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WQ Archive 71 - 80

Weekly Question Archive 71 - 80

by Remy Landau


Question 71

Which molad of Tishrei is the second to be repeated in the full Hebrew calendar cycle of 689472 years?

Answer

In the full Hebrew calendar cycle of 689472 years, the second molad of Tishrei to be repeated is 2d 20h 385p corresponding to year 4H.

Rosh Hashannah 4H occurred on Tue 5 Sep -3757g.


The medieval scholar Al-Biruni claimed that 2 deficient years could not follow each other because there are more 30 day months than 29 day months in the Hebrew calendar's 19 year cycle.

As stated on page 153 lines 15-18, in Dr. E.C. Sachau's 1879g translation of Al-Biruni's year 1000g work The Chronology of Ancient Nations

The reason why two imperfect years cannot follow each other is this, that the perfect months among the months of the cycle (Enneadecateris) preponderate over the imperfect ones. For the small cycle comprises 6,940 days, i.e. 125 perfect months and only 110 imperfect ones.

Question 72

Is Al-Biruni correct in stating that 2 deficient years cannot follow each other because "the perfect months among the months of the cycle (Enneadecateris) preponderate over the imperfect ones"?

Answer

The medieval scholar Al-Biruni claimed that 2 deficient years could not follow each other because there are more 30 day months than 29 day months in the Hebrew calendar's 19 year cycle.

As stated on page 153 lines 15-18, in Dr. E.C. Sachau's 1879g translation of Al-Biruni's year 1000g work The Chronology of Ancient Nations

The reason why two imperfect years cannot follow each other is this, that the perfect months among the months of the cycle (Enneadecateris) preponderate over the imperfect ones. For the small cycle comprises 6,940 days, i.e. 125 perfect months and only 110 imperfect ones.

There is no real connection between the number of months in a 19 year cycle and the the inability of 2 deficient months to follow each other.

353 day years can begin only on Mondays or Saturdays. Two such years together would cause the 3rd year to begin either on Sunday or on Friday. So that cannot happen.

383 day years can begin on Mondays, Thursdays, and Saturdays. If such years are followed by a 353 day year, then the 3rd year could begin on either Tuesday, Friday, or Sunday.

The need exists to examine the possibility of 383+353 or 353+383 day years beginning on Monday.

The span of time of the moladot for 2 successive years of which one is leap, is 25*(29d 12h 793p) = 738d 16h 385p (= 3d 16h 385p).

Hence, the earliest possible molad for the beginning of the 3rd year would be
(0d 18h 0p) + (3d 16h 385p) = 4d 12h 385p.

On the other hand, a 383 day year beginning on Monday followed by a 353 day year would necessarily end on [2 + 383 + 353] modulo 7 = 3d, at least 1d 12h 385p too short of the earliest possible molad of Tishrei for that 3rd year.

Therefore two imperfect years cannot follow each other.

Question 73

Can this table be used to calculate the Tishrei moladot?

SUM OF YEAR LENGTHS
 1
 5d 21h  589p
 2
 3d  6h  385p
 3
 0d 15h  181p
 4
 6d 12h  770p
 5
 3d 21h  566p
 6
 2d 19h   75p
 7
 0d  3h  951p
 8
 4d 12h  747p
 9
 3d 10h  256p
10
 0d 19h   52p
11
 5d  3h  928p
12
 4d  1h  437p
13
 1d 10h  233p
14
 5d 19h   29p
15
 4d 16h  618p
16
 2d  1h  414p
17
 0d 22h 1003p
18
 5d  7h  799p
19
 2d 16h  595p

Answer

YES!

The table represents the offsets which must be added to any molad whose year is of the form 19*k+3.

For any year, first subtract 3. Then find the remainder after dividing the difference by 19. Use that remainder as an index to the table of offsets. Add the value that you get to the molad of the year specified by 19*k+3. The result will be the molad of Tishrei for the target year.

