AST 309-TIME
What is the day of the week, given any date?

This is really a nice trick. You can easily calculate the day of the week, given any date in history, and with a little practice you can even do it in your head. The method is based on one developed by John Horton Conway, and is described in Winning Ways, a book that he wrote with Berlekamp and Guy. It is described in Volume 2.


Part 1-Within a Given Year


The secret of the method is to have a way of knowing the day of the week for one day in each month of the year. Conway's method uses the fact that the following dates always fall on the same day of the week in any given year. They are easy to memorize, and once one has this down pat, some simple calculations allow you to do this for any year. These are the dates that always fall on the same day of the week.

This year, this special day of the week, which Conway calls "Doomsday," is a Friday.

4/4, 6/6, 8/8, 10/10 and 12/12 always fall on the same day of the week (Doomsday) in any year.

If you memorize the phrase "I went to my nine-to-five job at the seven-eleven," you can also remember easily that 5/9, 9/5, 7/11 and 11/7 also fall on Doomsday.

Also, 3/0 (the zeroth day of March, i.e, the last day of February) falls on Doomsday.

January and February are complicated by the existence of Leap Years. In ordinary years, 2/0 (the last day of January) also falls on Doomsday, as does 1/3; in Leap Years, 2/1 and 1/4 fall on Doomsday.

In 1997, which is not a leap year, therefore, the following dates fall on Doomsday (Friday): 1/3, 2/0 (1/31), 3/0 (2/28), 4/4, 5/9, 6/6, 7/11, 8/8, 9/5, 10/10, 11/7 and 12/12.

So, when is Christmas this year? If 12/12 is a Friday, then so are 12/19 and 12/26, so 12/25 has to be a Thursday.

When is Valentine's day? Since this is a not leap year, 2/0, 2/7 and 2/14 are Fridays. So, Valentine's day (2/14) also falls on a Friday.

On what date does Labor Day fall this year? Labor Day is the first Monday in September. If 9/5 is a Friday, then 9/1 is a Monday, and this must be Labor Day.

Practice: On what date does Thanksgiving fall this year? Thanksgiving is the fourth Thursday of November. You figure it out, then check with a calendar to see if you were right.

Practice: On what day of the week were you born? Calculate it, and check with a calendar (or with your parents). Did you get it right?


Part 2-What is Doomsday for any year of this century?


The next and hardest part of the trick (because it requires a small amount of calculation) is to determine Doomsday for the year in question. Here's how to do it.

If the year is 19xx, divide xx by 12 to get a quotient Q and a remainder R;

Divide the remainder R by 4 to get another quotient S (forget the new remainder, it is not used).

Add Q, R and S together, and count that many days starting from Wednesday (which is the special Doomsday that applies to this century). This gives you the Doomsday for the year 19xx. From this point on, just use Part 1.

Example: This year is 1997. 97/12 = 8 remainder 1 so Q=8, R=1 and S=0. Nine days from Wednesday is a Friday, so in 1997, Doomsday is Friday (which we already said).

Example: On what day of the week did D-Day, June 6, 1944, fall? Well, 44/12 = 3 remainder 8, and 8/4 = 2 with remainder 0 which we forget. 3+8+2=13, and counting 13 days from Wednesday is the same as counting -1 day, so Doomsday for 1944 is a Tuesday. Since 6/6 (June 6) is June's magic day, we now know that June 6, 1944 was a Tuesday.

Practice: On what day of the week was Pearl Harbor bombed? It was December 7, 1941.


Part 3-Other Centuries (Gregorian Calendar)


The Gregorian calendar, which is our civil calendar, was introduced in 1582 (1752 in English-speaking countries, and only in 1919 in Russia). So one has to know whether the Gregorian or the old Julian calendar is being used. The rule in this section applies only to the Gregorian Calendar.

The only thing that changes in other centuries is that instead of using Wednesday as in Part 2, we use another day that depends on the century. This century day cycles over 4 centuries, so that it is the same in 16xx, 20xx, 24xx etc. Specifically,

In years 15xx, 19xx, 23xx, etc., use Wednesday
In years 16xx, 20xx, 24xx, etc., use Tuesday
In years 17xx, 21xx, 25xx, etc., use Sunday
In years 18xx, 22xx, 26xx, etc., use Friday

Example: On what day of the week did July 4, 1776 fall? First, calculate Doomsday: 76/12 = 6 remainder 4; 4/4 = 1 remainder 0 (forget the 0). 6+4+1=11 and counting 11 days from Sunday, which is the Doomsday for years 17xx, we get Thursday, which is Doomsday for 1776. Now July 11 is 7/11, which is a Thursday, so July 4, which is 1 week earlier, is also a Thursday. The Declaration of Independence was signed on a Thursday.

Practice: The Civil War of the United States opened with the firing on Fort Sumter, which took place on April 12, 1861. What day of the week was that?


Part 4-Julian Calendar


Finally, the same principles can be used if the date is on the old Julian Calendar, which was introduced by Julius Caesar some 2000+ years ago. The only difference is that the different leap year rule of the Julian calendar means that the rule given in Part 3 has to be modified. In years ccxx on the Julian calendar, we get the century Doomsday by subtracting cc from Sunday (i.e., counting back one day for each century).

Example: In 1582 we find that the Julian Doomsday is as follows: 82/12 = 6 remainder 10; 10/4 = 2 forget the remainder. 6+10+2 = 18. Sunday + 18 - cc = Sunday+18-15 = Sunday+3 so count 3 days forward from Sunday to find Wednesday.

The last day of the old Julian calendar was October 4. Since 10/10/1582 (Julian) is a Wednesday, so is 10/3/1582 is a Wednesday, and the next day, October 4, 1582, was a Thursday.

The next day, October 15, 1582 (Gregorian), was the first day of the new Gregorian calendar. Take the number 18 that we just calculated and count 18 days from Wednesday to get Gregorian Doomsday: That gives us a Sunday. Now 10/10/1582 (Gregorian) would have been a Sunday if there were such a date...So 10/15/1582, five days later, was a Friday.

Note that this is the day after Thursday. This is important. So far as we know, the weekly cycle of the days of the week has not been broken for thousands of years.

We have found that Thursday, October 4, 1582 (Julian) was followed by Friday, October 15, 1582 (Gregorian).

People complained that 10 days were missing from their lives!

Practice: September 2, 1752 (Julian) was followed in the English-speaking world by September 14, 1752 (Gregorian). On what day of the week did each fall? (Do the calculations separately and verify that the weekly cycle was not broken when the calendars changed.)

A related page can be found at How to determine the day of the week, given the date.

Day of the Week Calculator

This JavaScript example calculates the day of the week on the Gregorian calendar for any date in the twentieth century. For date mm/dd/yy in this century, enter mm, dd, yy below and click "compute." To use it, you have to be using a JavaScript-aware browser such as Netscape v. 2.0 or later, or Microsoft Explorer. You may use it for practice.

Month mm

Day dd

Year 19yy

Doomsday

For Month

DayOfWeek


You can use the Document Source function in the View menu to look at the source of this HTML document (if you wish). In the source, you'll see various math and formatting functions using the JavaScript language.


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