How often is Sedrah Vayyelech only partially read during the course of a given Hebrew year?

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 reading 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.

Correspondent Larry Padwa noted that in 5761H (2000g/2001g) Sedrah Vayyelech will only be partially read.

This portion is Devorim (Deuteronomy) 31:1 to 31:30.

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 Chumash (Pentateuch) as translated by Alexander Harkavy, and published by the Hebrew Publishing Co. in New York (1928g).

The first few verses of each Sedrah are generally read at the minchah (afternoon) Shabbat service, and also at the Monday and Thursday shacharit ( morning) services.

The Harkavy tabulations show that Sedrah Vayyelech is always read completely on the first Shabbat of a new Hebrew year whenever Rosh Hashannah begins on either Monday or Tuesday.

Consequently, this Torah portion is only partially read at minchah on the very last Shabbat of the preceding year, and cannot be be read completely until the next Shabbat which follows the inauguration of a new Hebrew year.

In the full Hebrew calendar cycle of 689,472 years Rosh Hashannah falls on Mondays 193,280 times and on Tuesdays 79,369 times.

Therefore, Sedrah Vayyelech is partially read in 39.5% of the years.

Why does the longest possible period of 120 Hebrew years always begin on Thursday?

The answer lies in the arithmetic rules of the Hebrew calendar and can be derived using relatively simple algebraic constructs and the concepts of modulus arithmetic.

The following analysis will be made easier by assuming the non-existence of the Molad Zakein rule. This strategy is mathematically permissible.

For Hebrew year H(1),

Let the molad of Tishrei be m(1) = d(1) + f,

where d(1) is the day of molad m(1) and f is the hour and parts of the molad on that day

Let D(1) = the day of Rosh Hashannah for year H(1).

Then D(1) = d(1) + p(1)

where p(1) is the number of days Rosh Hashannah is postponed for year H(1).

Similarly, for Hebrew year H(2),

Let the molad of Tishrei = m(2)
the day of the molad = d(2)
and let D(2) = the day of Rosh Hashannah for year H(2).

Then D(2) = d(2) + p(2).

where p(2) is the number of days Rosh Hashannah is postponed for year H(2).

Let the difference between m(2) and m(1) = A + a
where a represents the hour and parts of that difference.

Then m(2) = d(1) + f + A + a Hence the day of the molad for H(2) is

d(2) = d(1) + A + INT(f + a)

and D(2) = d(1) + A + INT(f + a) + p(2)

The length L for that span is L = D(2) - D(1)

Hence D(2) - D(1) = A + INT(f + a) + p(2) - p(1)

The maximum value for p(1) and p(2) is 2
The maximum value for INT(f + a) is 1

The minimum value for p(1) and p(2) is 0
The minimum value for INT(f + a) is 0

Consequently, the maximum possible value for the expression INT(f + a) + p(2) - p(1) is 3.

For the 120 Hebrew year period, the largest possible difference
m(2) - m(1) = 43,852d 22h 405p.

This difference modulus 7d is 4d 22h 405p.

To get the maximum number of days for the 120 year period, it is then necessary that
p(2) = 2 and that f + a > 1 + 15h 203p.

This is possible whenever D(2) is Thursday and f > 15h 204p.

In that case the expression D(2) - D(1) modulus 7d = 4 + 3. Which means that D(1) must also be Thursday.

Consequently, the longest possible period of 120 Hebrew years must always begin on Thursday.

Applying all of the rules of the Hebrew calendar, it is to be noted that whenever the molad of Tishrei occurs for a Hebrew leap year beginning on Thursday past 9h 204p it will be the starting day of the longest possible 120 Hebrew year span.

Can the analysis of the answer to Weekly Question 112 be used to explain the start of the longest and shortest possible 19 year periods?

YES! For that reason the analysis is repeated.

The answer lies in the arithmetic rules of the Hebrew calendar and can be derived using relatively simple algebraic constructs and the concepts of modulus arithmetic.

The following analysis will be made easier by assuming the non-existence of the Molad Zakein rule. This strategy is mathematically permissible.

For Hebrew year H(1),

Let the molad of Tishrei be m(1) = d(1) + f,

where d(1) is the day of molad m(1) and f is the hour and parts of the molad on that day

Let D(1) = the day of Rosh Hashannah for year H(1).

Then D(1) = d(1) + p(1)

where p(1) is the number of days Rosh Hashannah is postponed for year H(1).

Similarly, for Hebrew year H(2),

Let the molad of Tishrei = m(2)
the day of the molad = d(2)
and let D(2) = the day of Rosh Hashannah for year H(2).