Suppose the molad was for the year 5760H. This year is of the form 19*k+3, and its molad is 6d 21h 801p. Suppose that you now wanted the molad for 5765H. Subtracting 3 from 5765H and then dividing by 19 leaves a remainder of 5. The molad corresponding to that index is 3d 21h 566p. Adding the two moladot together yields 3d 19h 287p as the molad of Tishrei for 5765H.

The problem with the method is that the calculations are limited to 19 years.

Question 74

Using the table in Question 73, how can the calculations be extended to more than 19 years?

Answer

SUM OF YEAR LENGTHS
 1
 5d 21h  589p
 2
 3d  6h  385p
 3
 0d 15h  181p
 4
 6d 12h  770p
 5
 3d 21h  566p
 6
 2d 19h   75p
 7
 0d  3h  951p
 8
 4d 12h  747p
 9
 3d 10h  256p
10
 0d 19h   52p
11
 5d  3h  928p
12
 4d  1h  437p
13
 1d 10h  233p
14
 5d 19h   29p
15
 4d 16h  618p
16
 2d  1h  414p
17
 0d 22h 1003p
18
 5d  7h  799p
19
 2d 16h  595p

The table represents the offsets which must be added to any molad whose year is of the form 19*k+3.

For any year, first subtract 3. Then find the remainder after dividing the difference by 19. Use that remainder as an index to the table of offsets. Add the value that you get to the molad of the year 3H, which is 3d 22h 876p.

Multiply the integer part of the quotient by 2d 16h 595p. Add that product to the sum above. Then apply the usual reductions of the parts to less than 1080, the hours to less than 24, and the days to less than 7.

The result will be the molad of Tishrei for the target year.

Suppose the molad was for the year 5765H. Subtracting 3 from 5765H and then dividing by 19 leaves a remainder of 5.

The molad corresponding to the index of 5 is 3d 21h 566p. Adding it to the molad for year 3H gives 20h 362p after reduction.

The integer quotient is 303.

303 * (2d 16h 595p) => 2d 22h 1005p after reduction.

Adding that value to the previous sum gives 20h 362p + 2d 22h 1005p
which after reduction is 3d 19h 287p, the molad of Tishrei 5765H.

Question 75

How often are both parshiyot Vayyeshev and Miketz read on Hanukah?

Answer

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.

One of these two portions will always be read on Shabbat Hanukah.

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 (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.

These tables also indicate that when the year is shelemah, that is, either 355 or 385 days long, and begins on Shabbat, then Hanukah will both begin and end on Shabbat. In those years then, parshah Vayyeshev will be read on the first Shabbat of Hanukah and parshah Miketz will be read on the second Shabbat. Hence both parshiyot Vayyeshev and Miketz are read on Hanukah in 127139/689472 = 18.44% of the years.

Question 76

Which year, or years, of the mahzor qatan (19 year cycle) cannot begin a 19 year period of 6,942 days?

Last Week's Question

Which year, or years, of the mahzor qatan (19 year cycle) cannot begin a 19 year period of 6,942 days?

Answer

Measured from the first day of Tishrei, 19 year periods can be either 6938, 6939, 6940, 6941, or 6942 days long.

The longest of the 19 year periods, 6942 days, cannot occur if the first year of the 19 year period is a LEAP year.

Correspondent Larry Padwa not only sent the correct answer, but he also backed it up with a mathematical proof, which will be shown next week. Larry Padwa's proof also touches on the next Weekly Question.

Thank you Larry Padwa for sharing with us your great insights.

Question 77

Measured from the 1st day of Tishrei, on which day, or days, of the week can the longest period of 19 years begin?

Answer

The longest possible periods of 19 years are launched from the closing 1h 21m 12p of Shabbat's molad of Tishrei.

Correspondent Winfried Gerum provided the correct answer.

The answer to Q77 is, that the longest 19-year periods (6942 days), as measured from Tishrei 1 to Tishrei 1, always commence on a shabbat.

Correspondent Larry Padwa not only provided the correct answer, but also proved that answer!

