Xref: utzoo sci.math:2781 comp.sys.ibm.pc:11542
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From: Robert_A_Freed@cup.portal.com
Newsgroups: sci.math,comp.sys.ibm.pc
Subject: Re: perpetual calendar (also day of week)
Message-ID: <3026@cup.portal.com>
Date: 8 Feb 88 01:46:33 GMT
References: <1217@mtuxo.UUCP>
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In article <1217@mtuxo.UUCP> khreb@mtuxo.UUCP (01274-K.ROSEN) writes
to sci.math:

>The day of the week of day k, of month m (where a year is
>considered to start in March, so that March is month 1 of that year and
>January is considered the 11th month of the previous year and
>February the twelfth month of that year), of year
>100C + Y, where C is the century and Y is the year of the
>century, is given by the remainder of
>
>      k + [2.6m - 0.2] - 2C + Y  + [Y/4] + [C/4]
>
>when divided by 7, where Sunday is represented by a remainder of 0,
>Monday by 1, etc.
>
>For instance, for January 1, 1900 we have C=18, Y=99, m=11, and k=1
>(since January is considered the 11th month of 1899 for this
>calculation).   The formula becomes 1+28-36+99+4+24 which leaves a
>remainder of 1 when divided by 7.  Hence this day was a Monday.
>
>For a derivation of this formula see my book ELEMENTARY NUMBER THEORY
>AND ITS APPLICATIONS, Addison-Wesley, either first edition 1984 or
>the new second edition, 1988.  The formula is messy because of the
>problem of different numbers of days in months and leap years occurring
>every 4 years, except for even century years not divisible by 400.
>
>                                 Ken Rosen, AT&T
>                                 201-957-3691

(A similar, but slightly less elegant, formula was recently discussed in
comp.sys.ibm.pc as well.)

I believe this is known as Zeller's congruence.  It's also cited in the
text book, Elementary_Number_Theory, by Uspensky and Heaslet.  I first 
noticed this little gem in a 1965 book, Problems_for_Computer_Solution, 
by Fred Gruenberger and George Jaffray.  (The subject computer was the 
IBM 1620.  If that doesn't date me, I don't know what will! :-)

Two small comments about the above formula:

1.  [2.6m - 0.2] can be calculated as [(13m - 1)/5] to avoid the 
    overhead (and possible inaccuracy) of floating-point arithmetic.
    (Although not stated in the formulation, the square brackets [...]
    imply integer truncation.)

2.  This applies only to the current (Gregorian) calendar, which was
    established October 15, 1582, but was not adopted by Great Britain
    (and the American colonies) until 1752.

The formula is particularly efficient for computer solution since it 
involves only short integer arithmetic and a single conditional test 
(for January-February).  And, unlike many other day-of-the-week 
solutions I've seen, this one handles non-leap century years (such as 
1900 and 2100) correctly.

Bob Freed            Internet:  Robert_A_Freed@cup.portal.com
Newton Centre, MA        UUCP:  sun!portal!cup.portal.com!Robert_A_Freed