Then D(2) = d(2) + p(2).

where p(2) is the number of days Rosh Hashannah is postponed for year H(2).

Let the difference between m(2) and m(1) = A + a
where a represents the hour and parts of that difference.

Then m(2) = d(1) + f + A + a Hence the day of the molad for H(2) is

d(2) = d(1) + A + INT(a + f)

and D(2) = d(1) + A + INT(a + f) + p(2)

The length L for that span is L = D(2) - D(1)

Hence D(2) - D(1) = A + INT(f + a) + p(2) - p(1)

The maximum value for p(1) and p(2) is 2
The maximum value for INT(f + a) is 1

The minimum value for p(1) and p(2) is 0
The minimum value for INT(f + a) is 0

Consequently,

the maximum possible value for the expression INT(f + a) + p(2) - p(1) is 3.

and the minimum possible value for the expression INT(f + a) + p(2) - p(1) is -2.

For the 19 Hebrew year period, the only possible difference
m(2) - m(1) = 6939d 16h 595p.

This difference modulus 7d is 2d 16h 595p.

To get the maximum number of days for the 19 year period, it is then necessary that p(2) = 2.

This is possible whenever

d(2) = 3 (ie Tuesday) and f + a > 1 + 15h 203p.

Hence, f > 1 + 15h 203p - 16h 595p (= 22h 698p).

In that case the expression D(2) - D(1) modulus 7d = 2 + 3.

Which means that D(1) must be Saturday (ie 0d) since D(2) = 5.

Consequently, the longest possible period of 19 Hebrew years must always begin on Saturday.

Since in this case D(2) is the result of a postponement from Tuesday, H(2) is a 12 month Hebrew year.

Since H(1) and H(2) are 19 years apart, H(1) must also be a 12 month Hebrew year.

Applying all of the rules of the Hebrew calendar, it is to be noted that whenever the molad of Tishrei occurs for a 12 month Hebrew year beginning on Saturday past 16h 698p it will be the starting day of the longest possible 19 Hebrew year span.

The analysis is just a little bit more complicated for the minimum length of the 19 Hebrew year period. So this will be dealt with next week.

How can the analysis of Weekly Question 113 be used to derive the minimum length of the 19 Hebrew year period?

The analysis is repeated.

The answer lies in the arithmetic rules of the Hebrew calendar and can be derived using relatively simple algebraic constructs and the concepts of modulus arithmetic.

The following analysis will be made easier by assuming the non-existence of the Molad Zakein rule. This strategy is mathematically permissible.

For Hebrew year H(1),

Let the molad of Tishrei be m(1) = d(1) + f,

where d(1) is the day of molad m(1) and f is the hour and parts of the molad on that day

Let D(1) = the day of Rosh Hashannah for year H(1).

Then D(1) = d(1) + p(1)

where p(1) is the number of days Rosh Hashannah is postponed for year H(1).

Similarly, for Hebrew year H(2),

Let the molad of Tishrei = m(2)
the day of the molad = d(2)
and let D(2) = the day of Rosh Hashannah for year H(2).

Then D(2) = d(2) + p(2).

where p(2) is the number of days Rosh Hashannah is postponed for year H(2).

Let the difference between m(2) and m(1) = A + a
where a represents the hour and parts of that difference.

Then m(2) = d(1) + f + A + a Hence the day of the molad for H(2) is

d(2) = d(1) + A + INT(a + f)

and D(2) = d(1) + A + INT(a + f) + p(2)

The length L for that span is L = D(2) - D(1)

Hence D(2) - D(1) = A + INT(f + a) + p(2) - p(1)

The maximum value for p(1) and p(2) is 2
The maximum value for INT(f + a) is 1

The minimum value for p(1) and p(2) is 0
The minimum value for INT(f + a) is 0

Consequently,

the maximum possible value for the expression INT(f + a) + p(2) - p(1) is 3.

and the minimum possible value for the expression INT(f + a) + p(2) - p(1) is -2.

For the 19 Hebrew year period, the only possible difference
m(2) - m(1) = 6939d 16h 595p.

This difference modulus 7d is 2d 16h 595p.

To get the least number of days for the 19 year period, it is then necessary that p(1) = 2.

This is possible whenever

d(1) = 3 (ie Tuesday) and f + a < 1.

These conditions cannot be satisfied since the two day postponement is triggered only if
f > 15h 203p, which would make f + a > 1d 7h 798p.

Therefore, 19 Hebrew year periods cannot be 6,937 days long.

The next larger value possible for INT(f + a) + p(2) - p(1) is -1.