1) Since 6942 is congruent to 5 (mod 7), then if year x+19 begins 6942
days later than year x, then the day of the week that begins year x+19
must be 5 days later (or 2 days earlier) than that of year x.

Consider the four cases:

a) Year x begins on Monday. Then year x+19 must begin on Saturday. b) Year x begins on Tuesday. Then year x+19 must begin on Sunday. This is impossible. c) Year x begins on Thursday. Then year x+19 must begin on Tuesday. d) Year x begins on Saturday. Then year x+19 must begin on Thursday.

At this point, we are left with cases a, c, and d.

2) The molad of year x+19 is 2d 16h 595p later than molad of year x. (This is always the case).

Case a: If year x begins on Monday, then molad year x is no later than 2d 17h 1079p. (else Dehiyyah Molad Zaken would postpone the beginning of year x to Tuesday). Therefore molad of year x+19 would be no later than 5d 10h 595p which would mean RH of year x+19 would be Thursday. But case a) requires RH of x+19 to be Saturday, so case a) is impossible.

Case c: By reasoning exactly similar to case a), if year x begins on Thursday, then year x+19 would begin on Monday (not Tuesday as required by case c). Thus case c is impossible.

Case d: If year x begins on Saturday, then molad year x is no later than 0d 17h 1079p (else Dehiyyah Molad Zaken would postpone the beginning of year x to Monday).

Now consider a year x whose molad is on Saturday after 16h 689p and before 18h. This would leave RH for year x on Saturday, and the molad of year x+19 will be on Tuesday after 9h 204p. If years x and x+19 are leap years, then RH for year x+19 will be on Tuesday, and the requirement for case d) fails.

However, if years x and x+19 are common years, then Dehiyyah GaTaRad kicks in and RH for year x+19 will be on Thursday, satisfying the requirement of case d.

Thus the only time that a 19 year interval has a number of days which is congruent to 5 (mod 7) is when the starting year is a common year whose molad is between 0d 16h 689p, and 0d 18h--a period of about an hour and twenty-two minutes!

Finally, of the possible lengths of 19 year intervals (6939-6942 days), only 6942 is congruent to 5 (mod 7). Thus when case d) is satisfied, the number of days in the interval is in fact 6942.

QED

Nice work Larry and Winfried!

Correspondents Winfried Gerum and Larry Padwa both noticed and shared additional facts governing the 19 year periods as related to the shortest periods of 6,938 days.

Question 78

Measured from the first day of Tishrei, which year, or years, of the mahzor qatan (19 year cycle) can begin a 19 year period of 6,938 days?

Answer

Measured from the first day of Tishrei, 19 year periods can be either 6938, 6939, 6940, 6941, or 6942 days long.

The shortest of the 19 year periods, 6938 days, only can occur if the first year of the 19 year period is a LEAP year.

Correspondents Larry Padwa, Winfried Gerum, and Dwight Blevins provided correct answers to this question.

Correspondent Larry Padwa not only sent the correct answer, but he also backed it up with a mathematical proof, which will be shown next week. Larry Padwa's proof also touches on the next Weekly Question.

Thank you Larry Padwa for sharing with us your great insights.

Question 79

Measured from the 1st day of Tishrei, on which day, or days, of the week can the shortest period of 19 years begin?

Answer

Measured from the first day of Tishrei, 19 year periods can be either 6938, 6939, 6940, 6941, or 6942 days long.

The shortest of the 19 year periods, 6938 days, only can occur if the first year of the 19 year period is a LEAP year and begins on a Monday.

Correspondents Larry Padwa, Winfried Gerum, and Dwight Blevins provided correct answers to this question.

Correspondent Larry Padwa not only sent the correct answer, but he also backed it up with a mathematical proof, which is as follows:

Since 6938 is congruent to 1 (mod 7), then if year x+19 begins 6938 days later than year x, then the day of the week that begins year x+19 must be one day later than that of year x. Since the only days on which a year can begin are (Mon, Tue, Thu, Sat), the only possibility of two consecutive days are that year x begins on a Monday and year x+19 begins on a Tuesday. Furthermore, since 6938 is the only number in the set {6938,6939,6940,6941,6942} of allowable days for 19 year intervals that is congruent to 1 (mod 7), it follows that x beginning on Monday and x+19 beginning on Tuesday is necessary and sufficient for the 19 year interval to contain 6938 days.

