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WQ Archive 141 - 150

Question 141

Which is the shortest span of Hebrew years whose lengths in days give all of the possible remainders after division by 7?

Answer

10 years is the shortest span of Hebrew years whose lengths in days give all of the possible remainders after division by 7.

As shown in 34 Day Variance Hebrew Year Periods, dividing the possible lengths in days by 7 gives the remainders of 4, 5, 6, 0, 1, 2, and 3.

10 YEAR SPANS
    123 months =     3,632d  6h  339p
    124 months =     3,661d 19h   52p
M'+/-DAYSMOD 7dOCCURSM"+/-DAYSMOD 7dOCCURS
-2
     3,630d
4
   3,236
-2
         0d
0
       0
-1
     3,631d
5
  76,946
-1
         0d
0
       0
0
     3,632d
6
  22,866
0
     3,661d
0
 252,669
1
     3,633d
0
 114,680
1
     3,662d
1
  47,195
2
         0d
0
       0
2
     3,663d
2
 162,407
3
         0d
0
       0
3
     3,664d
3
   9,473
The maximum variance is 34 days


Correspondent Dr. John Stockton not only answered this question but also found a list of Hebrew year spans which also showed a complete set of remainders after diving the possible lengths in days by 7.

Thank you correspondent Dr. John Stockton for sharing your most fascinating result.


Question 142

Which unique property is most likely common to spans of Hebrew years whose lengths in days give all of the possible remainders after division by 7?

Answer

10 years is the shortest span of Hebrew years whose lengths in days give all of the possible remainders after division by 7.

One of the 10 year span's properties is that its maximum variance, that is, the difference between its longest possible number of days and its shortest possible number of days is 34 days.

As shown in Properties of Hebrew Year Periods - Part 2, 34 days is the absolute maximum difference between the shortest and longest possible lengths for any span of Hebrew years.

It is therefore quite interesting that 34 days may also be the unique property most likely common to all spans of Hebrew years whose lengths in days give all of the possible remainders after division by 7.

As shown by a number of entries taken randomly from 34 Day Variance Hebrew Year Periods, dividing the possible lengths in days by 7 gives the remainders of 4, 5, 6, 0, 1, 2, and 3.

10 YEAR SPANS
    123 months =     3,632d  6h  339p
    124 months =     3,661d 19h   52p
M'+/-DAYSMOD 7dOCCURSM"+/-DAYSMOD 7dOCCURS
-2
     3,630d
4
   3,236
-2
         0d
0
       0
-1
     3,631d
5
  76,946
-1
         0d
0
       0
0
     3,632d
6
  22,866
0
     3,661d
0
 252,669
1
     3,633d
0
 114,680
1
     3,662d
1
  47,195
2
         0d
0
       0
2
     3,663d
2
 162,407
3
         0d
0
       0
3
     3,664d
3
   9,473
The maximum variance is 34 days

237 YEAR SPANS
  2,931 months =    86,554d  4h  123p
  2,932 months =    86,583d 16h  916p
M'+/-DAYSMOD 7dOCCURSM"+/-DAYSMOD 7dOCCURS
-2
    86,552d
4
  11,645
-2
         0d
0
       0
-1
    86,553d
5
 167,294
-1
         0d
0
       0
0
    86,554d
6
  47,376
0
    86,583d
0
 117,938
1
    86,555d
0
 245,429
1
    86,584d
1
  22,866
2
         0d
0
       0
2
    86,585d
2
  74,774
3
         0d
0
       0
3
    86,586d
3
   2,150
The maximum variance is 34 days

660 YEAR SPANS
  8,163 months =   241,058d  5h  819p
  8,164 months =   241,087d 18h  532p
M'+/-DAYSMOD 7dOCCURSM"+/-DAYSMOD 7dOCCURS
-2
   241,056d
4
   5,792
-2
         0d
0
       0
-1
   241,057d
5
 189,047
-1
         0d
0
       0
0
   241,058d
6
  58,842
0
   241,087d
0
  55,772
1
   241,059d
0
 326,927
1
   241,088d
1
  11,433
2
         0d
0
       0
2
   241,089d
2
  39,518
3
         0d
0
       0
3
   241,090d
3
   2,141
The maximum variance is 34 days

1,993 YEAR SPANS
 24,650 months =   727,929d  3h  530p
 24,651 months =   727,958d 16h  243p
M'+/-DAYSMOD 7dOCCURSM"+/-DAYSMOD 7dOCCURS
-2
   727,927d
4
  12,730
-2
         0d
0
       0
-1
   727,928d
5
 182,846
-1
         0d
0
       0
0
   727,929d
6
  51,941
0
   727,958d
0
 100,300
1
   727,930d
0
 260,515
1
   727,959d
1
  19,055
2
         0d
0
       0
2
   727,960d
2
  60,966
3
         0d
0
       0
3
   727,961d
3
   1,119
The maximum variance is 34 days

The Weekly Question welcomes any finding of any year span that does not have a 34 day maximum variance but whose lengths in days give all of the possible remainders after division by 7.


