\left(
| y+11(y\bmod2) | |
| 2 |
\bmod2\right)\right]\bmod7
Although this expression looks daunting and complicated, it is actually simple because of a common subexpression that only needs to be calculated once.
Anytime adding 11 is needed, subtracting 17 yields equivalent results. While subtracting 17 may seem more difficult to mentally perform than adding 11, there are cases where subtracting 17 is easier, especially when the number is a two-digit number that ends in 7 (such as 17, 27, 37, ..., 77, 87, and 97).
Doomsday is related to the dominical letter of the year as follows.
| Doomsday | Dominical letter | ||
|---|---|---|---|
| Common year | Leap year | ||
| Sunday | C | DC | |
| Monday | B | CB | |
| Tuesday | A | BA | |
| Wednesday | G | AG | |
| Thursday | F | GF | |
| Friday | E | FE | |
| Saturday | D | ED | |
Look up the table below for the dominical letter (DL).
| Hundreds of Years | D L | Remaining Year Digits | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Julian (r ÷ 7) | Gregorian (r ÷ 4) | ||||||||
| r5 19 | 16 20 r0 | A | 00 06 17 23 | 28 34 45 51 | 56 62 73 79 | 84 90 | 0 | ||
| r4 18 | 15 19 r3 | G | 01 07 12 18 | 29 35 40 46 | 57 63 68 74 | 85 91 96 | 1 | ||
| r3 17 | align=center | N/A | F | 02 13 19 24 | 30 41 47 52 | 58 69 75 80 | 86 97 | 2 | |
| r2 16 | 18 22 r2 | E | 03 08 14 25 | 31 36 42 53 | 59 64 70 81 | 87 92 98 | 3 | ||
| r1 15 | align=center | N/A | D | 09 15 20 26 | 37 43 48 54 | 65 71 76 82 | 93 99 | 4 | |
| r0 14 | 17 21 r1 | C | 04 10 21 27 | 32 38 49 55 | 60 66 77 83 | 88 94 | 5 | ||
| r6 13 | align=center | N/A | B | 05 11 16 22 | 33 39 44 50 | 61 67 72 78 | 89 95 | 6 |
| Month | Dates | Week numbers * | |
|---|---|---|---|
| January (common years) | 3, 10, 17, 24, 31 | 1–5 | |
| January (leap years) | 4, 11, 18, 25 | 1–4 | |
| February (common years) | 7, 14, 21, 28 | 6–9 | |
| February (leap years) | 1, 8, 15, 22, 29 | 5–9 | |
| March | 7, 14, 21, 28 | 10–13 | |
| April | 4, 11, 18, 25 | 14–17 | |
| May | 2, 9, 16, 23, 30 | 18–22 | |
| June | 6, 13, 20, 27 | 23–26 | |
| July | 4, 11, 18, 25 | 27–30 | |
| August | 1, 8, 15, 22, 29 | 31–35 | |
| September | 5, 12, 19, 26 | 36–39 | |
| October | 3, 10, 17, 24, 31 | 40–44 | |
| November | 7, 14, 21, 28 | 45–48 | |
| December | 5, 12, 19, 26 | 49–52 |
* In leap years the th doomsday is in ISO week . In common years the day after the th doomsday is in week . Thus in a common year the week number on the doomsday itself is one less if it is a Sunday, i.e. in a common year starting on Friday (such as 2010, 2021, & 2027).
For computer use, the following formulas for the anchor day of a year are convenient.
For the Gregorian calendar:
anchorday=Tuesday+y+\left\lfloor
| y | |
| 4 |
\right\rfloor-\left\lfloor
| y | |
| 100 |
\right\rfloor+\left\lfloor
| y | |
| 400 |
\right\rfloor=Tuesday+5 x (y\bmod4)+4 x (y\bmod100)+6 x (y\bmod400)
Saturday(6)\bmod7=Tuesday(2)+2009+\left\lfloor
| 2009 | |
| 4 |
\right\rfloor-\left\lfloor
| 2009 | |
| 100 |
\right\rfloor+\left\lfloor
| 2009 | |
| 400 |
\right\rfloor
As another example, the doomsday 1946 is Thursday, since
Thursday(4)\bmod7=Tuesday(2)+1946+\left\lfloor
| 1946 | |
| 4 |
\right\rfloor-\left\lfloor
| 1946 | |
| 100 |
\right\rfloor+\left\lfloor
| 1946 | |
| 400 |
\right\rfloor
For the Julian calendar:
anchorday=Sunday+y+\left\lfloor
| y | |
| 4 |
\right\rfloor=Sunday+5 x (y\bmod4)+3 x (y\bmod7)
The formulas apply also for the proleptic Gregorian calendar and the proleptic Julian calendar. They use the floor function and astronomical year numbering for years BC.
