WQ Archive 181 - 190

Question 181

In what proportion does any long and short period of H Hebrew years occur over the full Hebrew calendar cycle of 689,472 years?

Properties of Hebrew Year Periods - Part 2 shows that except for exact multiples of 19 years, all Hebrew year periods have 2 possible length in months, differing from each other by exactly one molad period.

Periods of H Hebrew years, where H is not a multiple of 19 therefore are either M months or M + 1 months long.

Over any period of 19 Hebrew years, the number of times that a year in that period will inaugurate the period of M months is 12 * H MOD 19.

Consequently, periods of M + 1 months will be inaugurated 19 - (12 * H MOD 19) times in that same period of 19 Hebrew years.

Therefore, short periods of H Hebrew years are inaugurated (12 * H MOD 19) * 36,288 times over the full Hebrew calendar cycle of 689,472 years.

And, periods of H Hebrew years are inaugurated [19 - (12 * H MOD 19)] * 36,288 times over the full Hebrew calendar cycle of 689,472 years.

Winfried Gerum's answer to Weekly Question 180 indicated that the 420 Hebrew year periods of 5,194 months occur 5 out 19 times, and periods of 5,195 months occur 14 out of 19 times in the full Hebrew calendar cycle of 689,472 years.

Simple calculation shows that 12 * 420 MOD 19 = 5, which pretty well corresponds to Winfried Gerum's statistical observations.

Correspondent Winfried Gerum sent another solution to Weekly Question 181.

```Dear Remy,

My answer to Weekly Question 180 is as follows:

The number of Hebrew month of a period if H hebrew years is always

( H * 235  ) / 19

if H is an exact multiple of 19.

Otherwise the length is either

Amin = ( H * 235 ) DIV 19

or

Amax = ( ( H * 235 ) DIV 19 ) + 1

Periods of Amax occur with a frequency of

Fmax =  H * 235 - ( ( ( H * 235 ) DIV 19 ) * 19 )

out of 19. Consequently periods of Amin occur with

Fmin = 19 - Fmax

out of 19. DIV in this notation mean integer division, i.e. division

The proof is left as an exercise to the reader.

Shabbat shalom

Winfried
```

Winfried Gerum's anwer is quite correct. Checking it out with the 420 year span, we find that Fmax = 14, which is the number of times out of 19 that any 420 Hebrew year span displays the larger number of months.

Thank you Winfried Gerum for sharing your excellent solution.

Noting that Rosh Hodesh Shevat 5763H (Sat 4 Jan 2003g) coincided with Shabbat, correspondent Jerrold Landau suggested Weekly Question 182.

Question 182

How frequently does Rosh Hodesh occur on Shabbat?

Noting that Rosh Hodesh Shevat 5763H (Sat 4 Jan 2003g) coincided with Shabbat, correspondent Jerrold Landau suggested Weekly Question 182.

There exists a very real distinction between the first day of the month and Rosh Hodesh.

The festival of Rosh Hodesh may be either one or two days long. The observance will always begin on the 30th day of a Hebrew month. If the Hebrew month is 29 days long, then that 30th day is the first day of the subsequent month. If the month is 30 days long, then the next day is the first day of the subsequent month and Rosh Hodesh will once again be observed, thus making the festival two days long.

For purposes of discussion, all references to Rosh Hodesh will imply the first day of the festival.

Also, even though the first day of Tishrei is Rosh Hashannah and practically not viewed as being Rosh Hodesh, it is considered to also be Rosh Hodesh in the ensuing statistical tables.

The full Hebrew calendar cycle of 689,472 years consists of 8,527,680 months. This total can be easily derived by multiplying the number of months in every 19 year cycle with the number of these cycles in the full Hebrew calendar cycle. This gives 235 months * 36,288 = 8,527,680 months.

Each day of the week will coincide with Rosh Hodesh. However, due to the initial Rosh Hashannah rule of Lo ADU it is not possible for some of the Rashei Hadashim to begin on certain days of the week. For example, Tishrei cannot begin on either Sunday, Wednesday, or Friday. Similarly, Rosh Hodesh Tevet never falls on a Sunday, while Rosh Hodesh Shevat avoids Sunday and Friday.

In here, the 30 day month of Adar is called Leap Adar and the weekday distribution of its Roshei Hadashim is appended to the folowing table.