Correspondent Dr. John Stockton answered this week's question and also provided another rather interesting observation.

Thank you correspondent Dr. John Stockton for sharing your most fascinating result.


Question 143

Which is the smallest span of years consisting of all of the possible single year lengths?

Answer

Not too surprisingly, 6 years is the smallest span of Hebrew years consisting of all of the single years that are possible in the Hebrew calendar.

The first such period of 6 years is found to begin with year 22H which began on
Shabbat 17 August -3739g.

The 6 consecutive year lengths were each 383, 355, 354, 385, 353, and 384 days long.

Correspondent Dr. John Stockton brought this intriguing observation to my attention as follows

I guess you've already asked about the shortest span of years that
contains all lengths, and the longest which does not ?

I find that it is possible for six consecutive years to contain all
possible year lengths; 

Correspondent Robert H. Douglass answered most correctly as follows:

A: Six consecutive years can include all six possible year lengths.

First occurrence: Hebrew Years 22 through 27.

22 -- 383d
23 -- 355d
24 -- 354d
25 -- 385d
26 -- 353d
27 -- 384d

Recent occurrences:  1918-1924 CE, 1945-1951 CE


Thank you correspondents Dr. John Stockton and Robert H. Douglass for sharing your answers.


Question 144

For the 6 year spans which contain 6 different single year lengths, in how many different ways can the single years be arranged over the full Hebrew calendar cycle of 689,472 years?

Answer

Not too surprisingly, 6 years is the smallest span of Hebrew years consisting of all of the single years that are possible in the Hebrew calendar.

The first such period of 6 years is found to begin with year 22H which began on
Shabbat 17 August -3739g.

The 6 consecutive year lengths were each 383, 355, 354, 385, 353, and 384 days long.

Rather surprisingly, there are only 4 ways in which the year lengths can be arranged in these 6 year spans.

These are the only 4 arrangements, showing also their 1st occurences, and the frequency of these arrangements over the full Hebrew calendar cycle of 689,472 years.

       1st year         Arrangements       Freq
    -------------- ----------------------- ----
      22H (-3739g) 383 355 354 385 353 384 3806
     188H (-3573g) 383 354 385 353 355 384 3806
     242H (-3519g) 383 354 355 385 353 384 2746
     340H (-3421g) 383 354 385 355 353 384 2746
247 Hebrew Year Periods shows that the possible lengths for any span of 6 Hebrew years are

6 YEAR SPANS
     74 months =   2,185d  6h  362p
     75 months =   2,214d 19h   75p
M'+/-DAYSMOD 7dOCCURSM"+/-DAYSMOD 7dOCCURS
-2
       0d
0
       0
-2
       0d
0
       0
-1
   2,184d
0
 192,950
-1
       0d
0
       0
0
   2,185d
1
  64,125
0
   2,214d
2
  52,131
1
   2,186d
2
 255,205
1
   2,215d
3
  56,716
2
   2,187d
3
  32,040
2
   2,216d
4
  32,952
3
       0d
0
       0
3
   2,217d
5
   3,353
The maximum variance is 33 days

The length of the 6 Hebrew years which satisfy the conditions of this Weekly Question is
2214 days, and contain the extra leap month in their length.

Consequently, these periods also satisfy the conditions shown in 27a. Extra Month Period Starts and must always start with a leap year and end with a leap year.

As predicted by Property 27a (above) the possible leap year starts are years 3, 6, 14, and 17 of the GUChADZaT mahzor katan (19 year cycle).

In his reply to the Weekly Question Dr. John Stockton also noted that these 6 year periods all began an equal number of times at each one of these 4 leap years.

However, it is not presently known why these 6 year spans always begin on Shabbat.

Correspondents Dr. John Stockton and Winfried Gerum provided these excellent analyses:-

Dr. John Stockton:-

Programmed in haste; I may have the question wrong.  Four orders are
found; each has a descriptor X, and the years for the first of each
order are shown.

HEBCLNDR   jrs@merlyn.demon.co.uk   >= 2001-10-30
 689472 Hebrew years is always 251827457 days; 35975351 weeks plus 0 
days;
 36288 19-year leap-cycles.  Year lengths 353..385 map to 1..6 as 
needed.
 Sextuplets     
 X = 5851  5706 383  5707 354  5708 385  5709 355  5710 353  5711 384
 X = 5941  5733 383  5734 355  5735 354  5736 385  5737 353  5738 384
 X = 5841  5801 383  5802 354  5803 385  5804 353  5805 355  5806 384
 X = 5761  5855 383  5856 354  5857 355  5858 385  5859 353  5860 384
    Total = 13104
 X =  5761  2746 times
 X =  5841  3806 times
 X =  5851  2746 times
 X =  5941  3806 times


Each of the four possible orders can start either in the first half or
the second half of the 19-year cycle; and does so equally often.
X is 5500 smaller now, for convenience.