For comparison, see the calculation of a Julian day number.
| Julian centuries | -1600J -900J -200J 500J 1200J 1900J 2600J 3300J | -1500J -800J -100J 600J 1300J 2000J 2700J 3400J | -1400J -700J 0J 700J 1400J 2100J 2800J 3500J | -1300J -600J 100J 800J 1500J 2200J 2900J 3600J | -1200J -500J 200J 900J 1600J 2300J 3000J 3700J | -1100J -400J 300J 1000J 1700J 2400J 3100J 3800J | -1000J -300J 400J 1100J 1800J 2500J 3200J 3900J | |||
|---|---|---|---|---|---|---|---|---|---|---|
| valign="top" scope="col" style="text-align:right" | -1600 -1200 -800 -400 0 400 800 1200 1600 2000 2400 2800 3200 3600 | valign="top" scope="col" style="text-align:right" | -1500 -1100 -700 -300 100 500 900 1300 1700 2100 2500 2900 3300 3700 | valign="top" scope="col" style="text-align:right" | -1400 -1000 -600 -200 200 600 1000 1400 1800 2200 2600 3000 3400 3800 | valign="top" scope="col" style="text-align:right" | -1300 -900 -500 -100 300 700 1100 1500 1900 2300 2700 3100 3500 3900 | |||
| 00 | 28 | 56 | 84 | Tue. | Mon. | Sun. | Sat. | Fri. | Thu. | Wed. |
| 01 | 29 | 57 | 85 | Wed. | Tue. | Mon. | Sun. | Sat. | Fri. | Thu. |
| 02 | 30 | 58 | 86 | Thu. | Wed. | Tue. | Mon. | Sun. | Sat. | Fri. |
| 03 | 31 | 59 | 87 | Fri. | Thu. | Wed. | Tue. | Mon. | Sun. | Sat. |
| 04 | 32 | 60 | 88 | Sun. | Sat. | Fri. | Thu. | Wed. | Tue. | Mon. |
| 05 | 33 | 61 | 89 | Mon. | Sun. | Sat. | Fri. | Thu. | Wed. | Tue. |
| 06 | 34 | 62 | 90 | Tue. | Mon. | Sun. | Sat. | Fri. | Thu. | Wed. |
| 07 | 35 | 63 | 91 | Wed. | Tue. | Mon. | Sun. | Sat. | Fri. | Thu. |
| 08 | 36 | 64 | 92 | Fri. | Thu. | Wed. | Tue. | Mon. | Sun. | Sat. |
| 09 | 37 | 65 | 93 | Sat. | Fri. | Thu. | Wed. | Tue. | Mon. | Sun. |
| 10 | 38 | 66 | 94 | Sun. | Sat. | Fri. | Thu. | Wed. | Tue. | Mon. |
| 11 | 39 | 67 | 95 | Mon. | Sun. | Sat. | Fri. | Thu. | Wed. | Tue. |
| 12 | 40 | 68 | 96 | Wed. | Tue. | Mon. | Sun. | Sat. | Fri. | Thu. |
| 13 | 41 | 69 | 97 | Thu. | Wed. | Tue. | Mon. | Sun. | Sat. | Fri. |
| 14 | 42 | 70 | 98 | Fri. | Thu. | Wed. | Tue. | Mon. | Sun. | Sat. |
| 15 | 43 | 71 | 99 | Sat. | Fri. | Thu. | Wed. | Tue. | Mon. | Sun. |
| 16 | 44 | 72 | Mon. | Sun. | Sat. | Fri. | Thu. | Wed. | Tue. | |
| 17 | 45 | 73 | Tue. | Mon. | Sun. | Sat. | Fri. | Thu. | Wed. | |
| 18 | 46 | 74 | Wed. | Tue. | Mon. | Sun. | Sat. | Fri. | Thu. | |
| 19 | 47 | 75 | Thu. | Wed. | Tue. | Mon. | Sun. | Sat. | Fri. | |
| 20 | 48 | 76 | Sat. | Fri. | Thu. | Wed. | Tue. | Mon. | Sun. | |
| 21 | 49 | 77 | Sun. | Sat. | Fri. | Thu. | Wed. | Tue. | Mon. | |
| 22 | 50 | 78 | Mon. | Sun. | Sat. | Fri. | Thu. | Wed. | Tue. | |
| 23 | 51 | 79 | Tue. | Mon. | Sun. | Sat. | Fri. | Thu. | Wed. | |
| 24 | 52 | 80 | Thu. | Wed. | Tue. | Mon. | Sun. | Sat. | Fri. | |
| 25 | 53 | 81 | Fri. | Thu. | Wed. | Tue. | Mon. | Sun. | Sat. | |
| 26 | 54 | 82 | Sat. | Fri. | Thu. | Wed. | Tue. | Mon. | Sun. | |
| 27 | 55 | 83 | Sun. | Sat. | Fri. | Thu. | Wed. | Tue. | Mon. | |
Since in the Gregorian calendar there are 146,097 days, or exactly 20,871 seven-day weeks, in 400 years, the anchor day repeats every four centuries. For example, the anchor day of 1700–1799 is the same as the anchor day of 2100–2199, i.e. Sunday.
