Rosh Hashana Calculations

[Hans Havermann]

> {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},

... snip ...

> {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.

I asked:

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

There's an online pdf document called "The Doomsday Rule" by S.W.  
Graham (July 1995):

http://www.cst.cmich.edu/users/graha1sw/Pub/Doomsday/Doomsday.pdf

... which is "adapted from Winning Ways for Your Mathematical Plays"  
and gives (on page 6) a formula for computing Rosh Hashana that I was  
able to implement in Mathematica. All of the calculated dates (except  
two) matched the Webster "begins" dates. The two exceptions are:

1907 - Webster, begins Sept. 6. Formula suggests Sept. 9.
1933 - Webster, begins Sept.23. Formula suggests Sept.21.

If these two Webster dates are in fact misprints and replaced with  
their formula suggestions, it goes a ways to explaining the anomalous  
+2 & -4 days-discrepancy (in the table above), replacing them with  
values of -1 & -2, respectively. Furthermore, matching the formula  
calculations with some known recent Rosh Hashana dates, I surmise  
that my assumption about the Webster dates being the day prior to  
Rosh Hashana is also incorrect: They appear to be, in fact, the day  
of Rosh Hashana.

I had suggested that the Mathematica JewishNewYear[year] function  
appeared "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 believe now that that appearance was  
somewhat superficial and that the Mathematica implementation, however  
it was derived, is somehow wrong.

Using the formula to recalculate A118661:

24, 14, 2, 22, 10, 30, 20, 9, 26, 16, 4, 23, 12, 2, 21, 9, 28, 17, 7,  
25, 13, 3, 23, 11, 29, 19, 9, 27, 15, 5, 23, 12, 1, 21, 10, 28, 17,  
6, 26, 14, 3, 22, 12, 30, 18, 8, 26, 15, 4, 24, 12, 1, 20, 10, 28,  
17, 6, 26, 15, 3, 22, 11, 29, 19, 7, 27, 15, 5, 23, 13, 1, 20, 9, 27,  
17, 6, 25, 13, 2, 22, 11, 29, 18, 8, 27, 16, 4, 24, 12, 30, 20, 9,  
28, 16, 6, 25, 14, 2, 21, 11, 30, 18, 7, 27, 16, 4, 23, 13, 30, 19,  
9, 29, 17, 5, 25, 14, 3, 21, 10, 30, 19, 7, 26, 16, 3, 23, 12, 2, 21,  
10, 28, 18, 6, 24, 14, 4, 22, 10, 30, 19, 8, 26, 15, 5, 22, 12, 1,  
21, 8, 27, 17, 7, 24, 13, 3, 23, 11, 29, 19, 8, 25, 15, 5, 24, 11, 1,  
20, 10, 27, 16, 6, 24, 13, 2, 22, 10, 28, 18, 8, 26, 14, 4, 24, 13,  
30, 20, 9, 27, 16, 5, 25, 13, 2, 21, 11, 29, 17, 7, 27, 15, ...

and A118662:

9, 9, 10, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 10, 9, 9, 9, 9, 9, 9, 9, 10,  
9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 10, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 9,  
9, 9, 9, 9, 10, 9, 9, 10, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 9, 9, 9, 9,  
9, 10, 9, 9, 10, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 9, 9, 9, 9, 9, 10, 9,  
9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 9, 9,  
9, 9, 9, 9, 9, 9, 10, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 10, 9, 9, 9, 9,  
9, 9, 9, 10, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 10, 9, 9, 9, 9, 9, 9, 9,  
10, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 10, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9,  
9, 9, 9, 9, 9, 10, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 9, 9, 9,  
9, 9, ...

I've stayed up way too late working on this and I hope I haven't made  
(further) silly assumptions/mistakes.

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