 Sextuplets covering all lengths - X is a sequence "label", first 
observed:
 X=351  5706 383  5707 354  5708 385  5709 355  5710 353  5711 384
 X=441  5733 383  5734 355  5735 354  5736 385  5737 353  5738 384
 X=341  5801 383  5802 354  5803 385  5804 353  5805 355  5806 384
 X=261  5855 383  5856 354  5857 355  5858 385  5859 353  5860 384
                  Starting year can be year of cycle
        0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18    times
 X=261           +                                +              1373  
1373
 X=341                    +                                +     1903  
1903
 X=351                    +                                +     1373  
1373
 X=441           +                                +              1903  
1903
  Starting day can be  Sun  Mon  Tue  Wed  Thu  Fri  Sat
 X=261   2746 times                                   +
 X=341   3806 times                                   +
 X=351   2746 times                                   +
 X=441   3806 times                                   +
 Total  13104 times in 689472 years, of which a full sextuplet starts in
 year  3 of a cycle 3276 times
 year  6 of a cycle 3276 times
 year 14 of a cycle 3276 times
 year 17 of a cycle 3276 times
 
--------------------------------------------------------------------

Winfried Gerum:-

all possible year lengths that may occur in six consecutive years:
there are only four possible sequences of six consecutive years that
encompass all six possible year lengths. All of these begin with a
deficient leap (383) year and end with a regular leap year (384).

sequence: 383,354,355,385,353,384
occurs 2746 times in a full calendar cycle.
first    occurence:  242.. 247
previous occurence: 5608..5613
next     occurence: 5855..5860

sequence: 383,354,385,353,355,384
occurs 3806 times in a full calendar cycle.
first    occurence:  188.. 193
previous occurence: 5554..5559
next     occurence: 5801..5806

sequence: 383,354,385,355,353,384
occurs 2746 times in a full calendar cycle.
first    occurence:  340.. 345
previous occurence: 5706..5711
next     occurence: 5953..5958

sequence: 383,355,354,385,353,384
occurs 3806 times in a full calendar cycle.
first    occurence:   22..  27
previous occurence: 5733..5738
next     occurence: 5980..5985



Thank you correspondents Dr. John Stockton and Winfried Gerum for sharing your amazing analyses.

Dr. John Stockton made another very interesting observation...


Question 145

What is the largest possible number of consecutive Hebrew years which do not include all of the single Hebrew year lengths?

Answer

The largest possible number of consecutive Hebrew years which do not include all of the single Hebrew year lengths is 43.

Measured from 1 Tishrei to 1 Tishrei the very first such span of 43 years begins at
126H (Mon 6 Sep -3635g).

The arrangement of the 43 lengths of days in that span is

 355  355  383  354  355  385  354  383  355  354  383
 355  354  385  353  385  354  355  383  354  355  385 
 353  354  385  355  383  354  355  383  354  355  385 
 354  383  355  354  385  353  354  385  355  353  

The 384 day year is the only year length missing in those 43 years. However, it finally reappears as the 44th year.

Correspondents Dr. John Stockton, Winfried Gerum, and Dwight Blevins all found the correct answer.

In particular, I am grateful to Dr. John Stockton who first brought out this intriguing phenomenon to my attention, and sent a number of interesting messages portions of which will be quoted in subsequent issues of the Weekly Question.

From Dwight Blevins:-

Well....now that you peaked my curiosity, I found a span from 1853 to
1897 which does not contain a 384 year day.  

The year 1847 seems to begin a six year minimum span, which couples to
the 44 year maximum leading to 1897, yielding a total of 50 years, or a
perfect seven octave Jubilee.

Do you think these maximum-minimum spans are often or always grouped
together?

-----------------------

From Winfried Gerum:-

The longest periods that do not contain all six types of years are 43 
years.

Within a full calendar cycle there are 2149 such periods.


Thank you correspondents Dr. John Stockton, Winfried Gerum, and Dwight Blevins for sharing your amazing analyses.

Dwight Blevins observed that a particular instance of the deficient 43 year periods seemed to have been preceded by an occurrence of a complete 6 year period and wanted to know if that was or was not a general rule for the deficient 43 year periods.


Question 146

Are all 43 year periods missing a 384 day year preceded by 6 year periods containing all six possible year lengths?

Answer

NO!

The largest possible number of consecutive Hebrew years which do not include all of the single Hebrew year lengths is 43.

It is interesting to note that of the 2,149 such spans, 1074 are preceded by 6 year spans which include all of the possible single year lengths, while the remaining 1,075 spans are
preceded by 6 year spans which do not contain all 6 possible single year lengths.

