Determination of the day of the week

Corresponding days

Every seventh day in a month has the same name as the previous:

Corresponding months

"Corresponding months" are those months within the calendar year that start on the same day of the week. For example, September and December correspond, because 1 September falls on the same day as 1 December (as there are precisely thirteen 7-day weeks between the two dates). Months can only correspond if the number of days between their first days is divisible by 7, or in other words, if their first days are a whole number of weeks apart. For example, February of a common year corresponds to March because February has 28 days, a number divisible by 7, 28 days being exactly four weeks. In a leap year, January and February correspond to different months than in a common year, since adding 29 February means each subsequent month starts a day later.

January corresponds to October in common years and April and July in leap years. February corresponds to March and November in common years and August in leap years. March always corresponds to November, April always corresponds to July, and September always corresponds to December. August does not correspond to any other month in a common year. October doesn't correspond to any other month in a leap year. May and June never correspond to any other month.

In the months table below, corresponding months have the same number, a fact which follows directly from the definition.

Corresponding years

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There are seven possible days that a year can start on, and leap years will alter the day of the week after 29 February. This means that there are 14 configurations that a year can have. All the configurations can be referenced by a dominical letter, but as 29 February has no letter allocated to it, a leap year has two dominical letters, one for January and February and the other (one step back in the alphabetical sequence) for March to December.

2021 is a common year starting on Friday, which means that it corresponds to the 2010 calendar year. The first two months of 2021 correspond to the first two months of 2016. 2022 is a common year starting on Saturday, which means that it corresponds to the 2011 calendar year. The last ten months of 2022 correspond to the last ten months of 2016. 2023 is a common year starting on Sunday, which means that it corresponds to the 2017 calendar year. 2024 is a leap year starting on Monday, which means that it corresponds to the 1996 calendar year. The first two months of 2024 correspond to the first two months of 2018. The last ten months of 2024 correspond to the last ten months of 2019.

Each leap year repeats once every 28 years, and every common year repeats once every 6 years and twice every 11 years. For instance, the last occurrence of a leap year starting on Wednesday was 2020 and the next occurrence will be 2048. Likewise, the next common years starting on Friday will be 2027, 2038, and then 2049. Both of these statements are true unless a leap year is skipped, but that will not happen until 2100.

For details see the table below.

Notes:

• Black means all the months of Common Year
• Red means the first 2 months of Leap Year
• Blue means the last 10 months of Leap Year

Corresponding centuries

"Year 000" is, in normal chronology, the year 1 BC (which precedes AD 1). In astronomical year numbering the year 0 comes between 1 BC and AD 1. In the proleptic Julian calendar, (that is, the Julian calendar as it would have been if it had been operated correctly from the start), 1 BC starts on Thursday. In the proleptic Gregorian calendar, (so called because it wasn't devised until 1582), 1 BC starts on Saturday.

Complete table: Julian and Gregorian calendars

For Julian dates before 1300 and after 1999 the year in the table which differs by an exact multiple of 700 years should be used. For Gregorian dates after 2299, the year in the table which differs by an exact multiple of 400 years should be used. The values "r0" through "r6" indicate the remainder when the Hundreds value is divided by 7 and 4 respectively, indicating how the series extend in either direction. Both Julian and Gregorian values are shown 1500-1999 for convenience. Bold figures (e.g., 04) denote leap year. If a year ends in 00 and its hundreds are in bold it is a leap year. Thus 19 indicates that 1900 is not a Gregorian leap year, (but 19 in the Julian column indicates that it is a Julian leap year, as are all Julian x00 years). 20 indicates that 2000 is a leap year. Use Jan and Feb only in leap years.

For determination of the day of the week (1 January 2000, Saturday)

• the day of the month: 1 ~ 31 (1)
• the month: (6)
• the year: (0)
• the century mod 4 for the Gregorian calendar and mod 7 for the Julian calendar (0).
• adding 1+6+0+0=7. Dividing by 7 leaves a remainder of 0, so the day of the week is Saturday.

The formula is w = (d + m + y + c) mod 7.

Revised Julian calendar

Note that the date (and hence the day of the week) in the Revised Julian and Gregorian calendars is the same from 14 October 1923 to 28 February AD 2800 inclusive and that for large years it may be possible to subtract 6300 or a multiple thereof before starting so as to reach a year which is within or closer to the table.

