Doomsday rule

Short description: Way of calculating the day of the week of a given date

John Conway, inventor of the Doomsday algorithm

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, drawing inspiration from Lewis Carroll's perpetual calendar algorithm. 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.

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:

Days with the Doomsday rule for years from 1800 through 2100

Years during which a Doomsday rule occurs in the day to the right	Day with its Doomsday rule	Year length and weekday of January the 1st	Year's dominical letter
1802, 1813, 1819, 1824, 1830, 1841, 1847, 1852, 1858, 1869, 1875, 1880, 1886, 1897, 1909, 1915, 1920, 1926, 1937, 1943, 1948, 1954, 1965, 1971, 1976, 1982, 1993, 1999, 2004, 2010, 2021, 2027, 2032, 2038, 2049, 2055, 2060, 2066, 2077, 2083, 2088, 2094 and 2100.	Sunday	365: Friday 366: Thursday	C DC
1803, 1808, 1814, 1825, 1831, 1836, 1842, 1853, 1859, 1864, 1870, 1881, 1887, 1892, 1898, 1904, 1910, 1921, 1927, 1932, 1938, 1949, 1955, 1960, 1966, 1977, 1983, 1988, 1994, 2005, 2011, 2016, 2022, 2033, 2039, 2044, 2050, 2061, 2067, 2072, 2078, 2089 and 2095.	Monday	365: Saturday 366: Friday	B CB
1809, 1815, 1820, 1826, 1837, 1843, 1848, 1854, 1865, 1871, 1876, 1882, 1893, 1899, 1905, 1911, 1916, 1922, 1933, 1939, 1944, 1950, 1961, 1967, 1972, 1978, 1989, 1995, 2000, 2006, 2017, 2023, 2028, 2034, 2045, 2051, 2056, 2062, 2073, 2079, 2084 and 2090.	Tuesday	365: Sunday 366: Saturday	A BA
1804, 1810, 1821, 1827, 1832, 1838, 1849, 1855, 1860, 1866, 1877, 1883, 1888, 1894, 1900, 1906, 1917, 1923, 1928, 1934, 1945, 1951, 1956, 1962, 1973, 1979, 1984, 1990, 2001, 2007, 2012, 2018, 2029, 2035, 2040, 2046, 2057, 2063, 2068, 2074, 2085, 2091 and 2096.	Wednesday	365: Monday 366: Sunday	G AG
1805, 1811, 1816, 1822, 1833, 1839, 1844, 1850, 1861, 1867, 1872, 1878, 1889, 1895, 1901, 1907, 1912, 1918, 1929, 1935, 1940, 1946, 1957, 1963, 1968, 1974, 1985, 1991, 1996, 2002, 2013, 2019, 2024, 2030, 2041, 2047, 2052, 2058, 2069, 2075, 2080, 2086 and 2097.	Thursday	365: Tuesday 366: Monday	F GF
1800, 1806, 1817, 1823, 1828, 1834, 1845, 1851, 1856, 1862, 1873, 1879, 1884, 1890, 1902, 1913, 1919, 1924, 1930, 1941, 1947, 1952, 1958, 1969, 1975, 1980, 1986, 1997, 2003, 2008, 2014, 2025, 2031, 2036, 2042, 2053, 2059, 2064, 2070, 2081, 2087, 2092 and 2098.	Friday	365: Wednesday 366: Tuesday	E FE
1801, 1807, 1812, 1818, 1829, 1835, 1840, 1846, 1857, 1863, 1868, 1874, 1885, 1891, 1896, 1903, 1908, 1914, 1925, 1931, 1936, 1942, 1953, 1959, 1964, 1970, 1981, 1987, 1992, 1998, 2009, 2015, 2020, 2026, 2037, 2043, 2048, 2054, 2065, 2071, 2076, 2082, 2093 and 2099.	Saturday	365: Thursday 366: Wednesday	D ED

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 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, 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).

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	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 4, 11, 18, 25, 32
February	February 28 (common years), February 29 (leap years)	2/28 OR 2/29	last day of February	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.

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.

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: c = ⌊year/100⌋.

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

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 the sum of the last two digits of the year taken collectively plus the floor of those collective digits 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.

[math]\displaystyle{ \begin{matrix}\left({\left\lfloor{\frac{y}{12}}\right\rfloor+y \bmod 12+\left\lfloor{\frac{y \bmod 12}{4}}\right\rfloor}\right) \bmod 7+\rm{anchor}=\rm{Doomsday}\end{matrix} }[/math]

For the twentieth-century year 1966, for example:

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

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

[math]\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} }[/math]

So doomsday in 1966 fell on Monday.

Similarly, doomsday in 2005 is on a Monday:

[math]\displaystyle{ \left({\left\lfloor{\frac{5}{12}}\right\rfloor+5 \bmod 12+\left\lfloor{\frac{5 \bmod 12}{4}}\right\rfloor}\right) \bmod 7+\rm{Tuesday}=\rm{Monday} }[/math]

Why it works

Doomsday rule

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

[math]\displaystyle{ \left(y + \left\lfloor \frac{y}{4} \right\rfloor\right) \bmod 7. }[/math]

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

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

which gives the same answer as in the example above.

Where 12 comes in is that the pattern of (y + ⌊y/4⌋) mod 7 almost repeats every 12 years. After 12 years, we get (12 + 12/4) mod 7 = 15 mod 7 = 1. If we replace y by y mod 12, we are throwing this extra day away; but adding back in ⌊y/12⌋ compensates for this error, giving the final formula.

The "odd + 11" method

A simple flowchart showing the Odd+11 method to calculate the anchor day

A simpler method for finding the year's anchor day was discovered in 2010 by Chamberlain Fong and Michael K. Walters, 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

[math]\displaystyle{ \left(y + \left\lfloor \frac{y}{4} \right\rfloor\right) \bmod 7 }[/math].

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:

[math]\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 }[/math].

Although this expression looks daunting and complicated, it is actually simple 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 2023, the dominical letter is A - 0 = A.

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.

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:

[math]\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 100) + 6\times (y\bmod 400) }[/math]

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

[math]\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 }[/math]

As another example, the doomsday 1946 is Thursday, since

[math]\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 }[/math]

For the Julian calendar:

[math]\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) }[/math]

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
colspan=4 Template:Diagonal split header 2	-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). 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 99 days after Thursday, or, in other words, Friday (calculated as (18 + 1) × 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.

See also

• Ordinal date
• Computus – Gauss algorithm for Easter date calculation
• Zeller's congruence – An algorithm (1882) to calculate the day of the week for any Julian or Gregorian calendar date.
• Mental calculation

External links

• Encyclopedia of Weekday Calculation by Hans-Christian Solka, 2010
• Doomsday calculator that also "shows all work"
• World records for mentally calculating the day of the week in the Gregorian Calendar
• National records for finding Calendar Dates
• World Ranking of Memoriad Mental Calendar Dates (all competitions combined)
• What is the day of the week, given any date?
• Doomsday Algorithm
• Finding the Day of the Week