Thank you correspondent Dwight Blevins for having asked this very interesting question.


Question 147

How many different arrangements are possible for the years in the longest 43 Hebrew year spans excluding at least one single year length?

Answer

Considering the number of ways that 43 years may be arranged in terms of their lengths it is surprising to dicover that only 8 possible arrangements are possible for the years in the longest 43 Hebrew year spans excluding at least one single year length.

The following identifies the 8 possible arrangements and their first occurence in the full Hebrew calendar cycle of 689,472 years.


Pattern 1 first begins at      126H (  -3,635g)
Pattern 1       occurs          535 times
Pattern 1 has a length of    15,681 days

 355  355  383  354  355  385  354  383  355  354  383 
 355  354  385  353  385  354  355  383  354  355  385 
 353  354  385  355  383  354  355  383  354  355  385 
 354  383  355  354  385  353  354  385  355  353  

Pattern 2 first begins at    1,757H (  -2,004g)
Pattern 2       occurs          539 times
Pattern 2 has a length of    15,681 days

 355  355  383  354  355  385  354  353  385  354  383 
 355  354  385  353  354  385  355  383  354  355  383 
 355  354  385  355  354  383  355  383  354  355  385 
 354  353  385  354  383  355  354  385  353  355  

Pattern 3 first begins at    2,498H (  -1,263g)
Pattern 3       occurs          295 times
Pattern 3 has a length of    15,681 days

 355  355  383  354  355  383  354  355  385  354  383 
 355  354  385  353  354  385  355  383  354  355  383 
 355  354  385  355  354  383  355  383  354  355  385 
 354  353  385  354  383  355  354  385  353  355  

Pattern 4 first begins at    2,745H (  -1,016g)
Pattern 4       occurs          236 times
Pattern 4 has a length of    15,681 days

 355  355  383  354  355  383  354  355  385  354  383 
 355  354  385  353  354  385  355  383  354  355  383 
 354  355  385  353  354  385  355  383  354  355  385 
 354  353  385  354  383  355  354  385  353  355  

Pattern 5 first begins at    3,242H (    -519g)
Pattern 5       occurs          241 times
Pattern 5 has a length of    15,681 days

 355  355  383  354  355  385  354  383  355  354  385 
 353  354  385  355  383  354  355  383  354  355  385 
 353  354  385  355  383  354  355  385  354  353  385 
 354  383  355  354  385  353  354  385  355  353  

Pattern 6 first begins at    3,489H (    -272g)
Pattern 6       occurs          294 times
Pattern 6 has a length of    15,681 days

 355  355  383  354  355  385  354  383  355  354  383 
 355  354  385  353  385  354  355  383  354  355  385 
 353  354  385  355  383  354  355  385  354  353  385 
 354  383  355  354  385  353  354  385  355  353  

Pattern 7 first begins at   75,040H (  71,279g)
Pattern 7       occurs            4 times
Pattern 7 has a length of    15,681 days

 355  355  383  354  355  383  354  355  385  354  383 
 355  354  385  353  354  385  355  383  354  355  383 
 355  354  385  353  354  385  355  383  354  355  385 
 354  353  385  354  383  355  354  385  353  355  

Pattern 8 first begins at   75,784H (  72,023g)
Pattern 8       occurs            5 times
Pattern 8 has a length of    15,681 days

 355  355  383  354  355  385  354  383  355  354  385 
 353  354  385  353  385  354  355  383  354  355  385 
 353  354  385  355  383  354  355  385  354  353  385 
 354  383  355  354  385  353  354  385  355  353  

Correspondents Winfried Gerum and Dr. John Stockton sent the following very well documented analyses:-

From Winfried Gerum in response to Q145

The longest periods that do not contain all six types of years are 43 
years.

Within a full calendar cycle there are 2149 such periods.

All these periods commence on a Monday.
All these periods have
  an equal number (15) of leap      years (383, 385 days),
  an equal number (11) of deficient years (353, 383 days)
  an equal number (12) of normal    years (354      days)
  an equal number (20) of full      years (355, 385 days) and
  none of these periods has a normal leap year (384 days).
Hence all these periods are of equal length (15681 days).

These sequences start almost equally often
   on the  9th (1074)
or on the 12th (1075) year of a mazor katan.
Subsequent periods usually start on the same place in a mazor katan,
and their beginnings are 247 years apart.
Usually after 6 periods, sometimes after five periods there are more 
than 247
years between (the beginnings of) two periods and the beginning switches 
to
a different place in the mazor katan.