In that case, D(2) - D(1) = A + INT(f + a) + p(2) - p(1) = 2 - 1 = 1

The only possible pairs of year start days to satisfy this equation is D(2) = 3 and D(1) = 2 .

Since f + a < 1 and a = 16h 595p, f < 7h 495p.

Since a > 15h 203p, f + a would trigger a two day postponement
unless D(2) is a leap year Tuesday.

Since H(1) and H(2) are 19 years apart, H(1) must also be a leap Hebrew year.

Since D(1) = d(1) + p(1), the day of the molad for year H(1) must be Sunday.

Applying all of the rules of the Hebrew calendar, it is to be noted that whenever the molad of Tishrei for a Hebrew leap year occurs prior to 1h 495p on Sunday the following Monday will be the start of the shortest possible 19 Hebrew year span.

What do the Tishrei moladot of the 11th, 13th, and 15th years of the mahzor katan (19 year cycle) have in common?

The following observations are not found in the prior literature on the Hebrew calendar.

All Tishrei moladot that are at the start of either the 11th, 13th, or 15th years of the mahzor katan (19 year cycle) known as GUChADZaT are repeated exactly 3 times within the full Hebrew calendar cycle of 689,472 years.

Here are a couple of examples

The molad of Tishrei 11H (-3750g) is 1d 11h 543p.

Within the 689,472 year cycle, that particular molad reoccurs two more times as the molad of Tishrei, once for Tishrei 190716H (186957g), the 13th year of the mahzor katan, and again for Tishrei 381421H (377665g), the 15th year of the mahzor katan.

Weekly Question 58 noted that

The moladot of Tishrei which correspond to the value 4d 20h 408p are for the Hebrew years

       117357H (Thu 15 Jan 113598g)
       308062H (Thu 27 Apr 304305g)
       616124H (Thu  3 Dec 612370g)

These Hebrew years are respectively, the 13th, 15th, and 11th years of the mahzor katan using the leap year distribution GUChADZaT.

Weekly Question 68 noted the fact that in the full Hebrew calendar cycle of 689472 years the moladot of Tishrei were repeated exactly either 3 or 4 times.

In the full Hebrew calendar cycle of 689472 years, how often will a molad of Tishrei be repeated if it is not for either the 11th, 13th, or 15th years of the mahzor katan (19 year cycle)?

The following observations are not found in the prior literature on the Hebrew calendar.

All Tishrei moladot that are at the start of either the 11th, 13th, or 15th years of the
mahzor katan (19 year cycle) known as GUChADZaT are repeated exactly 3 times within the full Hebrew calendar cycle of 689,472 years.

If the Tishrei moladot are at the start of any other year in the mahzor katan then they are repeated exactly 4 times in the full Hebrew calendar cycle of 689,472 years.

Weekly Question 69 showed that 3/19 of the Tishrei moladot were repeated exactly 3 times.

Let T = the number of moladot that are repeated exactly 3 times
Let F = the number of moladot that are repeated exactly 4 times
Then

  T +   F = 181,440 (the total number of possible moladot)
3*T + 4*F = 689,472 (the number of moladot of Tishrei in the full cycle)

Hence,   T = 36,288 which = 1/19 of 689,472
And    3*T = 108864 which = 3/19 of 689,472

Since the 11th, 13th, and 15th years of the mahzor katan represent exactly 3/19 of all of the possible years in the full Hebrew calendar cycle of 689472 years all other years of the
mahzor katan have their Tishrei moladot repeated exactly 4 times in the full calendar cycle.

The following observations are not found in the prior literature on the Hebrew calendar.

All Tishrei moladot that are at the start of either the 11th, 13th, or 15th years of the
mahzor katan (19 year cycle) known as GUChADZaT are repeated exactly 3 times within the full Hebrew calendar cycle of 689,472 years.

If the Tishrei moladot are at the start of any other year in the mahzor katan then they are repeated exactly 4 times in the full Hebrew calendar cycle of 689,472 years.

Surprisingly, the remaining moladot are grouped into 4 sets each aligned to 4 different years of the mahzor katan.

These sets, numbered relatively to GUChADZaT, are

 1 14 16 18
 2  4 17 19
 3  5  7  9
10 12  6  8

The LSUM tables found in the section Cycles and Moladot of the Additional Notes bring to light the phenomenon.

In the LSUM tables, the multiple moladot repetitions can be seen to be located at the offsets which contain molad values that are multiples of 5 halakim.

Correspondent Robert Douglass sent a very thorough analysis of this problem, and also noted some rather interesting facts. Here is part of that correspondence.

