# Doomsday rule

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]

## Anchor days for some contemporary years

Doomsday's anchor day for the current year in the Gregorian calendar (2024) is Thursday. For some other contemporary years:

Anchor days for years from 1796 through 2105
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.

## Memorable dates that always land on 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).[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][7][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[9] (or: last day of January, pretending leap years have a January 32nd[7]) 3, 10, 17, 24, 31 OR 4, 11, 18, 25, 32[7]
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[7] (or: last day of February) 0, 7, 14, 21, 28 OR 1, 8, 15, 22, 29
March "March 0," March 14 3/0 AND 3/14 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-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

January and February can be treated as the last two months of the previous year.

### Example

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).

## 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" 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

There are some languages, such as Slavic languages, Chinese, Greek, Portuguese, Galician and Hebrew, that base some of the names of the week days in their positional order. The Slavic and Chinese agree with the table above; the other languages mentioned count from Sunday as day one.

## Finding a year's anchor 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)

For the Gregorian calendar:

Mathematical formula
5 × (c mod 4) mod 7 + Tuesday = anchor.
Algorithmic
Let r = c mod 4
if r = 0 then anchor = Tuesday
if r = 1 then anchor = Sunday
if r = 2 then anchor = Friday
if r = 3 then anchor = Wednesday

For the Julian calendar:

6c mod 7 + Sunday = anchor.

Note: ${\displaystyle c={\biggl \lfloor }{{\text{year}} \over 100}{\biggr \rfloor }}$.

Next, find the year's anchor day. To accomplish that according to Conway:[11]

1. Divide the year's last two digits (call this y) by 12 and let a be the floor of the quotient.
2. Let b be the remainder of the same quotient.
3. Divide that remainder by 4 and let c be the floor of the quotient.
4. Let d be the sum of the three numbers (d = a + b + c). (It is again possible here to divide by seven and take the remainder. This number is equivalent, as it must be, to y plus the floor of y divided by four.)
5. Count forward the specified number of days (d or the remainder of d/7) from the anchor day to get the year's one.
${\displaystyle {\begin{matrix}\left({\left\lfloor {\frac {y}{12}}\right\rfloor +y{\bmod {1}}2+\left\lfloor {\frac {y{\bmod {1}}2}{4}}\right\rfloor }\right){\bmod {7}}+{\rm {{anchor}={\rm {Doomsday}}}}\end{matrix}}}$

For the twentieth-century year 1966, for example:

${\displaystyle {\begin{matrix}\left({\left\lfloor {\frac {66}{12}}\right\rfloor +66{\bmod {1}}2+\left\lfloor {\frac {66{\bmod {1}}2}{4}}\right\rfloor }\right){\bmod {7}}+{\rm {Wednesday}}&=&\left(5+6+1\right){\bmod {7}}+{\rm {Wednesday}}\\\ &=&{\rm {Monday}}\end{matrix}}}$

As described in bullet 4, above, this is equivalent to:

${\displaystyle {\begin{matrix}\left({66+\left\lfloor {\frac {66}{4}}\right\rfloor }\right){\bmod {7}}+{\rm {Wednesday}}&=&\left(66+16\right){\bmod {7}}+{\rm {Wednesday}}\\\ &=&{\rm {Monday}}\end{matrix}}}$

So doomsday in 1966 fell on Monday.

Similarly, doomsday in 2005 is on a Monday:

${\displaystyle \left({\left\lfloor {\frac {5}{12}}\right\rfloor +5{\bmod {1}}2+\left\lfloor {\frac {5{\bmod {1}}2}{4}}\right\rfloor }\right){\bmod {7}}+{\rm {{Tuesday}={\rm {Monday}}}}}$

### 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 365y + y/4 (rounded down). But 365 equals 52 × 7 + 1, so after taking the remainder we get just

${\displaystyle \left(y+\left\lfloor {\frac {y}{4}}\right\rfloor \right){\bmod {7}}.}$

This gives a simpler formula if one is comfortable dividing large values of y by both 4 and 7. For example, we can compute

${\displaystyle \left(66+\left\lfloor {\frac {66}{4}}\right\rfloor \right){\bmod {7}}=(66+16){\bmod {7}}=82{\bmod {7}}=5}$

which gives the same answer as in the example above.

Where 12 comes in is that the pattern of ${\displaystyle {\bigl (}y+{\bigl \lfloor }{\tfrac {y}{4}}{\bigr \rfloor }{\bigr )}{\bmod {7}}}$ almost repeats every 12 years. After 12 years, we get ${\displaystyle {\bigl (}12+{\tfrac {12}{4}}{\bigr )}{\bmod {7}}=15{\bmod {7}}=1}$. If we replace y by y mod 12, we are throwing this extra day away; but adding back in ${\displaystyle {\bigl \lfloor }{\tfrac {y}{12}}{\bigr \rfloor }}$ 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

${\displaystyle (100+24){\bmod {7}}=2+3=5}$

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

${\displaystyle \left(5c+{\biggl \lfloor }{c \over 4}{\biggr \rfloor }\right){\bmod {7}}}$,

but this is equivalent to

${\displaystyle (5(c{\bmod {4}})){\bmod {7}}}$.

Four centuries move the doomsday forward by

${\displaystyle -2-2-2-1=-7,\qquad -7\equiv 0\quad {\pmod {7}}}$;

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,[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[12] to computing

${\displaystyle \left(y+\left\lfloor {\frac {y}{4}}\right\rfloor \right){\bmod {7}}}$.

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 T in six steps, as follows:

1. Let T be the year's last two digits.
2. If T is odd, add 11.
3. Now let T = T/2.
4. If T is odd, add 11.
5. Now let T = 7 − (T mod 7).
6. Count forward T days from the century's anchor day to get the year's anchor day.