There are eight different sequences of years that do not contain all six
types of year: These occur with the folling frequencies:
(1=deficient year,      2=normali year, 3=full year
 4=deficient leap year, 5=,    6=full leap year)

types of year:                                                 
frequency:
33423 4236243 26 126 3423 432 6 126 3 423 62 162432 6 13 occurs   4 
times
33423 4236243 26 126 3423 423 6 126 3 423 62 162432 6 13 occurs 236 
times
33423 4236243 26 126 3423 432 6 324 3 423 62 162432 6 13 occurs 295 
times
33423 6216243 26 126 3423 432 6 324 3 423 62 162432 6 13 occurs 539 
times

33423 6243243 26 162 3423 612 6 342 3 423 62 432612 6 31 occurs 535 
times
33423 6243243 26 162 3423 612 6 342 3 621 62 432612 6 31 occurs 294 
times
33423 6243261 26 342 3423 612 6 342 3 621 62 432612 6 31 occurs 241 
times
33423 6243261 26 162 3423 612 6 342 3 621 62 432612 6 31 occurs   5 
times 

The first six years of a period are always the same sequence of year 
types.

-------------------------------------------------------------------------------

From Dr. John Stockton

Lines starting + are added comment; the rest is from the program.

+ as before :
 Doing  Run   Min First Count   Max First Count   SMTWTFS   Span
                6    22 13104    43   126  2149   .^.....  15681
+             Yrs    AM         Yrs    AM         Weekday   Days


        Prev  First   Pattern ... Last Next
Min  6     3     22   432615     27   3
Max 43     5    126   3342362432432616234236126342342362432612631    168   
5
+  Yrs        Start                                                  End 

+  Number    1st HY   Year Lengths, 1=Min to 6=Max
        5*    75784   3342362432612616234236126342362162432612631
        4*    75040   3342342362432612634234326126342362162432613
      294*     3489   3342362432432616234236126342362162432612631
      241*     3242   3342362432612634234236126342362162432612631
      236*     2745   3342342362432612634234236126342362162432613
      295*     2498   3342342362432612634234326324342362162432613
      539*     1757   3342362162432612634234326324342362162432613
      535*      126   3342362432432616234236126342342362432612631

Correspondent Winfried Gerum also pointed out in his answer that these sequences start 1,074 times on the 9th year of the mahzor katan (19 year cycle) and 1,075 times on the 12th year of that cycle.

It is interesting to note that of the 2,149 such spans, 1074 are preceded by 6 year spans which include all of the possible single year lengths, while the remaining 1,075 spans are preceded by 6 year spans which do not contain all 6 possible single year lengths.

Thank you correspondents Winfried Gerum and Dr. John Stockton for having shared your penetrating analyses.


The next question first appeared as Weekly Question 27.


The question was asked by Rabbi Steven Saltzman of the Adath Israel Congregation in Downsview, Ontario.

The question involves the frequency of a particular Torah reading on Shabbat Hanukah.

One of the more prevalent practices, among the Jewish people, is that of reading the entire Mosaic text of their scriptures (Torah) over the course of one Hebrew year. At Simchat Torah, the last few verses are read, and then the entire cycle is repeated once again from Bereshit (Genesis).

The scriptural readings are divided into contiguous weekly portions, which are read in their entirety each Shabbat morning. Each division is known as a Parshah or Sedrah. These portions are arranged so as to be completely read over the course of one Hebrew year.

Each portion is given a special name. The two portions whose readings tend to coincide with Shabbat Hanukah are Vayyeshev and Miketz.

These portions are Genesis 37:1 to 40:23 and Genesis 41:1 to 44:17 respectively.

Since one of these two portions will always be read on Shabbat Hanukah, Rabbi Saltzman asked the following question.


Question 27

How often does the reading of Parshah Vayyeshev coincide with Shabbat Hanukah?

Answer

Since there are 14 ways of laying out the Hebrew years (14 keviyyot), there exist only 14 ways of dividing the annual Torah reading cycle. As a result, the 14 different divisions can be easily tabulated in very compact form.

One such tabulation may be found at the back of certain editions of the Humash (Pentateuch) as translated by Alexander Harkavy, and published by the Hebrew Publishing Co. in New York (1928).

Shabbat Hanukah is any Shabbat which occurs anywhere from Kislev 25 through Tevet 2 or 3 (if the year is deficient, ie, 353 or 383 days).

From the Torah reading tables, it can be easily found that Parshah Vayyeshev is read on Shabbat Hanukah only when the preceding Rosh Hashannah began on Shabbat!

Since exactly 2/7 of all of the Hebrew years begin on Shabbat, Parshah Vayyeshev is read on Shabbat Hanukah in two out of every seven years, or on 28.57% of all of the Hanukah's.

Correspondent Larry Padwa sent in the correct answer as follows:-

If I am incorrect about it being in years in which RH begins on Shabbat,
then I await your answer on Thursday.

However if I am correct about that, but it is the frequency with which 
you take issue, then please realize that the 29% is a rounding (to the 
nearest full per cent) of the fraction 196992/689472.

