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Formula for the Day of the Week


Date: 05/21/97 at 11:08:07
From: Christine Durling
Subject: Declaration of Independence

Would you please tell us what day of the week the Declaration of
Independence was signed on as well the  formula to determine such? 
Thank you very much!

Christine Durling, Education Media Specialist
Bordentown Regional High School
Bordentown, NJ 08505


Date: 05/27/97 at 14:53:12
From: Doctor Rob
Subject: Re: Declaration of Independence

This is a trick question, of course.  The date at the top of the 
document is July 4th, 1776.  I believe it was actually signed on 
July 2nd, however.

Ignoring this historical technicality, the day of the week upon which
July 4th, 1776, fell was Thursday.

The rule is quite complicated.  It goes like this:

Let k be the day of the month.  In this case, k = 4.

Let m be the month, counting March as 1 and February as 12.  (Here 
January and February are considered as the last months of the 
preceding year.  This is to make Feb. 29th [if any] be the last day 
of the year.  This also means that the values of C and D are those 
for the preceding year, so, for example, 1 Jan 2000 would have C = 19 
and D = 99.)  In this case, m = 5 (July).

Let D be the last two digits of the year.  In this case, D = 76.

Let C be the first two digits of the year (the century).  In this 
case, C = 17.

For any real number x, let [x] be the greatest integer less than or 
equal to x, which you get by truncating any fractional part.  Then 
compute:

f = k + [(13*m-1)/5] + D + [D/4] + [C/4] - 2*C.

Once you have this, then  f - 7*[f/7] will give you the day of the
week, with Sunday = 0, Monday = 1, and so on.

In your case
f = 4 + [64/5] + 76 + [76/4] + [17/4] - 34
  = 4 + 12 + 76 + 19 + 4 - 34
  = 81
and f - 7*[f/7] = 81 - 7*[81/7] = 81 - 7*11 = 81 - 77 = 4, or 
Thursday.

This rule was given by a certain Rev. Zeller, and so is called 
Zeller's Rule.  This works for the Gregorian calendar only.  There is 
a simpler version for the Julian calendar.  Recall that English-
speaking countries used the Gregorian calendar beginning 14 Sep 1752, 
and before that used the Julian calendar.

A good reference for calendar matters is the _Encyclopedia 
Britannica_, under the heading "Calendar".

-Doctor Rob,  The Math Forum
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