The analysis I sent in earlier today correctly identified (a priori)
the key repetition interval of 190705 years, and suggested that the  
next interval should be twice that (+13 and +26 cycles of 14669.6170
years), and that counting backward should also be allowed (full Calendar 
cycle is 47, so take 47 - 13 cycles of 14669.6170).

Right after sending this in, I realized that each resulting position
can be seen as its own starting position, allowing the matrix to be
extended further than I had anticipated.  Since x + 13 and x + 26 may
be valid repetition intervals, then x + 39 (and possibly x + 52) must
be considered as well, likewise from x - 13 one must consider x - 26,
x - 39 (and possibly x - 52) as well.

Once the full matrix is thus revealed, it is truly remarkable, just as
you had stated.  Positions 11, 13, and 15 in any 19-year cycle each
have exactly three occurrances of the identical Molad Tishrei (day, 
hour, and parts) somewhere in the full 689472-year Calendar repetition
cycle.  All other positions have exactly four occurrances.

Counting from the start of each series, going forward by 190705 years
adds two to the cycle position.  Identical values of Molad Tishrei
occur as follows over this single interval:

3 -- 5 -- 7 -- 9

6 -- 8 -- 10 -- 12

11 --  13   -- 15

14 -- 16 -- 18 -- 1

17 -- 19 -- 2 -- 4

This accounts for all 19 positions in the mahzor katan.

In my first submission today, I explained how the correct number of 
extra months was necessary to each repetition.  This becomes clearer
now that the full pattern has been revealed.

Each "ending position" (9, 12, 15, 1, 4) is a non-leap year occurring
two years before a leap year.  So going forward from here (eg from 
position 1 to 3) fails to insert the extra month required for the
repetition of Molad Tishrei to occur.  The insertion later in year 3
does not affect Tishrei of year 3.

Each "starting position" (3, 6, 11, 14, 17) is a leap year occurring
two years after a  non-leap year.  So each of these is the end of the same
failed intervals just listed (eg position 1 to 3).

All other positions do insert the extra leap year. Eg from position 2 to 4,
and also from 3 to 5, the interval benefits from the leap year in year 3, 
likewise from 6 to 8 because year 6 is a leap year. (The leap year in year 
8 comes after Tishrei of year 8 so doesn't count here).  

As we see, each chain has four members except for 11 - 13 - 15, with only 
three.  This chain cannot extend in either direction, because position 9 is 
an ending position, and position 17 is an initial position. (All the ending 
and starting positions match this way by two's, representing the five failed 
intervals 1 - 3, 4 - 6, 9 - 11, 12 - 14, and 15 - 17 over which the exact 
repetition does not occur.

It is directly relevant that position 9 is the latest occurrance of Tishrei in 
the season, and position 17 is the earliest.  This symmetry of 11, 13, and 15 
occurring between the earliest and latest seasonalities was the only 
distinguishing feature of the three positions that I could discover during the 
week, but I could not see how to develop any interesting conclusions from 
this.  Now however you have shown us a much richer view of the relevant 
patterns.

Thank you Robert Douglass for this fascinating insight into the years of the mahzor katan.

120 Hebrew years can be either as short as 43,822 days or as long as 43,855 days. Is this 33 day difference the longest period of time between the lower and upper duration of any period of Hebrew years?

For any period of H Hebrew years, what is the maximum possible difference of days between its shortest and longest durations?

The following observations are not found in the prior literature on the Hebrew calendar.

120 Hebrew years can be either as short as 43,822 days or as long as 43,855 days. However, this 33 day difference between its lower and upper duration is not the largest possible number of days for any other period of Hebrew years.

For some given period of H Hebrew years

Let its lunar length be A + a
where A is the integral number of days in that length
and   a is the outstanding fraction of a day in that length

Then Ls, the shortest possible INTEGRAL length of that
period,  is A - 2 days.

If the H years are not a multiple of 19 years, then
the alternate length includes an additional month and the
longer lunar length is A + a + 29d 12h 793p.

Then Lm, the maximum possible INTEGRAL length of that
longer period, is A + 29 + 3 days = A + 32 days.

The difference between the maximum length and the shortest possible
length is Lm - Ls = A + 32 - (A - 2) = 34 days.

Therefore, for any period of H Hebrew years, the maximum possible difference of days between its shortest and longest durations is 34 days.

Correspondent Robert Douglass confirmed this result as follows:

Concerning the maximum possible variation in length for any interval with 
a fixed whole number of Hebrew years.