Thank you Larry Padwa for sharing your excellent analysis.


Question 148

Is it possible to calculate the length of a single Hebrew year using no more than the postponement rule Lo ADU Rosh?

Answer

YES!

Pages 91-92 of Calendric Calculations by Nachum Dershowitz and Edward M. Reingold (Cambridge University Press 1997) shows a remarkable strategy that can be used to eliminate the 356 and 382 day years without looking at the postponement rules GaTaRaD and BeTUTeKaPoT.
(See Properties of Hebrew Year Periods - Part 1 for a complete analysis of these two rules).

The authors of the book suggest that if we wish to find the length of Hebrew year H, then the Tishrei moladot for Hebrew years H-1, H, and H+1 must first be calculated.

Then the Molad Zakein postponement rule is bypassed by adding 6 hours to each of those Tishrei molad times.

Dechiyyah Lo Adu Rosh is then applied by adding 1 to the integer portion of any of the moladot whose remainder, after dividing by 7 is either 1, 4, or 6.

Let D0 = the resulting number of days up to year H-1
Let D1 = the resulting number of days up to year H 
Let D2 = the resulting number of days up to year H+1 
The next step requires that
2 days be added to D1 IF D2-D1 = 356  
1 day  be added to D1 IF D1-D0 = 382  
0 days be added to D1 OTHERWISE 
After these steps the length of year H is NOT ALWAYS given by D2-D1.


The Weekly Question tested this algorithm with the years 5765H (Thu 16 Sep 2004g) and 5788H (Sat 2 Oct 2027g) and discussed the results with author Nachum Dershowitz who very kindly and patiently explained the obvious.

As a result, the Weekly Question invites its readers to experiment with this algorithm and to suggest what that obvious matter should be.


Question 149

In order to always determine the length of a single Hebrew year, what would be the obvious steps needed in the algorithm shown on pages 91-92 of the book Calendrical Calculations by Nachum Dershowitz and Edward Reingold?

Answer

Pages 91-92 of Calendric Calculations by Nachum Dershowitz and Edward M. Reingold (Cambridge University Press 1997) shows a remarkable strategy that can be used to eliminate the 356 and 382 day years without looking at the postponement rules GaTaRaD and BeTUTeKaPoT.
(See Properties of Hebrew Year Periods - Part 1 for a complete analysis of these two rules).

The authors of the book suggest that if we wish to find the length of Hebrew year H, then the Tishrei moladot for Hebrew years H-1, H, and H+1 must first be calculated.

Then the Molad Zakein postponement rule is bypassed by adding 6 hours to each of those Tishrei molad times.

Dechiyyah Lo Adu Rosh is then applied by adding 1 to the integer portion of any of the moladot whose remainder, after dividing by 7 is either 1, 4, or 6.


Let D0 = the resulting number of days up to year H-1
Let D1 = the resulting number of days up to year H 
Let D2 = the resulting number of days up to year H+1 
The next step requires that

2 days be added to D1 IF D2-D1 = 356  
1 day  be added to D1 IF D1-D0 = 382  
0 days be added to D1 OTHERWISE 
After these steps the length of year H is NOT ALWAYS given by D2-D1.

However, the starting day of year H will always be correctly determined.

If the molad of Tishrei

is between  9h 204p and 17h 1079p on Tuesday at the start of a 12 month year 
or between 15h 589p and 17h 1079p on Monday  at the end   of a 13 month  year 

then the 6 hour addition to the time of the molad of Tishrei will not trigger the postponements needed for these years.

That is why length of year H is NOT ALWAYS given by D2-D1.

In order to determine the length of year H, it is then needed to develop the start day of year H+1.

So the OBVIOUS step omitted by the authors was simply to apply their algorithm to year H so as to determine its correct starting day D1 and then once again to year H+1 so as to determine its correct starting day D2.

The Weekly Question tested this algorithm with the years 5765H (Thu 16 Sep 2004g) and 5788H (Sat 2 Oct 2027g) and discussed the results with author Nachum Dershowitz who very kindly and patiently explained the obvious.

As a result, the Weekly Question invites its readers to experiment with this algorithm.

Correspondent Robert H. Douglass shared the following absolutely correct answer.


I enjoyed your discussion of a simplified approach to determining the
length of a Hebrew Year.  The special delay rules are only needed to
avoid any occurrance of a year length of 356 or 382 days.

Obviously, when looking at a particular Molad of Tishrei and comparing
it to the preceding and following years, the actual lengths of the years
in question CANNOT BE KNOWN until we know whether either of those years
in turn may have been affected by the special delay rules.

So it is necessary to inspect not only years H, H+1, and H-1, but also
H+2 and H-2.

For example, HY 5766 begins on Tuesday, October 4, 2005 with a special
delay to avoid the previous year's having only 382 days.  This needs to
be known to determine the length of year HY 5766.

