Re: Friday 13th, try it yourself (was Language Wars..)

[Randy MacDonald <randy@godin.on.ca>]

I've reposted this message from Roger Hui, since his access to
the newsgroup didn't allow for posting to the other groups.
With apologies to the comp.lang.apl folk...

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Subject: Re: Triskaidekanalysis
       Date: 21 Sep 96 05:35:19 GMT
       From: Roger Hui <Roger.Hui@SYMPATICO.CA>
Organization: UofNB News Gateway, Fredericton, CANADA
  Newsgroups:comp.lang.apl


A more recent and complete leap year rule specifies that years
divisible by 4000 are common (not leap) years.  In J, the following
verb is 1 if the argument is a leap year, and 0 if it is common:

   leapyear=: 0&(~:/ . =) @ (4000 400 100 4&(|/))

In words: Compute the remainder of division by 4000, 400, 100,
and 4.  If the number of 0 remainders is odd, then it is
a leap year; if even, then it is a common year.

The additional 4000-year rule is described in Guy Ottewell's
"The Astronomical Companion", 1979, Furman University, Greenville,
SC, USA, Telephone (803) 294-2208.  The reason for having the rule
is to bring the average calendar year closer to the tropical year
(within one day in 20000 years):

   1460969 days % 4000 years = 365.24225 days/year  (4000 year rule)
   146097  days % 400  years = 365.2425  days/year  (400  year rule)

Therefore, to determine the distribution of the 13-th day over
the days of the week, it is necessary to consider not just a
400 year cycle, but a 28000 year cycle.  (Actually the first
should be a 2800 year cycle, with a short cut to 400 years made
possible by the fact that 146097 is divisible by 7.)

The analysis can be done in J as follows:

   p=: 31 28,10$5$31 30       NB. days in month in common years
   q=: 31 29,10$5$31 30       NB. days in month in leap   years
   +/,(leapyear i.4000){p,:q  NB. # days in 4000 years
1460969

   y=: i.28000                NB. year numbers from 0 to 27999.

   d=: (leapyear y){p,:q      NB. # of days in each month in y
   $d                         NB. d is a 28000 by 12 matrix
28000 12
   5{.d                       NB. first 5 rows of d
31 28 31 30 31 30 31 31 30 31 30 31
31 28 31 30 31 30 31 31 30 31 30 31
31 28 31 30 31 30 31 31 30 31 30 31
31 28 31 30 31 30 31 31 30 31 30 31
31 29 31 30 31 30 31 31 30 31 30 31

   w=: 7| +/\ ,d              NB. day-of-week of every 13-th day of
month
   20 {. w                    NB. first 20 entries in w
3 3 6 1 4 6 2 5 0 3 5 1 4 4 0 2 5 0 3 6

   t=: /:~ (~.w) ,. #/.~ w    NB. sorted table of day-of-week, #
occurrences
   t
0 48000
1 48000
2 48000
3 48000
4 48000
5 48000
6 48000

The next example shows that a 4000-year cycle exhibits uneven
distribution.
Since the number of days in 4000 years (1460969) is not divisible by 7,
the unevenness is smoothed out when the cycle is lengthen by a factor of
7,
as above.

   /:~ (~.w) ,. #/.~ w=: 7 | +/\ , (leapyear i.4000){p,:q
0 6880
1 6840
2 6869
3 6851
4 6850
5 6870
6 6840

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