Any interval of 19 years or a multiple thereof will contain a constant 
number of leap years, hence a constant number of months, limiting the 
variation to a few days.  This follows from the structure of the Hebrew 
Calendar.  Any other interval may vary in length by exactly one month, 
depending on the number of leap years included.  This can be quickly 
verified by examining intervals from 1 to 9 years, and considering the 
possible sequences of leap and common year allowed.  By subtracting from 
the fixed 19 year interval, the same can be shown for intervals from 
10 to 18 years.  By adding some number of 19-year intervals, all longer 
intervals can be shown to have the same property.

A given number of months will span that same number of Molads, whose 
interval is always constant.  The actual interval from Molad to Molad 
can vary by one day on the calendar, depending on the time of day the 
Molads occur.  Eg, if one follows the other by eight hours and some 
number of days, that eight hour excess may all occur on one day or may 
cover parts of two days, lengthing the Calendar interval by one day.

In considering the length of some number of years, only the dates of 
First Tishrei of the start and end of the interval need be considered.  
All year-lengths internal to the interval are irrelevant.  These first 
and last dates may occur on the same day as the respective Molad, or up 
to two calendar days later (depending which delay rules come into play).  
The two day variation at each end makes four days, plus the one day 
uncertainty concerning the Molad interval itself, resulting in five days 
maximum of variation, plus the one month which adds a maximum of thirty 
days to the Molad interval.  Therefore the theoretical maximum cannot 
exceed 35 days.

The possible variation in the days of the Molad interval is not 
independent of the positioning within the time frames necessary to 
produce the varying delays of the start and end dates, so the full 
35 day variation is not realized.  The observed maximum variation is 
34 days.  Careful inspection of the interplay of the time frames 
reveals that the 34 day limit cannot be exceeded.

First consider the extra month that must be added between the shortest 
and longest intervals. If that extra month (In Calendar days) extends 
from a Saturday to a Saturday, then only 28 days are added. (Eg, first 
Molad on Friday which counts as Saturday for the Calendar). From a Monday 
to a Thursday, 31 days are added. This occurs when the first Molad of the 
added month is naturally on a Monday, and the final Molad (29d 12h 793p 
later) is naturally on a Wednesday, or a Tuesday after 9h 204p (which 
becomes Thursday on the Calendar).  So the maximum number of Calendar 
days added by the extra month is 31 days... 29 days between Molads plus 
two days of Calendar delay, or 30 days between Molads plus one day of 
Calendar delay.  (Never can both aspects maximize together.)

Next consider the rest of the interval, without adding the extra month.  
From Molad to Molad, some constant interval is involved, and the number 
of Calendar days depends on the exact time of the first Molad.  Consider 
an interval that covers one day more than an exact number of weeks.  Then 
first and second Molads will occur on successive days of the week... eg 
Thursday-Friday, Friday-Saturday, etc.  This will give Calendar days 
Thursday-Saturday, Saturday-Saturday, etc.  The day-count variation here 
is two.  If the interval starts on a Saturday, it will cover an exact 
number of weeks.  If it starts on a Thursday, it is exactly two days longer.

The only way to get more variation is to let the interval cover some 
fraction of an additional day (as nearly every real interval must do).  
For example, starting early Monday might end late Monday, and starting 
late Monday might end on Tuesday, giving one day of variation.  Or starting 
early Monday might end early Tuesday, and starting late Monday might end 
late Tuesday (which becomes Thursday for the Calendar if not in a leap year). 
This gives two days of variation.  

The maximum variation that can be achieved this way is three days.  For 
example, a Molad interval Sunday to Monday would count as Monday to Monday 
for the Calendar, and the same Molad interval running from Monday to Tuesday 
would count as Monday to Thursday for the Calendar.  Three days variation in 
Calendar days from the same Molad interval.

The maximum variation therefore is three days (from the above) plus 31 days 
(for the extra month), giving a total of 34 days.

Thank you correspondent Robert Douglass for sharing your very thorough analysis.

What is the minimum period of Hebrew years that can have the maximum 34 day difference between its shortest and longest durations?

The smallest number of Hebrew years that can have the maximum 34 day difference between its shortest and longest durations is 10 years.

The shortest period of 10 Hebrew years is 3,630 days and occurs 3,237 times in the full Hebrew calendar cycle of 689,472 years.

The longest period of 10 Hebrew years is 3,664 days and occurs 9,473 times in the full Hebrew calendar cycle of 689,472 years.

Correspondent Robert Douglass confirmed this result as follows:

Here are the actual results found.

The smallest interval of interest is Ten Hebrew Years.

When a ten year interval starts on a Monday and ends on a Thursday,
and includes 4 intercalary months, it lasts for 3664 days.

