Calculating the day of the week

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This article details a mathematical algorithm to calculate on which day of the week any particular date fell or will fall. There are a number of other algorithms to do this, but they are all variants of each other, simply using different rules to achieve the same result.

A typical application is to calculate the day of the week on which someone was born.

Table of contents

1 Introduction
2 Useful Concepts

2.1 Corresponding months
2.2 Corresponding years

3 The Algorithm to Calculate the Day of the Week

3.3 Examples
3.4 Centuries Table
3.5 Months Table
3.6 Days Table

4 References

Introduction

The basis of all the algorithms to calculate the day of the week is:

To number the days of the week from 0 to 6 so that we can then use arithmetic modulo 7 to add the number of days elapsed since the start of a known period (usually in practice a century).
To look up or calculate using a known rule what day the given century started on.
To look up or calculate what day the given year in that century started on.
To look up or calculate what day the given month in that year in that century started on.
To then add on the day of the month - this of course being the days elapsed since the month started.

Put simply, using arithmetic modulo 7 means ignoring multiples of 7 during calculations. Thus we can treat 7 as 0, 8 as 1, 9 as 2, 18 as 4 and so on; the interpretation of this being that if we signify Sunday as day 0, then 7 days later (i.e. day 7) is also a Sunday, and day 18 will be the same as day 4, which is a Thursday since this falls 4 days after Sunday. Some algorithms do all the additions first and then cast out sevens whereas others cast them out at each step. Either way is quite permissible; the former is better for when using calculators and in computer algorithms, the latter for mental calculation, since it is quite possible to do all the calculations in one's head with a little practice.

Useful Concepts

Corresponding months

'Corresponding months' are those months within the calendar year that start on the same day. For example, September and December correspond, because September 1 falls on the same day as December 1. 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 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 Feb 29 means each subsequent month starts a day later.

Here's how the months correspond:

Common-year January and October.
Leap year January, April and July.
Common-year February, March and November.
Leap-year February and August.
September and December.
May
June
August

Note that May, June and August correspond with no other month. Also note that in the months table below, corresponding months have the same number, a fact which follows directly from the definition.

Corresponding years

There are 7 possible days that a year can start on, and leap years will alter the day of the week after February 29. This means that there are 14 configurations that a year can have. All the configurations are referenced in the article on Dominical letter. For example, 2003 was a common year starting on Wednesday, meaning that 2003 corresponds to the 1997 calendar year. 2004, on the other hand, is a leap year starting on Thursday, meaning that the year starts off corresponding to 1998 and ends corresponding to 1999.

The Algorithm to Calculate the Day of the Week

The algorithm is valid for the Gregorian Calendar. This began in Britain and her colonies on September 14, 1752. The area now forming the United States changed at different times depending on the colonial power; Spain and France had changed in 1582 and Russia had not changed by 1867 when Alaska was purchased by the U.S. from Russia.

It should be noted that, in this algorithm, the days on which the century, year, and month start are the "zeroth" day. This allows us to add the day of the month directly (without subtracting 1). For example, 1900 starts on day 0 which corresponds to a Sunday; however, we still need to add 1 for the 1 January which brings the day on which 1 January, 1900 fell to day 1, which is Monday, the correct day.

Centuries First, we can either refer to the centuries table below or use the rule: Divide centuries-figure by 4, take the remainder from 3, then multiply the result by 2.
- For example, for years beginning 18 (1800-1899), 18/4 gives remainder 2, 3-2=1 then 2*1 = 2.
Years Because there are 365 days in a common year, which is 52 weeks plus 1 day, each year will start on the day of the week after that starting the preceding year. Each leap year has of course one more day than a common year. Assuming we know on which day a century starts (from above), if we add the number of years elapsed since the start of the century, plus the number of leap years that have elapsed since the start of the century, we get the day of the week on which the year starts.
- Taking 1978 as an example, 78 common years would add 78 to the start-day of the century, but there have been 78/4 = 19 leap years since 1900 (we can ignore the remainder since the 2 years since 1976 contribute no extra leap-day). So 1978 started on day 0 + 78 + 19 = 97 which is the same as day 6.
Months We refer to the months table below to work out on which day of the week a month starts. Notice that January starts on day 0, which is simply another way of saying that the year and January of that year start on the same day. The months table shown allows for leap years; other algorithms leave the correction to the end and then deduct 1 from the final figure if the month is a January or February of a leap year.
Day of the Month Once we know on which day of the week the month starts, we simply add the day of the month to find the final result (noting that as mentioned above, we've been working with the "zeroth" day of the month as the start). For example, January 22 will be 22 days after the start of January, so we add 22.

Examples

Now for an example of the complete algorithm, let's use April 24, 1982.

Look up the 1900s in the centuries table: 0
Note the last two digits of the year: 82
Divide the 82 by 4: 82/4 = 20.5 and drop the fractional part: 20
Look up April in the months table: 6
Add all numbers from steps 1-4 to the day of the month (in this case, 24): 0+82+20+6+24=132.
Divide the sum from step 5 by 7 and find the remainder: 132/7=18 remainder 6
Find the remainder in the days table: 6=Saturday.

Now let's try September 18, 1783.

Look up the 1700s in the centuries table: 4
Note the last two digits of the year: 83
Divide the 83 by 4: 82/4 = 20.75 and drop the fractional part: 20
Look up September in the months table: 5
Add all numbers from steps 1-4 to the day of the month (in this case, 18): 4+83+20+5+18=130.
Divide the sum from step 5 by 7 and find the remainder: 130/7=18 remainder 4
Find the remainder in the days table: 4=Thursday.

Finally, let's try June 19, 2054

Look up the 2000s in the centuries table: 6
Note the last two digits of the year: 54
Divide the 54 by 4: 54/4 = 13.5 and drop the fractional part: 13
Look up June in the months table: 4
Add all numbers from steps 1-4 to the day of the month (in this case, 19): 6+54+13+4+19=96.
Divide the sum from step 5 by 7 and find the remainder: 96/7=13 remainder 5
Find the remainder in the days table: 5=Friday.

Centuries Table

1752-1799     4

1800-1899     2

1900-1999     0

2000-2099 6

Months Table

January      0 (6 in leap year)

February     3 (2 in leap year)

March        3

April        6

May          1

June         4

July         6

August       2

September    5

October      0

November     3

December 5

Days Table

Sunday 0

Monday 1

Tuesday 2

Wednesday 3

Thursday 4

Friday 5

Saturday 6