To look up the weekday of any date for any year using the table, subtract 100 from the year, divide the difference by 100, multiply the resulting quotient (omitting fractions) by seven and divide the product by nine. Note the quotient (omitting fractions). Enter the table with the Julian year, and just before the final division add 50 and subtract the quotient noted above.

Example: What is the day of the week of 27 January 8315?

8315−6300=2015, 2015−100=1915, 1915/100=19 remainder 15, 19×7=133, 133/9=14 remainder 7. 2015 is 700 years ahead of 1315, so 1315 is used. From table: for hundreds (13): 6. For remaining digits (15): 4. For month (January): 0. For date (27): 27. 6 + 4 + 0 + 27 + 50 − 14 = 73. 73/7=10 remainder 3. Day of week = Tuesday.

Dominical Letter

To find the Dominical Letter, calculate the day of the week for either 1 January or 1 October. If it is Sunday, the Dominical Letter is A, if Saturday B, and similarly backwards through the week and forwards through the alphabet to Monday, which is G.

Leap years have two Sunday Letters, so for January and February calculate the day of the week for 1 January and for March to December calculate the day of the week for 1 October.

Leap years are all years which divide exactly by four with the following exceptions:

In the Gregorian calendar – all years which divide exactly by 100 (other than those which divide exactly by 400).

In the Revised Julian calendar – all years which divide exactly by 100 (other than those which give remainder 200 or 600 when divided by 900).

The "doomsday"

This is an artefact of recreational mathematics. See doomsday rule for an explanation.

Check the result

Use this table for finding the day of the week without any calculations.

Examples:

• For common method

26 December 1893 (GD)

December is in row F and 26 is in column E, so the letter for the date is C located in row F and column E. 93 (year mod 100) is in row D (year row) and the letter C in the year row is located in column G. 18 ([year/100] in the Gregorian century column) is in row C (century row) and the letter in the century row and column G is B, so the day of the week is Tuesday.

13 October 1307 (JD)

October 13 is a F day. The letter F in the year row (07) is located in column G. The letter in the century row (13) and column G is E, so the day of the week is Friday.

1 January 2000 (GD)

January 1 corresponds to G, G in the year row (00) corresponds to F in the century row (20), and F corresponds to Saturday.

A pithy formula for the method: "Date letter (G), letter (G) is in year row (00) for the letter (F) in century row (20), and for the day, the letter (F) become weekday (Saturday)".

The Sunday Letter method

Each day of the year (other than 29 February) has a letter allocated to it in the recurring sequence ABCDEFG. The series begins with A on 1 January and continues to A again on 31 December. The Sunday letter is the one which stands against all the Sundays in the year. Since 29 February has no letter, this means that the Sunday Letter for March to December is one step back in the sequence compared to that for January and February. The letter for any date will be found where the row containing the month (in black) at the left of the "Latin square" meets the column containing the date above the "Latin square". The Sunday letter will be found where the column containing the century (below the "Latin square") meets the row containing the year's last two digits to the right of the "Latin square". For a leap year, the Sunday letter thus found is the one which applies to March to December.

So, for example, to find the weekday of 16 June 2020:

Column "20" meets row "20" at "D". Row "June" meets column "16" at "F". As F is two letters on from D, so the weekday is two days on from Sunday, i.e. Tuesday.

Rata Die

The Rata Die method works by adding up the number of days d that has passed since a date of known day of the week D. The day of-the-week is then given by (D + d) mod 7, conforming to whatever convention was used to encode D.

For example, the date of 13 August 2009 is 733632 days from 1 January AD 1. Taking the number mod 7 yields 4, hence a Thursday.

Gauss's algorithm

Carl Friedrich Gauss described a method for calculating the day of the week for 1 January in any given year in a handwritten note in a collection of astronomical tables. ^[5] He never published it. It was finally included in his collected works in 1927. ^[6] Compared to Rata Die, the result helps simplify the counting of years.

Gauss's method was applicable to the Gregorian calendar. He numbered the weekdays from 0 to 6 starting with Sunday. He defined the following operation.

Inputs

Year number A, month number M, date number D.

Output

Day of year.