When it starts on a Thursday and ends on a Monday, and includes
3 intercalary months, it lasts for only 3630 days... 34 days less.

Thank you correspondent Robert Douglass for sharing your very thorough analysis.

The morning services for the festival days that are neither the High Holidays nor Purim include the recitation of a set Psalms that collectively are known as Hallel.

Tachanun is a set of penitential formulations that are recited during the weekday morning services.

Correspondent Larry Padwa makes a very interesting calendrical observation related to these practices.

What is the only date in the Hebrew Calendar that is sometimes celebrated as a holiday (with Hallel added to the service), and sometimes as an ordinary day (with Tachanun included in the service).

Correspondent Larry Padwa suggested this question which requires some additional religious information in order to be properly understood.

The morning services for the festival days that are neither the High Holidays nor Purim include the recitation of a set Psalms that collectively are known as Hallel.

Tachanun is a set of penitential formulations that are recited during the weekday morning services.

The month of Kislev can be either 29 or 30 days long. When it is 29 days long, as happened in 5761H (2000g/2001g), then the last day of the 8 day festival of Hanukah coincides with the 3rd day of the month of Tevet, thus making it a festival day.

When the month of Kislev is 30 days, then the last day of Hanukah coincides with the 2nd day of Tevet. This causes the 3rd day of Tevet to be an ordinary, non-festive day.

In the entire Hebrew calendar, only the 3rd day of Tevet is either a festive or a non-festive day.

Thank you correspondent Larry Padwa for having provided this excellent observation.

The next question first appeared as Weekly Question 4.

According to Talmudic tradition, which high officers were barred from the Sod Haibbur?

The Sod Haibbur (the secret calendar council) also had to determine whether or not to declare a leap year and thereby intercalate an extra month.

The Talmud, in tractate Sanhedrin 18b, relates that due to this function both the King and the Cohein Gadol (the High Priest) had to be excluded from the Sod Haibbur.

The King was disqualified because he paid his armies on an annual basis and therefore would favour the leap years of 13 months.

The Cohein Gadol, on the other hand, would more likely favour the 12 month years. During the High Holidays, the Cohein Gadol had to immerse himself several times in fresh spring waters as part of the Temple rituals. And so, he probably would have preferred an earlier time of the year when these waters were a bit warmer.

The next question first appeared as Weekly Question 10.

In which year or years of the 19 year Hebrew calendar cycle is Rosh Hashannah most likely to advance by another day in the Gregorian calendar?

The last time that Rosh Hashannah advanced in the Gregorian calendar was at the start of 5576H corresponding to Thursday 5 October 1815g. The next time that this will happen will be at the start of 5975H corresponding to Thursday 6 October 2214g.

Both of these years inaugurate the 9th year of the 19 year Hebrew calendar cycle. Carefully examining the calendar drift tables shown in the Additional Notes, it becomes apparent that after a certain distance into the drift the advancement of the Hebrew year into the Gregorian year only occurs at the start of the 9th year of a 19 year cycle.

However, not every 9th year of the cycle will cause the Hebrew year to be advanced.

If the 9th year of a 19 year cycle seems to cause the latest arrival of Rosh Hashannah, it seems reasonable to conclude that some other year in the cycle might see the earliest possible arrivals of Rosh Hashannah.

The next question first appeared as Weekly Question 11.

In which year or years of the 19 year Hebrew calendar cycle is Rosh Hashannah most likely to occur at its earliest point in the Gregorian year?

The clue to the answer can be found under the topic of the Calendar Drift in the Additional Notes.

The first Rosh Hashannah shown is that for 17H corresponding to Monday 11 August -3744g.

The table also indicates that it was also the last occurrence of Rosh Hashannah on that date until 77,491H corresponding to Saturday 11 August 73,731g.

Closer to our times, the earliest Rosh Hashannah cannot occur any earlier than September 5. This coincidence last occurred for the year 5660H corresponding to Tuesday 5 September 1899g.

It won't happen again until 5774H corresponding to Thursday 5 September 2013g.

The whole of the 20th century missed out on a September 5 Rosh Hashannah!

Common to all these Hebrew years is that they are the 17th year of the 19 year cycle.

In point of fact, it is the 17th year of the 19 year cycle that will most likely host the earliest possible Rosh Hashannah's.

However, not all of those 17th years will see the earliest possible Gregorian date for the festival.

The next question first appeared as Weekly Question 13.

After the molad of Tevet 29,340H when next will BaHaRaD occur?

There are 181,440 months in the cycle of the moladot.

