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Formula for the Day of the Month


Date: 04/05/2001 at 11:05:05
From: AJ Valadez
Subject: Day of Week Formula

I have read the Dr. Math FAQ on finding the day of the week from the 
date. 

  The Calendar and the Days of the Week 
  http://mathforum.org/dr.math/faq/faq.calendar.html   

How do you get the part of the equation that deals with the month?


Date: 04/05/2001 at 13:39:09
From: Doctor Rob
Subject: Re: Day of Week Formula

Thanks for writing to Ask Dr. Math, AJ.

Rev. Zeller came up with this formula about 200 years ago.

Here is a partial explanation of why [(13*m-1)/5] appears.

    m  Month  # Days   # Extra   Total Extra
   ==========================================
    1  March     31        3          0
    2  April     30        2          3
    3  May       31        3          5
    4  June      30        2          8
    5  July      31        3         10

    6  August    31        3         13
    7  September 30        2         16
    8  October   31        3         18
    9  November  30        2         21
   10  December  31        3         23

   11  January   31        3         26
   12  February  28/29     0/1       29

Notice that the pattern in the 4th column repeats every 5 months,
and the total number of extra days beyond 4*m weeks in one of
those cycles is 3+2+3+2+3 = 13.  That is why 13/5 appears in this
part.

Now look at 13*m/5, and its greatest integer (because the number
of days is an integer):

   m     1    2    3    4    5    6    7    8    9   10   11   12 
13*m/5  2.6  5.2  7.8 10.4  13  15.6 18.2 20.8 23.4  26  28.6 31.2 
[13*m/5] 2    5    7   10   13   15   18   20   23   26   28   31
 diff      3    2    3    3    2    3    2    3    3    2    3
      ^^^^^^^^^^^^^^^^    ^^^^^^^^^^^^^^^^^^^^^    ^^^^^^^^^^^^^^^

Notice the pattern 3,2,3,2,3 appearing in the differences, but at an 
offset from where we want it. This is almost the right set of numbers.  
It can be adjusted to be the right set by looking at
X = [(13*m-1)/5] - 2:

   m     1    2    3    4    5    6    7    8    9   10   11   12 
   X     0    3    5    8   10   13   16   18   21   23   26   29
 diff      3    2    3    2    3    3    2    3    2    3    3
           ^^^^^^^^^^^^^^^^^^^^^    ^^^^^^^^^^^^^^^^^^^^^    ^^^^^

Again, notice the 3,2,3,2,3 pattern appearing in the differences, but 
at the correct offset. That is what adding a constant in the numerator 
inside the brackets does: it shifts the difference pattern. Adding an 
integer outside the brackets leaves the difference pattern unchanged.  
This X duplicates the last column in the first table above.

This means that [(13*m-1)/5] - 2 equals the contribution to the day of 
the week made by the month part of the date.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   
    
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