## 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.

In 2015, this special day of the week, which Conway calls "Doomsday," was a Saturday.

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 9/5, 7/11, 5/9 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.

Alternatively, you can use the Doomsday for the previous year to handle January and February, without worrying about leap years. Simply note that 1/2 and 2/6 fall on the Doomsday for the previous year.

In 2015, which is a not leap year, therefore, the following dates fall on Doomsday (Saturday): 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 was Christmas 2015? If 12/12 is a Saturday, then so are 12/19 and 12/26, so 12/25 has to have been a Friday.

When was Valentine's day? Since this is a not leap year, 2/0, 2/7 and 2/14 are Saturdays. So, Valentine's day (2/14) also fell on a Saturday.

On what date did Labor Day fall in 2015? Labor Day is the first Monday in September. If 9/5 was a Saturday, then 9/7 was a Monday, and this must have been Labor Day.

Practice: On what date did Thanksgiving fall in 2015? Thanksgiving is the fourth Thursday of November. You figure it out, then check with a calendar to see if you were right.

### Part 2-What is Doomsday for any year of the 21st 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. (This is a new method, devised by Mike Walters, that is easier than the method that John Conway originally published):

If the year is 20xx, is xx even? If if is, set A=xx; if it is odd, set A=xx+11.

Set B=A/2 (A is always even so you can do this exactly).

If B is even set C=B; if it is odd, set C=B+11 (this is the same thing you did in the first step). C will be an even number.

Divide C by 7 and compute the remainder. Call this remainder R.

Count R days backwards starting from Tuesday (which is the special Doomsday that applies to this century). This gives you the Doomsday for the year 20xx. From this point on, just use Part 1.

Example: What was Doomsday in 2015? Well, xx=15, which is odd, so A=15+11=26. B=A/2=13. B is odd so C=B+11=24. If we divide C=24 by 7, we get 21 with remainder R=3. Count 3 days backwards from Tuesday to get Saturday. Doomsday in 2015 was a Saturday (a fact we used in Part 1 above).

Example: The World Trade Center and Pentagon suffered terrorist attacks on September 11, 2001. xx=1, which is odd, so A=12. B=A/2=6, which is even, so C=6. The remainder when dividing 6 by 7 is 6, so counting 6 days backwards from Tuesday (or equivalently, counting 1 day forwards) tells us that doomsday in 2001 was a Wednesday. Therefore, 9/5 was a Wednesday, 9/12 was a Wednesday, and 9/11 was a Tuesday.

In the 20th century, use the same rule, except count backwards from Wednesday instead of Tuesday. Wednesday is the special doomsday day for the 20th century.

Example: On what day of the week did D-Day, June 6, 1944, fall? Well, 44 is even so A=44, B= 22 and since B is even, C= 22. The remainder when dividing C by 7 is 1 so R=1. In the 20th century count 1 day backwards from Wednesday 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. Hint: This is also a 20th century date.

Practice: On what day of the week were you born? Calculate it using the Doomsday rule and check it on a calendar or (if you already know it) with your memory.

### Part 3-Other Centuries (Gregorian Calendar)

The Gregorian calendar, which is our civil calendar, was introduced in 1582 (1752 in English-speaking countries, and not until 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 Tuesday for the century doomsday as in Part 2 (for the 21st century), 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: xx=76, which is even, so A=xx and B=A/2=38, which is also even. So C=38, and C/7=35 Remainder 3 and R=3. The century doomsday for 17xx is Sunday, so counting backwards 3 days we find that doomsday for 1776 was a Thursday. 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 the Julian Doomsday as follows: xx= 82 which is even, so A=82 and B=41. B is odd, so C=B+11=52. C/7=7 Remainder 3 so R=3. cc=15 which is 2 weeks and 1 day, so we count 1 day backwards from Sunday to find that the (Julian) century doomsday for 15xx is Saturday, and since R=3 count another 3 days backwards from Saturday to find that the (Julian) doomsday for 15xx is Wednesday.

The last day of the old Julian calendar was October 4, 1582. Since 10/10/1582 (Julian) was a Wednesday, 10/3/1582 was also 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 R=3 that we just calculated and  count 3 days backwards from Wednesday (the Gregorian century doomsday for the 15xx dates) 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 (Gregorian), five days later, was a Friday, which is 1 day later than Thursday, 10/4/1582 (Julian).

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.)

Note: I wrote above, "if there were such a date as 10/10/1582." Actually, we can calculate backwards from the actual date of introduction of the Gregorian calendar using the Gregorian rules if we wish, and we designate such dates as being on the Gregorian Proleptic Calendar. "Proleptic" just means "before this calendar was officially introduced, but using the same calculation rules".

# 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. 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. The doomsday calculation for a given year used John Conway's original method, and not the method described here. However, they give identical results.