Procedure

1. First determine the day-of-week d₁ of 1 January.
   • For a Gregorian calendar, the weekday is ^[5]
        (1 + 5((A−1) % 4) + 4((A−1) % 100) + 6((A−1) % 400)) % 7. Alternatively, set C = A \ 100, Y = A % 100, and the value is
        (1 + 5((Y−1)%4) + 3(Y−1) + 5(C%4)) % 7.
   • For a Julian calendar, the weekday is
        (6 + 5((A−1)%4) + 3(A−1)) % 7 or
        (6 + 5((Y−1) % 4) + 3(Y−1) + 6C) % 7.
2. Now determine the month-related offset m by using the lookup table with M.
3. Return d = (d₁ + m + D) % 7.

The above procedure can be condensed into a single expression for the Gregorian case:
   (D + m + 5((A−1)%4) + 4((A−1)%100) + 6((A−1)%400))%7

Disparate variation

The following formula is an example of a version without a lookup table. The year is assumed to begin in March, meaning dates in January and February should be treated as being part of the preceding year. The formula for the Gregorian calendar is ^[8]

${\displaystyle w=\left(d+\lfloor 2.6m-0.2\rfloor +y+\left\lfloor {\frac {y}{4}}\right\rfloor +\left\lfloor {\frac {c}{4}}\right\rfloor -2c\right){\bmod {7}},}$

where

• d is the day of the month (1 to 31)
• m is the shifted month (March = 1,...,February = 12)
• Y is the year unless m is 11 = January or 12 = February which are considered part of the preceding year, giving Y = year − 1
• c is the century given by ${\displaystyle c=\lfloor Y/100\rfloor }$
• y is the year relative to the century, given by ${\displaystyle y=Y-100c}$, or simply the last 2 digits of Y
• w is the day of the week (0 = Sunday,...,6 = Saturday)

Table of month offsets (${\displaystyle \lfloor 2.6m-0.2\rfloor }$)

Months	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec	Jan	Feb
Offset	2	5	7	10	12	15	18	20	23	25	28	31

Zeller's algorithm

In Zeller's algorithm, the months are numbered from 3 for March to 14 for February. The year is assumed to begin in March; this means, for example, that January 1995 is to be treated as month 13 of 1994. ^[9] The formula for the Gregorian calendar is
$${\displaystyle w\equiv \left(\left\lfloor {\frac {13(m+1)}{5}}\right\rfloor +\left\lfloor {\frac {y}{4}}\right\rfloor +\left\lfloor {\frac {c}{4}}\right\rfloor +d+y-2c\right){\bmod {7}}}$$ where

• ${\displaystyle d}$ is the day of the month (1 to 31)
• ${\displaystyle m}$ is the shifted month (March=3,...January = 13, February=14)
• ${\displaystyle Y}$ is the year unless ${\displaystyle m}$ is 13 = January or 14 = February which are considered part of the preceding year giving ${\displaystyle Y={\textit {year}}-1}$
• ${\displaystyle c}$ is the century given by ${\displaystyle c=\lfloor Y/100\rfloor }$
• ${\displaystyle y}$ is the year relative to the century, given by ${\displaystyle y=Y-100c}$, or simply the last 2 digits of ${\displaystyle Y}$
• ${\displaystyle w}$ is the day of week (1 = Sunday,..0 = Saturday)

Table of month offsets (${\displaystyle \lfloor 2.6(m+1)\rfloor }$)

Months	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec	Jan	Feb
Offset	10	13	15	18	20	23	26	28	31	33	36	39

The only difference is one between Zeller's algorithm (Z) and the Disparate Gaussian algorithm (G), that is Z − G = 1 = Sunday.

${\displaystyle (d+\lfloor (m+1)2.6\rfloor +y+\lfloor y/4\rfloor +\lfloor c/4\rfloor -2c){\bmod {7}}-(d+\lfloor 2.6m-0.2\rfloor +y+\lfloor y/4\rfloor +\lfloor c/4\rfloor -2c){\bmod {7}}}$

${\displaystyle =(\lfloor (m+2+1)2.6-(2.6m-0.2)\rfloor ){\bmod {7}}}$ (March = 3 in Z but March = 1 in G)