The full 400-year cycle of doomsdays is given in the adjacent table. The centuries are for the Gregorian and proleptic Gregorian calendar, unless marked with a J for Julian. The Gregorian leap years are highlighted.
Negative years use astronomical year numbering. Year 25BC is −24, shown in the column of −100J (proleptic Julian) or −100 (proleptic Gregorian), at the row 76.
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Total | ||
|---|---|---|---|---|---|---|---|---|---|
| Non-leap years | 43 | 43 | 43 | 43 | 44 | 43 | 44 | 303 | |
| Leap years | 13 | 15 | 13 | 15 | 13 | 14 | 14 | 97 | |
| Total | 56 | 58 | 56 | 58 | 57 | 57 | 58 | 400 |
A leap year with Monday as doomsday means that Sunday is one of 97 days skipped in the 400-year sequence. Thus the total number of years with Sunday as doomsday is 71 minus the number of leap years with Monday as doomsday, etc. Since Monday as doomsday is skipped across 29 February 2000 and the pattern of leap days is symmetric about that leap day, the frequencies of doomsdays per weekday (adding common and leap years) are symmetric about Monday. The frequencies of doomsdays of leap years per weekday are symmetric about the doomsday of 2000, Tuesday.
The frequency of a particular date being on a particular weekday can easily be derived from the above (for a date from 1 January – 28 February, relate it to the doomsday of the previous year).
For example, 28 February is one day after doomsday of the previous year, so it is 58 times each on Tuesday, Thursday and Sunday, etc. 29 February is doomsday of a leap year, so it is 15 times each on Monday and Wednesday, etc.
Regarding the frequency of doomsdays in a Julian 28-year cycle, there are 1 leap year and 3 common years for every weekday, the latter 6, 17 and 23 years after the former (so with intervals of 6, 11, 6, and 5 years; not evenly distributed because after 12 years the day is skipped in the sequence of doomsdays). The same cycle applies for any given date from 1 March falling on a particular weekday.
For any given date up to 28 February falling on a particular weekday, the 3 common years are 5, 11, and 22 years after the leap year, so with intervals of 5, 6, 11, and 6 years. Thus the cycle is the same, but with the 5-year interval after instead of before the leap year.
Thus, for any date except 29 February, the intervals between common years falling on a particular weekday are 6, 11, 11. See e.g. at the bottom of the page Common year starting on Monday the years in the range 1906–2091.
For 29 February falling on a particular weekday, there is just one in every 28 years, and it is of course a leap year.
The Gregorian calendar is currently accurately lining up with astronomical events such as solstices. In 1582 this modification of the Julian calendar was first instituted. In order to correct for calendar drift, 10 days were skipped, so doomsday moved back 10 days (i.e. 3 days): Thursday 4 October (Julian, doomsday is Wednesday) was followed by Friday 15 October (Gregorian, doomsday is Sunday). The table includes Julian calendar years, but the algorithm is for the Gregorian and proleptic Gregorian calendar only.
Note that the Gregorian calendar was not adopted simultaneously in all countries, so for many centuries, different regions used different dates for the same day.
Suppose we want to know the day of the week of September 18, 1985. We begin with the century's anchor day, Wednesday. To this, add,, and above:
This yields . Counting 8 days from Wednesday, we reach Thursday, which is the doomsday in 1985. (Using numbers: In modulo 7 arithmetic, 8 is congruent to 1. Because the century's anchor day is Wednesday (index 3), and 3 + 1 = 4, doomsday in 1985 was Thursday (index 4).) We now compare September 18 to a nearby doomsday, September 5. We see that the 18th is 13 past a doomsday, i.e. one day less than two weeks. Hence, the 18th was a Wednesday (the day preceding Thursday). (Using numbers: In modulo 7 arithmetic, 13 is congruent to 6 or, more succinctly, −1. Thus, we take one away from the doomsday, Thursday, to find that September 18, 1985 was a Wednesday.)
Suppose that we want to find the day of week that the American Civil War broke out at Fort Sumter, which was April 12, 1861. The anchor day for the century was 94 days after Tuesday, or, in other words, Friday (calculated as ; or just look at the chart, above, which lists the century's anchor days). The digits 61 gave a displacement of six days so doomsday was Thursday. Therefore, April 4 was Thursday so April 12, eight days later, was a Friday.