Hence, the third cycle will require the elapsed time of

               181,440 * 3 = 544,320 months

There are 235 months in every cycle of 19 Hebrew years.

Hence there are 544,320 / 235 = 2316 cycles of 19 years in those moladot.

2316 * 235 = 544,260 months which is 60 short of the total.

Also, 2316 * 19 = 44004 years.

Consequently, the molad after the 3rd molad cycle will begin 60 months after Tishrei 44005H
(Mon 17 Mar 40245g).

Since 44005H is the first year in a 19 year cycle, we count the remaning months as follows

12 + 12 + 12 + 13 + 12 = 49 months with 11 left over.

This means that the molad following immediately after the 3rd molad cycle will be at the the end of the 11th month in the year 44009H.

Since 44009H is not a leap year, the date of the molad we are looking for is

Correspondent Robert Douglass confirmed the answer as follows

This is a good one to do on a hand calculator.  For the recurrence of a given 
molad, it is necessary to wait 1080 x 24 x 7 = 181440 months (= number of 
possible molads).  This is 772 cycles of 19 years (at 235 months each) plus 
another 20 months... or just under 14670 years.  The year 29340 is twice that, 
so the next occurrance will be at three times the repetition interval... third 
repetition (fourth occurrance) since the start of the Calendar.

Three times 181440 is 544320 months, or 2316 cycles of 19 years plus 60 months.
Sixty months is just under five years, so the interval is just under 44009 
years.  Measured from BaHaRaD at the start of year one, this means just before 
the start of Hebrew Year 44010.  The initial five years in any 19-year cycle 
take precisely 61 months, so 60 months is just one month shy of that interval.  
So we come to the start of Elul, the 12th month, in the year 44009 (the fifth 
year in the 2317th 19-year cycle).

544320 months is 16074093 days (at 29.53059414 days per month).  The precise 
date of that Molad will be 2296299 weeks, 2 days, 5 hours, 204 parts.  This 
is Hebrew Day number 16074095.  By subtracting 1373121 (to calibrate to the 
beginning of the Julian Calendar), we get 14700974 days.  Every four years 
takes 1461 days.  There are 10062 of these units plus 392 days... therefore 
40248 years plus 392 days, or 40249 years plus 27 days, therefore answer is 
March 27, 40249 on the Julian Calendar.  For those who prefer the Gregorian 
date, subtract two days (calendar difference at start of first century CE) 
then add one day for each time the Gregorian calendar has omitted a century 
leap year.  40000 years is 400 centuries, x 3/4 makes 300 days difference 
there, plus the final two centuries (two more days), so add 302 days.

March 27 - 2 + 302 = March 327, 40249 = Jan 21, 40250 Gregorian.

The next molad of Tishrei (Hebrew Year 44010,start of year 6 in cycle) will be 
after the completion of 2316 cycles plus 61 months (5 years), that is, 44009 
years after start of Year 1, or 544321 months, that is, 
2296303 weeks 1 d 12 h 793 p... add BaHaRaD and get 3d 17h 997p... a Tuesday 
before 18h. (Start of a year 6 ie Leap Year so no delay rules come into play).  
First of Tishrei is on this date, so first of preceding Elul is 29 days 
earlier... 4 wks and 1 day, therefore 2296299 wks 2d... same day as the Molad, 
so the Molad in question does actually occur on Monday, the First day of Elul, 
Hebrew Year 44009.

Thank you Robert Douglass for sharing with us this thorough analysis of the correct answer.

The next question first appeared as Weekly Question 15.

What is the significance to the Hebrew calendar of the Gregorian date September 16?

At the present time the earliest that Rosh Hashannah can occur in the Gregorian calendar is September 5. The latest that Rosh Hashannah can occur in the Gregorian calendar is October 5.

In this interval, September 16 is the separating date between the Hebrew years that are leap and non-leap.

Any Hebrew year that begins prior to September 16 is a leap year, that is a year of 13 months.

Any Hebrew year that begins after September 16 is a non-leap year consisting of 12 months.

This great separating date is also subject to the Calendar Drift, which topic is discussed in the Additional Notes.

When Rosh Hashannah will begin no earlier than September 6, the separating date between the leap years and the non-leap years will become September 17.

However, that won't actually happen for another 90 years or so.

The table below shows the Gregorian dates for Rosh Hashannah between the 60 years 5751H (1990g) and 5811H (2050g), and is ordered by day and month.

