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Chris' River Extension - Doomsday Maths


Doomsday Math

I was looking for examples of Christians burning people at the
stake /for my Easter observance/ and had started with Galileo (who
narrowly escaped here).  Followed various links in Wikipedia and
eventually came upon some calendar math that interested me.

In particular, apparently John Horton Conway has developed a rather
useful formulation for finding the day of the week for any date,
using the so-called Doomsday (the last day of February, equivalent
to 'March 0').

/ means integer division
% means the remainder of such a division
Y is 4-digit year of interest

y = Y % 100 (take the last two digits of the year)
a = y / 12
b = y % 12
c = b / 4
d = a + b + c
e = d % 7
Doomsday = Wednesday + e (for 1900-1999)

Example -- Y = 1959:

y = 1959 % 100   = 59
a =   59 /  12   =  4
b =   59 %  12   = 11
c =   11 /   4   =  2
d =   4 + 11 + 2 = 17
e =   17 %   7   =  3
Doomsday = Wednesday + e (for 1900-1999)
Doomsday = Wednesday + 3 = Saturday for 1959 /answer/.

That is, February 28 ("March 0") is a Saturday in 1959.


The Doomsday rotation is +5, +5, +5, +6 days for centuries.
(Tue + 5 = Sun; Sun + 5 = Fri; Fri + 5 = Wed; Wed + 6 = Tue).

Therefore,

Tue, Sun, Fri, Wed, Tue, Sun, Fri, Wed .. is the rotation
for centuries.

And so,

Doomsday = Tuesday   + e (for 2000-2099)


A second example -- Y = 2011:

y = 2011 % 100   =  11   "divide modulo 100" or "subtract the century"
a =   11 /  12   =   0   "divide integer 12" ('ZERO, remainder eleven')
b =   11 %  12   =  11   "divide modulo  12" ('zero, remainder ELEVEN')
c =   11 /   4   =   2   "divide integer  4" ('TWO remainder three')
d =   0 + 11 + 2 =  13   "sum"
e =   13 % 7     =   6   "divide modulo   7" ('one, remainder SIX')

Doomsday = Tuesday   + e (for 2000-2099)
Doomsday = Tuesday   + 6 = Monday for 2011 /answer/.

February 28 (or "March 0") in 2011 falls on a Monday.

(Mental calculation hint: hold only a, b and c in the mind; they
reduce to d, and can then be forgotten.  From then on, concentrate
solely on reduction of d into e.  Not much juggling there).


Indexing further into the calendar

Finding Doomsday for a year, this easily, would be neat enough in
itself, but someone (perhaps Conway) has looked a bit further and
noticed a strange thing; there are indices into the rest of the
calendar year that are easy to remember and point to the same day
of the week as the Doomsday.

They are:

4/4 (4 April)
6/6 (6 June)
8/8 (8 August)
10/10 (10 October)
12/12 (12 December).

And how about these:

5/9 (9 May)
9/5 (5 September)
7/11 (11 July)
11/7 (7 November).

That covers one index day per month, except for January (the real
Doomsday is in February, and is also 'March 0', so February and
March are already covered).

For January, how about:

 3rd for 3 years
 4th for 4th year (leap year).

All of those dates fall on the same day of the week as Doomsday.

And there you have it.


Copyright (C) 2008 Christopher W Hafey
Sun Mar 23 23:47:28 PDT 2008