John H. Conway was one of the greatest mathematicians of the 20th/21st century. His work is famous for brilliance and innovation. It is, therefore, unfortunate that he came up with a method of determining the day of the week for any date in the Julian/Gregorian calendar (the "Doomsday Rule") that has become very popular; popular due to his reputation. Unfortunate because it is, bluntly, an ugly, ugly mess.
Ugly, too, are all the methods shown in Wikipedia under "Determination of the day of the week." In fact, there is a fairly simple method for finding the day of the week for any date from 5 CE onwards. Let me show you....
First, I must explain "integer division". An integer is a.k.a. a "whole number"; that is, one with no fractional part. Interger division is where you say "seven divided by three is two and a remainder of one" or "fifteen divided by four is three and a remainder of three". That is, a whole number is divided by another, and the whole result and the remainder are given. In what follows, we will use the two parts (whole number and remainder) separately. I thus will use ÷ to mean "the whole part of the division" and mod to mean "the remainder". IOW, "22 ÷ 4 = 5" and "22 mod 4 = 2".
As well, we need the "month key"; that is, a number form 0 to 6 assigned to each month. This chart shows the month keys:
| J | F | M | 1 | 4 | 4 | |
| A | M | J | 0 | 2 | 5 | |
| J | A | S | 0 | 3 | 6 | |
| O | N | D | 1 | 4 | 6 |
OK, with integer division explained, and the month keys given, it's time to explain the method. It consists of a few formulas, etc., written one above the other, with the answers in a column. The number in the column are added up, some subtractions are made, and the answer divided and the remainder taken. Specifically:
| Year number ÷ 12 | ______ |
| Year number mod 12 | ______ |
| Previous number ÷ 4 | ______ |
| Month Key | ______ |
| Day of the month | ______ |
| Add all the above | ______ |
| Subtract 3 | -3 |
| If Jan. or Feb in leap year, subtract 1 | {-1} |
| If Gregorian, subtract difference from Julian | {-X} |
Take the final number and mod 7 (that is, find the remainer), then compare it to the following:
1 = Sunday
2 = Monday
3 = Tuesday
4 = Wednesday
5 = Thursday
6 = Friday
0 = Saturday
Let me give you an example. Let's find the day of the week for November 1, 1221 in the Julian calendar. By the chart, we see that November's key number is 4.
| 1221 ÷ 12 | 101 |
| 1221 mod 12 | 9 |
| 9 ÷ 4 | 2 |
| Month Key | 4 |
| Day of the month | 1 |
| Add all the above | 117 |
| Subtract 3 | -3 |
| If Jan. or Feb in leap year, subtract 1 | nope |
| If Gregorian, subtract difference from Julian | nope |
We end up with 114. 114 mod 7 = 2, so it's a Monday.
By the way, to determine the difference between Gregorian and Julian calendars, find
(year number ÷ 100) - (year number ÷ 400) - 2.
If you were using the "proleptic" Gregorian calendar for the above determination, the difference would be
12 - 3 - 2 = 7
and you would subtract 7 from 114, giving 107, which, mod 7 is also 2.
Now, that might seem kind of complex to you, but if you compare it to any of the methods I referred to at the beginning, you will find it so, so much simpler.
Play around with this, and have fun! :-)