The molad of year x+19 is always 2d 16h 595p later than the molad of year x.

For RH of year x to be on Monday, its molad can be as early as 0d 18h 0p, and as late as 2d 15h 588p (if x-1 is a leap year), or as late as 2d 17h 1079p (if x-1 is not a leap year).

This means that if RH for year x is on Monday, then the earliest that the molad of x+19 could be is 3d 10h 595p. Now, if x (and x+19) are common years, then Dehiyyah GaTaRad would cause RH of x+19 to be on Thursday, thus failing our requirement of a Tuesday RH. However if x (and x+19) are leap years, then GaTaRad would not apply, and RH for x+19 would be on Tuesday if the Molad of x+19 is no later than 3d 17h 1079p. This would occur if the Molad of x is no later than 1d 1h 484p (which would of course still leave RH of year x on Monday).

To summarize: If the molad for a leap year is between 0d 18h 0p and 1d 1h 484p, then the 19 years beginning with year x will contain 6938 days.

Correspondent Winfried Gerum made the following observations:

The answer to Q77 is, that the longest 19-year periods (6942 days) always commence on a shabbat.

Looking at a full calender cycle, one finds, that if one counts just
cycles starting at year 1H one gets

     cycles of  length 6939 commencing on Shabbat or Tuesday or
                                          Thursday
                length 6940 commencing on Shabbat or Monday
                length 6941 commencing on Thursday or Tuesday
                length 6942 commencing always on a Shabbat

If one considers 19-year periods starting with any year there are also
periods of length 6938 days commencing always on a Monday
(see 5790..5808)

Correspondent Dwight Blevins sent in the following answer:

In the upcoming discussions of the 6938 period question on the web--the only examples I've found occur in the 19th year of the cycle, and re-occur only at intervals of 247 years. Examples are 1464 - 1483 ce, 1711 - 1730, 1958 - 1977, 2205 - 2224, etc. The set-up limits are Monday for the first year of the period, and Tueday for the declaration of Tishrie 1, day one of the next period. Thus a 6938/7 = 991.14285 week, or a 0.14285 x 7 = 1 day rotation or advance from the first day of the period.
Thank you correspondents Larry Padwa, Winfried Gerum, and Dwight Blevins for these most intriguing answers.

Question 80

Which is the first molad of Tishrei not to have appeared prior to the repetition of the Tishrei moladot?

Answer

The time of any molad is usually given as day, hour, minute and part.

There are 24*60*1080 = 25,920 parts in one day.

Hence there are 7*25,920 = 181,440 parts in one week.

In the full Hebrew calendar cycle of 689472 years, any molad of Tishrei is repeated either exactly 3 times or exactly 4 times.

For example, the molad of Tishrei 1H, BaHaRad, (Monday, September 7, -3760g) is repeated 4 times in the full Hebrew calendar cycle. That molad will next occur for the following Hebrew years

117,358H04 Jan 113,599g
308,063H16 Apr 304,306g
498,768H26 Jul 495,013g
689,473H04 Nov 685,720g

Since BaHaRad is first repeated as a molad of Tishrei for the year 117,358H
(beginning on Mon 4 Jan 113,599g), and there are 181,440 possibilities for the Tishrei moladot, it implies that some of the Tishrei moladot have not yet been seen for the first time.

It is therefore interesting to note that the very first occurrence of 1d 2h 793p as a molad of Tishrei follows immediately after the very first repetition of BaHaRad as a molad of Tishrei. This first occurrence happens for Tishrei 117,359H (Mon 24 Jan 113,600g).


Correspondent Larry Padwa suggested the next question, which is very timely because
Tu B'Shevat 5760H (the 15th day of Shevat 5760H) occurs on Sat 22 Jan 2000g.


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  5 Nov 2004