The following table demonstrates the weekday distribution of the Roshei Hadashim for each month over the full Hebrew calendar cycle.

Roshei Hadashim Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 193280 79369 0 219831 0 196992 689472
Heshvan 196992 0 193280 79369 0 219831 0 689472
Kislev 219831 0 196992 0 193280 79369 0 689472
Tevet 0 151093 68738 69853 127139 79369 193280 689472
Shevat 0 193280 26677 124416 138591 0 206508 689472
Adar 206508 0 193280 26677 124416 138591 0 689472
v'Adar 85899 0 72576 0 68864 26677 0 254016
Nisan 79369 0 219831 0 196992 0 193280 689472
Iyar 193280 79369 0 219831 0 196992 0 689472
Sivan 196992 0 193280 79369 0 219831 0 689472
Tammuz 0 196992 0 193280 79369 0 219831 689472
Av 0 219831 0 196992 0 193280 79369 689472
Elul 79369 0 219831 0 196992 0 193280 689472
Totals 1258240 1033845 1463854 989787 1345474 1153940 1282540 8527680
Leap Adar 72576 0 68864 26677 0 85899 0 254016

The difficult task, given those statistics, is to develop the additional numbers required to account for the situations in which Shabbat is the second day of Rosh Hodesh. Logically, these days must occur when the first day of Rosh Hodesh is a Friday on the 30th day of the month.

This occurs as follows

```Year   Shabbat as
Length 2nd Day RH  Month
====== =========== =====
354         124416 Heshvan
354          43081 Iyar
355          22839 Heshvan
355          81335 Iyar
383          26677 Heshvan
383          40000 Iyar
385          45899 Heshvan
385          32576 Iyar
=====       ======
TOTAL       582091
```

Consequently, the number of times that Rosh Hodesh occurs on Shabbat over the full Hebrew calendar cycle is
(582,091 + 1,282,540) / 8,527,680 = 1,864,631 / 8,527,680 = 39,673 / 181,440.

Intriguingly, 39,673 = 1d 12h 793p and 181,440 = 7 * 24 * 1080.

Correspondent Winfried Gerum sent an answer that was very much appreciated because it corroborated the statistics found in The First Day of The Month.

Correspondent Jerrold Landau sent an answer that indicated exactly the question he had in mind.

Question 183

In any given Hebrew year, how frequently does Rosh Hodesh occur on Shabbat?

This kind of Hebrew calendar question is significantly complicated by the rules applied as to which day or days of the Hebrew months constitute Rosh Hodesh.

Adding to its complexity is the fact that Hebrew calendar literature has yet to produce any kind of analytical formula which will identify the days of Rosh Hodesh.

If anyone is aware of such formulae, reference to these will be greatly appreciated.

There exists a very real distinction between the first day of the month and Rosh Hodesh.

The festival of Rosh Hodesh may be either one or two days long. The observance will always begin on the 30th day of a Hebrew month. If the Hebrew month is 29 days long, then that 30th day is the first day of the subsequent month. If the month is 30 days long, then the next day is the first day of the subsequent month and Rosh Hodesh will once again be observed, thus making the festival two days long.

Consequently, the answer to this question requires finding all of the cases in which Shabbat is either the first or the second day of Rosh Hodesh.

If Shabbat is to be the first day of a two day Rosh Hodesh, then the first day of the new month, that is, the 2nd day of Rosh Hodesh, must be Sunday.

It is therefore possible to develop a table which will track all of the week days coinciding with the first day of each new month.

In the following table, the column headed by the letter 'F' indicates by an '*' that a month has 30 days.

Year Length 353 354 355 383 384 385
Year Start F02 F35 F025 F025 F3 F025
MonthMonth's Weekday Start
Tishrei *02 *35 *025 *025 *3 *025
Heshvan .24 .50 *240 .240 .5 *240
Kislev .35 *61 *462 .351 *6 *462
Tevet .46 .13 .614 .462 .1 .614
Shevat *50 *24 *025 *503 *2 *025
(v')Adar .02 .46 .240 .240 .6 .462
Nisan *13 *50 *351 *351 *0 *503
Iyar .35 .02 .503 .503 .2 .025
Sivan *46 *13 *614 *614 *3 *136
Tamuz .61 .35 .136 .136 .5 .351
Av *02 *46 *240 *240 *6 *462
Elul .24 .61 .462 .462 .1 .614
Shabbatot .32 .23 .323 .323 .3 .333

The Shabbatot and the Sundays preceded by Shabbat Rosh Rodesh are highlighted in blue.