${\displaystyle =(\lfloor 2.6m+7.8-2.6m+0.2\rfloor ){\bmod {7}}}$

${\displaystyle =8{\bmod {7}}=1}$

Wang's algorithm

Wang's algorithm ^[10] for human calculation of the Gregorian calendar is (the formula should be subtracted by 1 if m is 1 or 2 if the year is a leap year)
$${\displaystyle w=\left(d-d_{0}(m)+y_{0}-y_{1}+\left\lfloor y_{0}/4-y_{1}/2\right\rfloor -2\left(c{\bmod {4}}\right)\right){\bmod {7}},}$$ where

• ⁠${\displaystyle y_{0}}$⁠ is the last digit of the year (units)
• ⁠${\displaystyle y_{1}}$⁠ is the second last digit of the year (tens)
• ⁠${\displaystyle c}$⁠ is the century, given by ${\displaystyle c=\lfloor {\textit {year}}/100\rfloor }$
• ⁠${\displaystyle d}$⁠ is the day of the month (1 to 31)
• ⁠${\displaystyle m}$⁠ is the month (January=1,...,December=12)
• ⁠${\displaystyle w}$⁠ is the day of the week (0=Sunday,...,6=Saturday)
• ⁠${\displaystyle d_{0}(m)}$⁠ is the null-days function (month offset) with values listed in the following table

m	⁠${\displaystyle d_{0}(m)}$⁠
1	1	A day
3	5	m + 2
5	7
7	9
9	3	m + 1
11	12
2	12	m + 3
4	2	m − 2
6	4
8	6
10	8
12	10

An algorithm for the Julian calendar can be derived from the algorithm above $${\displaystyle w=\left(d-d_{0}(m)+y_{0}-y_{1}+\left\lfloor y_{0}/4-y_{1}/2\right\rfloor -c\right){\bmod {7}},}$$ where ⁠${\displaystyle d_{0}(m)}$⁠ is a doomsday.

m	⁠${\displaystyle d_{0}(m)}$⁠
1	3	C day
3	7	m + 4
5	9
7	11
9	5	m − 4
11	7
2	0	m − 2
4	4	m
6	6
8	8
10	10
12	12

Schwerdtfeger's method

In a partly tabular method by Schwerdtfeger, the year is split into the century and the two digit year within the century. The approach depends on the month. For m≥ 3,

${\displaystyle c=\left\lfloor {\frac {y}{100}}\right\rfloor \quad {\text{and}}\quad g=y-100c,}$

so g is between 0 and 99. For m = 1,2,

${\displaystyle c=\left\lfloor {\frac {y-1}{100}}\right\rfloor \quad {\text{and}}\quad g=y-1-100c.}$

The formula for the day of the week is ^[6]

${\displaystyle w=\left(d+e+f+g+\left\lfloor {\frac {g}{4}}\right\rfloor \right){\bmod {7}},}$

where the positive modulus is chosen. ^[6]

The value of e is obtained from the following table:

The value of f is obtained from the following table, which depends on the calendar. For the Gregorian calendar, ^[6]

For the Julian calendar, ^[6]

Lewis Carroll's method

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Charles Lutwidge Dodgson (Lewis Carroll) devised a method resembling a puzzle, yet partly tabular in using the same index numbers for the months as in the "Complete table: Julian and Gregorian calendars" above. He lists the same three adjustments for the first three months of non-leap years, one 7 higher for the last, and gives cryptic instructions for finding the rest; his adjustments for centuries are to be determined using formulas similar to those for the centuries table. Although explicit in asserting that his method also works for Old Style dates, his example reproduced below to determine that "1676, February 23" is a Wednesday only works on a Julian calendar which starts the year on January 1, instead of March 25 as on the "Old Style" Julian calendar.

Algorithm: ^[11]

Take the given date in 4 portions, viz. the number of centuries, the number of years over, the month, the day of the month.

Compute the following 4 items, adding each, when found, to the total of the previous items. When an item or total exceeds 7, divide by 7, and keep the remainder only.

Century-item: For 'Old Style' (which ended 2 September 1752) subtract from 18. For 'New Style' (which began 14 September 1752) divide by 4, take overplus [surplus] from 3, multiply remainder by 2.

Year-item: Add together the number of dozens, the overplus, and the number of 4s in the overplus.

