Tuesday, November 6, 2012 - , , 0 comments

Easiest method for solving the calendar question (The Doomsday rule)

The other day, I saw a really simple way to calculate the day of the week which minimizes (in my opinion) the amount of calculation done. Here's how it goes:

First, there is a day of the week that is shared in common by April 4th (4/4), 6/6, 8/8, 10/10, 12/12, and a few others every year. Let's call this day "Doomsday". It is, of course, different every year, but in a single year all of the dates above will fall on the same day.

It's really easy to remember: 4/4, 6/6, 8/8, 10/10, and 12/12 all fall on Doomsday.

Now, remember "I go to my nine-to-five job at the seven-eleven". 5/9, 9/5, 7/11, and 11/7 all fall on Doomsday.

3/0 (The last day of February) falls on Doomsday.

January and February take a little more practice to remember due to Leap Years. In ordinary years, 2/0 (the last day of January) and 1/3 fall on Doomsday; however, on Leap Years, 2/1 and 1/4 fall on Doomsday.

So the list of "Doomsday" dates to remember are:

1/3 (or 4), 2/0 (or 1), 3/0, 4/4, 5/9, 6/6, 7/11, 8/8, 9/5, 10/10, 11/7, and 12/12

For 2006, Doomsday is Tuesday (We'll talk about how to calculate Doomsday for any year in just a second). We can calculate the day of the week of any day in the year 2006 by picking the nearest Doomsday day to the date in question, and doing simple arithmetic to get to the day.

So, if I want to know what day Labor Day is on this year, I can take 9/5, since that is on a Tuesday, and it immediately follows that Labor Day is on September 4th, the day previous.

If I want to know what day of the week Christmas is on, I can start with 12/12, which is on a Tuesday this year, add two weeks and a day, giving me Wednesday for Christmas.

It's easy!

As a test, then, what day of the week will/did your birthday fall on in 2006?

Now, how do we know what Doomsday is for any given year? It is very easy. Each century has a "century day" that all of the Doomsday's cycle around. The 2000s century day is Tuesday as well (coincidentally).

If we take our century's century day, we can use it to determine any year's Doomsday in that century:

1. Take the last two digits of the year (06 for 2006).

2. Divide by 12 and note both the quotient (q) and the remainder (r).

3. Take the remainder, divide that by four, and note the quotient (s), but forget the second remainder.

4. Add q, r, and s to get the offset from the century day.

5. Add that to the century day, and that is the year's Doomsday!

For 2011, then:

1. The last two digits of 2011 is 11.
2. 11/12 = 0 remainder 11. So q = 0 and r = 11.
3. r/4 = 11/4 or 2 remainder 3, so s = 2 (we forget the 3).
4. q+r+s = 0 + 11 + 2 = 13.
5. Tuesday (the century day for the 2000s) + 13 days = Monday.

The Doomsday for 2011 is Monday!

Now, I can determine any day of the week for 2011 just like I did for 2006.

What day of the week will 01/23/2045 be?

Now, the century day is really easy, too, but it is simply memorization. Here is the table: (Note that the table only goes back to the 1500s because the Gregorian calendar was instituted in 1582. There is a link at the end of this comment describing what to do with Julian dates).

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

Notice that they cycle every 400 years.

One more example of the whole kitten caboodle. What day of the week was the Independence of India (15 Aug 1947)?

The century day for the 1900s is Wednesday. 47/12=3 rem 11, and 11/4 is 2 rem 3. 3+11+2=16. So the Doomsday for 1776 is Friday (Wednesday + 16 days). So, 8/8 is our closest Friday, which means that August 15th, seven days ahead, was a Friday!

Good luck amazing all your friends!

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