Applying this method to the year 2005, for example, the steps as outlined would be:

1. T = 5
2. T = 5 + 11 = 16 (adding 11 because T is odd)
3. T = 16/2 = 8
4. T = 8 (do nothing since T is even)
5. T = 7 − (8 mod 7) = 7 − 1 = 6
6. Doomsday for 2005 = 6 + Tuesday = Monday

The explicit formula for the odd+11 method is:

${\displaystyle 7-\left[{\frac {y+11(y\,{\bmod {2}})}{2}}+11\left({\frac {y+11(y\,{\bmod {2}})}{2}}{\bmod {2}}\right)\right]{\bmod {7}}}$.

Although this expression looks daunting and complicated, it is actually simple[12] because of a common subexpression y + 11(y mod 2)/2 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).

## Correspondence with dominical letter

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 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 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 N/A B 05 11 16 22 33 39 44 50 61 67 72 78 89 95 6

For the year 2024, the dominical letter is BA + 2 = GF.

## Overview of all Doomsdays

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 nth doomsday is in ISO week n. In common years the day after the nth doomsday is in week n. 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).

## Computer formula for the anchor day of a year

For computer use, the following formulas for the anchor day of a year are convenient.

For the Gregorian calendar:

${\displaystyle {\mbox{anchor day}}={\mbox{Tuesday}}+y+\left\lfloor {\frac {y}{4}}\right\rfloor -\left\lfloor {\frac {y}{100}}\right\rfloor +\left\lfloor {\frac {y}{400}}\right\rfloor ={\mbox{Tuesday}}+5\times (y{\bmod {4}})+4\times (y{\bmod {1}}00)+6\times (y{\bmod {4}}00)}$

For example, the doomsday 2009 is Saturday under the Gregorian calendar (the currently accepted calendar), since

${\displaystyle {\mbox{Saturday (6)}}{\bmod {7}}={\mbox{Tuesday (2)}}+2009+\left\lfloor {\frac {2009}{4}}\right\rfloor -\left\lfloor {\frac {2009}{100}}\right\rfloor +\left\lfloor {\frac {2009}{400}}\right\rfloor }$

As another example, the doomsday 1946 is Thursday, since

${\displaystyle {\mbox{Thursday (4)}}{\bmod {7}}={\mbox{Tuesday (2)}}+1946+\left\lfloor {\frac {1946}{4}}\right\rfloor -\left\lfloor {\frac {1946}{100}}\right\rfloor +\left\lfloor {\frac {1946}{400}}\right\rfloor }$

For the Julian calendar:

${\displaystyle {\mbox{anchor day}}={\mbox{Sunday}}+y+\left\lfloor {\frac {y}{4}}\right\rfloor ={\mbox{Sunday}}+5\times (y{\bmod {4}})+3\times (y{\bmod {7}})}$

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.

## 400-year cycle of anchor days

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
Gregorian
centuries

Years
-1600
-1200
-800
-400
0
400
800
1200
1600
2000
2400
2800
3200
3600
-1500
-1100
-700
-300
100
500
900
1300
1700
2100
2500
2900
3300
3700
-1400
-1000
-600
-200
200
600
1000
1400
1800
2200
2600
3000
3400
3800
-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.

Frequency of Gregorian Doomsday in the 400-year cycle per weekday and year type
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.

### 28-year cycle

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).[citation needed] 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.

### Julian calendar

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.

## Full examples

### Example 1 (1985)

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 a, b, and c above:

• a is the floor of 85/12, which is 7.
• b is 85 mod 12, which is 1.
• c is the floor of b/4, which is 0.

This yields a + b + c = 8. 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.)

### Example 2 (other centuries)

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 18 × 5 + ⌊18/4; 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.

## References

1. ^ John Horton Conway, "Tomorrow is the Day After Doomsday" (PDF). Eureka. October 1973. p. 28-32.
2. ^ 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, ISBN 0-12-091102-7.
3. ^ Lewis Carroll, "To Find the Day of the Week for Any Given Date", Nature, March 31, 1887. doi:10.1038/035517a0
4. ^ Martin Gardner, The Universe in a Handkerchief: Lewis Carroll's Mathematical Recreations, Games, Puzzles, and Word Plays, pages 24–26, Springer-Verlag, 1996.
5. ^ "What Day is Doomsday". Mathematics Awareness Month. April 2014.
6. ^ Alpert, Mark (1999-04-01). "Not Just Fun and Games". Scientific American. Retrieved 2024-04-18.
7. Torrence, Bruce; Torrence, Eve. "John H. Conway - Doomsday, part 1". YouTube. Mathematical Association of America. Archived from the original on 2021-12-21. Retrieved 14 April 2020.
8. ^ "The Doomsday Algorithm - Numberphile". YouTube. Retrieved 9 July 2023.
9. ^ a b Limeback, Rudy (3 January 2017). "Doomsday Algorithm". Retrieved 27 May 2017.
10. ^ "On what day of the week is Christmas? Use the Doomsday Rule". The Irish Times. Retrieved 2022-07-20.
11. ^ John Horton Conway, "Tomorrow is the Day After Doomsday" (PDF). Eureka. October 1973. p. 29-30. Each ordinary year has its Doomsday 1 day later than the previous year, and each leap year 2 days later. It follows that within any given century a dozen years advances Doomsday by 12 + 3 = 15 days = 1 day. ('A dozen years is but a day.') So we add to the Doomsday for the century year the number of dozens of years thereafter, the remainder, and the number of fours in the remainder.
12. ^ a b c Chamberlain Fong, Michael K. Walters: "Methods for Accelerating Conway's Doomsday Algorithm (part 2)", 7th International Congress on Industrial and Applied Mathematics (2011).