Month-item: If it begins or ends with a vowel, subtract the number, denoting its place in the year, from 10. This, plus its number of days, gives the item for the following month. The item for January is "0"; for February or March, "3"; for December, "12".

Day-item: The total, thus reached, must be corrected, by deducting "1" (first adding 7, if the total be "0"), if the date be January or February in a leap year, remembering that every year, divisible by 4, is a Leap Year, excepting only the century-years, in 'New Style', when the number of centuries is not so divisible (e.g. 1800).

The final result gives the day of the week, "0" meaning Sunday, "1" Monday, and so on.

Examples: ^[11]

1783, September 18

17, divided by 4, leaves "1" over; 1 from 3 gives "2"; twice 2 is "4". 83 is 6 dozen and 11, giving 17; plus 2 gives 19, i.e. (dividing by 7) "5". Total 9, i.e. "2" The item for August is "8 from 10", i.e. "2"; so, for September, it is "2 plus 31", i.e. "5" Total 7, i.e. "0", which goes out. 18 gives "4". Answer, "Thursday".

1676, February 23

16 from 18 gives "2" 76 is 6 dozen and 4, giving 10; plus 1 gives 11, i.e. "4". Total "6" The item for February is "3". Total 9, i.e. "2" 23 gives "2". Total "4" Correction for Leap Year gives "3". Answer, "Wednesday".

Dates before 1752 would in England be given Old Style with 25 March as the first day of the new year. Carroll's method however assumes 1 January as the first day of the year, thus he fails to arrive at the correct answer, namely "Friday".

Had he noticed that 1676, February 23 (with 25 March as New Year's Day) is actually 1677, February 23 (with 1 January as New Year's Day), he would have accounted for differing year numbers—just like George Washington's birthday differs—between the two calendars. Then his method yields:

1677 (corrected), February 23

16 from 18 gives "2" 77 is 6 dozen and 5, giving 11; plus 1 gives 12, i.e. "5". Total "7" The item for February is "3". Total 10, i.e. "3" 23 gives "2". Total "5". Answer, "Friday".

It is noteworthy that those who have republished Carroll's method have failed to point out his error, most notably Martin Gardner. ^[12]

In 1752, the British Empire abandoned its use of the Old Style Julian calendar upon adopting the Gregorian calendar, which has become today's standard in most countries of the world. For more background, see Old Style and New Style dates.

Methods in computer code

(d+=m<3?y--:y-2,23*m/9+d+4+y/4-y/100+y/400)%7

dayofweek(y,m,d)/* 1 <= m <= 12, y > 1752 (in the U.K.) */{staticintt[]={0,3,2,5,0,3,5,1,4,6,2,4};if(m<3){y-=1;}return(y+y/4-y/100+y/400+t[m-1]+d)%7;}

dow(m,d,y){y-=m<3;return(y+y/4-y/100+y/400+"-bed=pen+mad."[m]+d)%7;}

Gauss's method in MATLAB

% example date inputy1=2022;m1=1;d1=1;month_offset=[033614625035];% common year% offset if y1 leap yearifmod(y1,4)==0&&mod(y1,100)==0&&mod(y1,400)==0month_offset=[034025036146];% leap yearend% Gregorweekday_gregor=rem(d1+month_offset(m1)+5*rem(y1-1,4)+4*rem(y1-1,100)+6*rem(y1-1,400),7)% Julianweekday_julian=rem(6+5*rem(y1-1,4)+3*(y1-1),7)

0: Sunday 1: Monday .. 6: Saturday

Gauss''s method for Gregorian calendar in Python

fromnumpyimportremainderasremdefis_leap_year(year:int)->bool:""" Determine whether a year is a leap year. """returnyear%4==0and(year%100!=0oryear%400==0)defday_of_week(y:int,m:int,d:int)->str:''' return day of week of given date as string,    using Gauss's algorithm to find it '''ifis_leap_year(y):month_offset=(0,3,4,0,2,5,0,3,6,1,4,6)[m-1]else:month_offset=(0,3,3,6,1,4,6,2,5,0,3,5)[m-1]y-=1wd=int(rem(d+month_offset+5*rem(y,4)+4*rem(y,100) \              +6*rem(y,400),7))return('Sun','Mon','Tue','Wed','Thu','Fri','Sat')[wd]

Notes

Citations

Further reading