Likewise, HY 5789 begins on Thursday, September 21, 2028 with a special
delay to avoid the FOLLOWING year's having 356 days.  This needs to be
known to determine the length of year HY 5788.

If we only require the interval from H to H+1 (and not that from H-1 to
H), then we must know (as a minimal requirement) the Molads of H-1, H,
H+1, and H+2... four data points for full confidence concerning a single
one-year interval.

Robert H. Douglass

Thank you Robert H. Douglass for this very well expressed solution.


Two year ago, while the world anxiously awaited the much ballyhooed Y2K phenomenon on January 1, 2000g, exactly one week later, a very little known but highly significant event took place in the Hebrew calendar .

Question 150

What was the highly significant Hebrew calendar event that took place on Sat 8 Jan 2000g?

Answer

Understanding the Molad Zakein Rule explains that Dehiyah Molad Zakein was introduced so as to eliminate the problem of a calculated molad having a time exceeding the first day of a new month.

In the absence of Dehiyah Molad Zakein this particular excess, termed the Overpost by Hebrew Calendar Science and Myths, occurs in some of the 383 and 384 years at the start of the months of Kislev and Shevat. The overpost on the 2nd day of these months can be as high as 5h 422p.

Absent Dehiyah Molad Zakein the Overpost can occur in 383 and 384 day years whose subsequent molad of Tishrei exceeds 18h 656p, such as leap year 5760H (Sat 11 Sep 1999g) which ended with a molad of Thu 19h 310p.

If Dehiyah Molad Zakein had not been applied to the start of 5761H, then Rosh Hodesh Shevat 5760H would have taken place on Thur 6 Jan 2000g and its molad would have occurred 733p past the start of the second day of Shevat.

Therefore, it was highly significant, that as a result of Dehiyah Molad Zakein applied to the start of 5761H, Rosh Hodesh Shevat occurred on Sat 8 Jan 2000g, and its molad took place prior to the onset of the 2nd day of Shevat.

In his recently published work Calendar and Community, author Sacha Stern provides good historical grounds to suggest that Dehiyah Molad Zakein might have entered the fixed Hebrew calendar calculations sometimes in the 9th century c.e. His research into medieval Hebrew calendar artefacts does not appear to indicate a prior presence of that rule.

Correspondent Dennis Kluk pleasantly surprised the Weekly Question with another significance for Sat 8 Jan.

Your question asks what is significant about the date Sabbath, 8 January 2000, 1 Shebat, 5760. This date was Julian date 2451552. No other day was Sabbath and 8 January and 1 Shebat simultaneously from 1785g through 2100g, a 316 year span. The 5 previous first of Shebat’s did match again within a century. Sunday, February 29, 1824g and 30 Adar 5584 is also unique over the same span of years.

Correspondent Winfried Gerum had this comment regarding Weekly Question 149:

It simply is not correct that Dehiyyah Lo ADU Rosh may be used as the
sole postponement rule.

Dershowitz and Reingold use all four dehiyyot, albeit some in disguise!

The addition of 6 hours to the molad time *is* the molad zakein rule.
Elimination of years with 356 days *is* fully equivalent to GaTaRad
Elimination of years with 382 days *is* fully equivalent to BeTUTeKaPoT.
:-(

Have a nice day.

Thank you correspondents Dennis Kluk and Winfried Gerum for your very interesting contributions to the Weekly Question.


The next question first appeared as Weekly Question 30.


Question 30

What was the Hebrew acronym given to the once used leap year distribution
3 5 8 11 14 16 19 ?

Answer

The surprising Hebrew acronym given to this leap year distribution was gimel-bet-tet-bet-gimel.

A trace of the answer to this question can be found on page 65 in the 1879 Sachau translation of the 1000g Al-Biruni work The Chronology of Ancient Nations.

The Hebrew numbering system traditionally used a letter of the alphabet to represent a given number. Thus the Hebrew letters alef through yud were used to represent the numbers
1 through 10.

In representing the leap year distributions, the traditional practice was to use only the first nine letters of the alphabet, it being understood that, once the years had passed the 10 mark, 10 would be subtracted and the remainder used to identify the letter to be used.

Hence, the current leap year distribution, 3 6 8 11 14 17 19, has an acronym formed from the Hebrew letters gimel, vov, het, alef, daled, zayen, tet and is usually known as GUChADZaT.

Consequently, it would be expected to see the Hebrew acronym for the leap year distribution
3 5 8 11 14 16 19 written as gimel-heh-het-alef-daled-vov-tet.

What the Al-Biruni work shows is that as early as 1000g calendar scholars gave this leap year distribution a palindromic acronym formed from the Hebrew letters gimel-bet-tet-bet-gimel. The acronym, in Hebrew, actually reads the same backwards as forwards.

