Rosh Hashana Calculations

[Hans Havermann]

A follow-up observation (Re: Years when Ramadan and Rosh Hashana  
coincide)... It's extremely difficult to find any actual tabular  
information on the Gregorian dates for more than a few Rosh-Hashanas  
online. If the dates at the bottom of <http://www.everything2.com/ 
index.pl?node_id=146512> (supposedly taken from 'Webster 1913') are  
accurate then I get the following discrepancies with Mathematica's  
JewishNewYear[year] (1900 <= year < 2100) function:

{1900, 9,25},
{1901, 9,15}, { 9,14} -1
{1902,10, 3},
{1903, 9,23}, { 9,22} -1
{1904, 9,11},
{1905,10, 1}, { 9,29} -2
{1906, 9,21}, { 9,20} -1
{1907, 9, 7}, { 9, 9} +2
{1908, 9,27}, { 9,26} -1
{1909, 9,17}, { 9,16} -1
{1910,10, 5}, {10, 4} -1
{1911, 9,24},
{1912, 9,13}, { 9,12} -1
{1913,10, 3}, {10, 1} -2
{1914, 9,22}, { 9,21} -1
{1915, 9,10}, { 9, 9} -1
{1916, 9,29}, { 9,28} -1
{1917, 9,18}, { 9,17} -1
{1918, 9, 8}, { 9, 7} -1
{1919, 9,26}, { 9,25} -1
{1920, 9,14}, { 9,13} -1
{1921,10, 4}, {10, 3} -1
{1922, 9,24}, { 9,22} -2
{1923, 9,12},
{1924, 9,30}, { 9,29} -1
{1925, 9,20}, { 9,19} -1
{1926, 9,10}, { 9, 8} -2
{1927, 9,28},
{1928, 9,16},
{1929,10, 6}, {10, 4} -2
{1930, 9,24},
{1931, 9,13}, { 9,12} -1
{1932,10, 2}, {10, 1} -1
{1933, 9,24}, { 9,20} -4
{1934, 9,11}, { 9, 9} -2
{1935, 9,29}

The first column gives the Webster dates plus one (to allow for Rosh  
Hashana to "begin" the evening of the previous day). The second  
column gives the Mathematica JewishNewYear[year] {month, day} values,  
where they differ from the first column, followed by the discrepancy  
in days. Only 9 of the 36 years are in agreement.

The Mathematica Miscellaneous`Calendar` functions appear to be some  
15 years old, based on Ilan Vardi's 1991 'Computational Recreations  
in Mathematics' [Addison-Wesley, 0-201-52989-0] wherein there is no  
mention of the JewishNewYear function and the implementation of the  
Jewish calendar is left as an exercise for the reader. Vardi is given  
as the author of the Calendar package here:

http://computing.ee.ethz.ch/sepp/mathematica-5.0.1-rs.SEPP/scratch/ 
AddOns/StandardPackages/Miscellaneous/Calendar.m

In spite of a caveat ("I have not yet implemented the Jewish  
calendar"), the package contains the JewishNewYear[year] function and  
defines it thus:

JewishNewYear[y_] :=
             Block[{t,g,n,f,d},
                   g= Mod[12 (Mod[y,19] +1) , 19];
                   t= Quotient[y,100] -Quotient[y,400] -2  +
                      765433/ 492480 g + Mod[y,4] / 4 -
                      (313 y + 89091)/98496 ;
                   n = Floor[t];
                   f = t - n;
                   d = Switch[Mod[ DateToNumber[y,9,n] [[1]],7],
                          0, n+1,
                          1, If[ f >= 23269/25920 &&  g >11 , n+1,n],
                          2, If[ f>= 1367/2160 && g > 6, n+2,n],
                          3, n+1,
                          4, n,
                          5, n+1,
                          6,n];
                    If[d < 31,{y, 9, d},{y, 10, d - 30}]
                    ]

The procedure appears to be an implementation of Berlekamp/Conway/ 
Guy's "Jewish New Year (Rosh Hashana)" calculation from their 1982  
"Winning Ways For Your Mathematical Plays" (I have it on page 909 of  
their second-edition Volume 4).

Does anyone know if the Berlekamp/Conway/Guy formulation reproduces  
the "incorrect" Mathematica dates?