Introduction
To determine the day of the week from numerical operations, it is necessary to represent Sunday to Saturday as numbers (usually from 0 to 6, respectively, which is equivalent to ISO 8601's alternative usage of 1 = Monday to 7 = Sunday). This is achieved with arithmetic modulo 7. Modulo 7 is an operation that calculates the remainder of a number being divided by 7. 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.
The basic approach of nearly all of the methods to calculate the day of the week begins by starting from a known pair (such as January 1, 1800 as a Wednesday), determining the number of days between the known day and the day that you are trying to determine, and using arithmetic modulo 7 to find a new numerical day of the week.
One standard approach is to look up (or calculate, using a known rule) the value of the first day of the week of a given century, look up (or calculate, using a method of congruence) an adjustment for the month, calculate the number of leap years since the start of the century, and then add these together along with the number of years since the start of the century, and the day number of the month. Eventually, one ends up with a day-count on which one applies modulo 7 to determine the day of the week of the date.
Some methods do all the additions first and then cast out sevens, whereas others cast them out at each step, as in Charles Lutwidge Dodgson's method. Either way is quite viable: the former is easier for calculators and computer programs; the latter for mental calculation (it is quite possible to do all the calculations in one's head with a little practice).
None of the methods given here perform range checks, so that unreasonable dates will produce erroneous results.
Read more about this topic: Determination Of The Day Of The Week
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