                5774H Thu  5 Sep 2013g =385d
                5755H Tue  6 Sep 1994g =384d
                5793H Mon  6 Sep 2032g =383d
                5763H Sat  7 Sep 2002g =385d
                5782H Tue  7 Sep 2021g =384d
                5801H Sat  8 Sep 2040g =383d
                5809H Tue  8 Sep 2048g =384d
                5752H Mon  9 Sep 1991g =385d
                5771H Thu  9 Sep 2010g =385d
                5779H Mon 10 Sep 2018g =385d
                5790H Mon 10 Sep 2029g =383d
                5798H Thu 10 Sep 2037g =385d
                5760H Sat 11 Sep 1999g =385d
                5787H Sat 12 Sep 2026g =385d
                5806H Tue 12 Sep 2045g =384d
                5768H Thu 13 Sep 2007g =383d
                5757H Sat 14 Sep 1996g =383d
                5776H Mon 14 Sep 2015g =385d
                5795H Thu 14 Sep 2034g =385d
                5803H Mon 15 Sep 2042g =385d
                5773H Mon 17 Sep 2012g =353d
                5811H Sat 17 Sep 2050g =355d
                5762H Tue 18 Sep 2001g =354d
                5792H Thu 18 Sep 2031g =354d
                5770H Sat 19 Sep 2009g =355d
                5781H Sat 19 Sep 2020g =353d
                5800H Mon 19 Sep 2039g =355d
                5751H Thu 20 Sep 1990g =354d
                5759H Mon 21 Sep 1998g =355d
                5778H Thu 21 Sep 2017g =354d
                5789H Thu 21 Sep 2028g =354d
                5808H Sat 21 Sep 2047g =353d
                5797H Mon 22 Sep 2036g =353d
                5805H Thu 22 Sep 2044g =355d
                5767H Sat 23 Sep 2006g =355d
                5786H Tue 23 Sep 2025g =354d
                5794H Sat 24 Sep 2033g =355d
                5756H Mon 25 Sep 1995g =355d
                5775H Thu 25 Sep 2014g =354d
                5783H Mon 26 Sep 2022g =355d
                5802H Thu 26 Sep 2041g =354d
                5764H Sat 27 Sep 2003g =355d
                5810H Mon 27 Sep 2049g =355d
                5753H Mon 28 Sep 1992g =353d
                5791H Sat 28 Sep 2030g =355d
                5772H Thu 29 Sep 2011g =354d
                5750H Sat 30 Sep 1989g =355d
                5761H Sat 30 Sep 2000g =353d
                5769H Tue 30 Sep 2008g =354d
                5780H Mon 30 Sep 2019g =355d
                5799H Thu 30 Sep 2038g =354d
                5807H Mon  1 Oct 2046g =355d
                5758H Thu  2 Oct 1997g =354d
                5788H Sat  2 Oct 2027g =355d
                5777H Mon  3 Oct 2016g =353d
                5785H Thu  3 Oct 2024g =355d
                5766H Tue  4 Oct 2005g =354d
                5796H Thu  4 Oct 2035g =354d
                5804H Mon  5 Oct 2043g =353d

It is to be noted that September 16 is missing from this table.

The next question first appeared as Weekly Question 16.

Presently, which kind of Hebrew years begin on the Gregorian date
September 16?

At the present time the earliest that Rosh Hashannah can occur in the Gregorian calendar is September 5. The latest that Rosh Hashannah can occur in the Gregorian calendar is October 5.

In this interval, September 16 is the separating date between the Hebrew years that are leap and non-leap.

Any Hebrew year that begins prior to September 16 is a leap year, that is a year of 13 months.

Any Hebrew year that begins after September 16 is a non-leap year consisting of 12 months.

This great separating date is also subject to the Calendar Drift, which topic is discussed in the Additional Notes.

When Rosh Hashannah will begin no earlier than September 6, the separating date between the leap years and the non-leap years will become September 17.

However, that won't actually happen for another 90 years or so.

For the period extending from 5576H (1815g) to 5850H (2089g) Rosh Hashannah is seen to occur 10 times on September 16.

In that time, exactly half the Hebrew years beginning on September 16 are leap years, while the remainder are non-leap years.

It is to be noted that during this period all of the leap years started on September 16 are 383 days long, while none of the common years are 353 days long.

                5583H Mon 16 Sep 1822g =355d
                5602H Thu 16 Sep 1841g =354d
                5632H Sat 16 Sep 1871g =383d
                5670H Thu 16 Sep 1909g =383d
                5746H Mon 16 Sep 1985g =383d
                5754H Thu 16 Sep 1993g =355d
                5765H Thu 16 Sep 2004g =383d
                5784H Sat 16 Sep 2023g =383d
                5830H Mon 16 Sep 2069g =355d
                5849H Thu 16 Sep 2088g =354d

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