Understanding that the difference between the last year of this cycle (19), and the first year of the following cycle (3) is 3 years, it becomes easy to see that the acronym was formed from the differences between successive leap years in that distribution. The tet, representing 9, was an economy derived from the fact that the middle 3 differences were all 3.

The Encyclopedia Judaica, in its article on the Calendar shows the Hebrew acronym for the
3 5 8 11 14 16 19 distribution to be the expected gimel-heh-het-alef-daled-vov-tet with the equivalent, but less economical, notation of gimel-bet-gimel-gimel-gimel-bet-gimel.


The next question first appeared as Weekly Question 31.


Question 31

How was the Hebrew leap year distribution of 3 5 8 11 14 16 19 different than the currently used distribution of 3 6 8 11 14 17 19?

Answer

Admittedly, the leap year distribution 3 5 8 11 14 16 19 does indeed look different than the current distribution of
3 6 8 11 14 17 19. However,these two leap year distributions are exactly the same.

Examining the differences between the years in both of these cycles, we get 2 3 3 3 2 3 3 for the first cycle,
and 3 2 3 3 3 2 3 for the second cycle. These two chains of numbers are circularly the same, as can be seen when the last number in each is then followed once again by the first number.

The actual diffference between these two 19 year distribution lies in the year counting method used.

The leap year distribution GUChADZaT is used for a Hebrew year counting system in which the beginning of each 19 year cycle is defined as 19 * M + 1.

The distribution 3 5 8 11 14 16 19 is used for a Hebrew year counting system in which the beginning of each 19 year cycle is defined as 19 * M - 2.

The modulo 19 difference between these 2 year counting systems is -3. So, if you subtract 3 from each of the years in the leap year cycle 3 6 8 11 14 17 19 (GUChADZaT), you derive 0 3 5 8 11 14 16.

Since 0 is 19 modulo 19, the number 19 can be substituted for 0, giving us
3 5 8 11 14 16 19. Hence, both leap year distributions are exactly the same.

Correspondent Dwight Blevins demonstrates this idea even more forcefully.

It is apparent that in order for years 3, 8, 11, 14, and 19 of both past
and present leap year patterns to over lay, there would have to be a
four year displacement (inclusive count) as to the philosophical debate
about when time began.

The Encyclopedia Judaica suggests there have been at least four
different leap year patterns, including the present pattern of years
3, 6, 8, 11, 14, 17, and 19 of the 19 year cycle.  The other patterns
were (one) 2, 5, 7, 10, 13, 16, and 18, (two) 1, 4, 6, 9, 12, 15, and
17, (three) the one quoted in question 30 of years 3, 5, 8, 11, 14, 16,
and 19.  All are based on a different opinion of when time began, which
in effect is merely reduced to a philosophical debate, since a change in
patterns of intercalation does not result in a change of the dates of
the lunar calendar as to a month and day Gregorian.  As is said, in the
final outcome, the result becomes transparent, because its effect
occurs only "on paper."

All this you already know.  I have provided a chart to demonstrate how
this works.  Please observe that the X's, which mark the leap years,
form a straight column, indicating that nothing was ever altered as much
as one part per minute in lunar calendar declarations due to this
philosophical discussion on the date of creation.

The far left column indicates years (BCE) suggested as the beginning of
time.  Since the 19 year cycle would obviously be based on the year of
creation, beginning with year one of the cycle, any later opinion of
when time began would alter the numbered year identity of the 19 year
cycle, making it appear that the declarations of Tishrei 1 have been
moved around to suit the particular pattern.  This is total myth as the
chart demonstrates.  Tishrei 1, or the beginning of a new year does not
move to suit the pattern, rather the pattern is moved (i.e., leap years
receive a new number name) or superimposed over the leap years.


             Lines of Time in Leap Year Patterns (BCE)  
      
3761= 1  2  3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 (present)
3760= 19 1  2 3 4 5 6 7 8  9 10 11 12 13 14 15 16 17 18 19 (past)
3759= 18 19 1 2 3 4 5 6 7  8  9 10 11 12 13 14 15 16 17 18 (past)
3758= 17 1819 1 2 3 4 5 6  7  8  9 10 11 12 13 14 15 16 17 (past)
Leap yrs.   X     X   X       X        X        X     X

Again, please observe that the X's for leap years are fixed, while the
year number of the 19 year cycle changes for any given start date.  Many
lunar calendar enthusiasts do not understand how this works, therefore
the need for your question. Lacking this basic understanding caused me
much grief and frustration in my early years of interest in lunar
calendar calculations.

Dwight Blevins

Thank you Dwight Blevins for this excellent demonstration.


For other Additional Notes click here.
For other Archived Weekly Questions click here.
Hebrew Calendar Science and Myths

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Remy Landau

 First  Begun 21 Jun 1998 
First  Paged  5 Nov 2004
Next Revised 12 Nov 2004