From that display, an easy count down each column can be made of the days which lead to Shabbat Rosh Rodesh.

The count shows that Shabbat Rosh Rodesh can occur either 2 or 3 times in any given Hebrew year.

```Hi Remy.

Given that you posted the question in my name, it perhaps
behooves me to give an answer:

Rosh Chodesh of any month is always 29 days following the Rosh Chodesh
of a previous month. This is true regardless if the month is 29 or 30 days
long, since the 1st of a month is Rosh chodesh:  if the month has 30 days,
the 30th day will be Rosh Chodesh (thus with a 29 day period in between),
and if the month has 29 days, the 1st of the next month will be Rosh Chodesh
(thus with a 29 day period in between).  The 29 day gap between Rosh
Chodeshes is constant.  Dividing by 7 days of the week, there is a
remainder of 1.

Thus, a subsequent Rosh Chodesh will always fall 1 day in the week
following the preceding Rosh Chodesh.

4 possibilities:  (Maleh refers to a 30 day month, and chaser to a 29
day month)

maleh follows maleh:  e.g. Rosh Chodesh on Tuesday/ Wednesday,  Next
will be Thursday / Friday

maleh follows chaser.  e.g. Rosh Chodesh on Tuesday, next will be on
Wednesday / Thursday

chaser follows maleh.  e.g. Rosh Chodesh on Tuesday / Wednesday, Next
will be on Thursday chaser follows chaser.
e.g. Rosh Chodesh on Tuesday, next will be on Wednesday.

Note that the months always alternate maleh chaser, with a few
exceptions.

Cheshvan / Kislev: you can have 2 malehs consecutively, and (rarer) 2
chasers consecutively Adar I and Adar II in a leap year:  always 2 malesh consecutively.

So how often does this happen.  Generally every 4-5 months.

Longest gap, assume that there will be 2 chasers in a row in between:

Tishrei:  Sat
Cheshvan:  Sun Mon
Kislev Tue
Tevet Wed
Shvat Thurs
Adar  Fri Sat               :5 months elapsed)

Shortest gap:  assume that there are 2 malesh in a row in between:

Tishrei  Sat
Cheshvan: Sun Mon
Kislev: Tues Wed
Tevet Thurs Fri
Shvat:  Sat  4 months elapsed

I chose a Tishrei of Sat in the above case, since I knew RH on Sat can
result in a maleh or chaser year.

Other possibility of shortest gap:  leap year in between, continue with
above example  (this year)

Shvat: Sat
Nisan:  Thurs
Iyar:  Fri Sat:   4 months elapsed

Other examples:  a normal alternating cycle (month irrelevant)

I Fri Sat
II Sun
III Mon Tue
IV Wed
V  Thurs Fri
VI  Sat                    : 5 month gap

I Sat Sun
II MOn
III Tues Wed
IV  Thurs
V  Fri Sat:               4 month gap

I Sat
II Sun Mon
III Tue
IV Wed Thurs
V Fri
VI Sat Sun   :5 month gap

Thus, it will happen every 4-5 months.  Making it happen generally twice
a year, but sometimes 3 times a year (e..g if RH is on Saturday, it will
occur 3 times in the year -- of course, RH is not observed as Rosh
Chodesh, but it is.  I believe that if RC Cheshvan falls on Shabbat,
and you have a leap year (or perhaps even if not), you will end up with 3 occurrences.
```

Correspondent Larry Padwa chose the pragmatic approach using a simple spreadsheet calculation. Due to technical details, the Weekly Question apologizes to correspondent Larry Padwa for not being able to include the spreadsheet. Here are his summary observations.

```Depending on the year, there are either two or three months with a
Shabbat Rosh Chodesh.

Ten of the fourteen year types have three. The remaining four year types
have two.

The full picture is summarized on the attached file.
```

Thank you correspondents Jerrold Landau and Larry Padwa for having shared your wonderful and very correct responses.

Weekly Question 181 was interested in finding the frequency of the shorter spans of H Hebrew years.

Question 184

Which Hebrew year or years begin the shorter periods of H Hebrew years?

Properties of Hebrew Year Periods - Part 2 explains and demonstrates the fact that all periods of Hebrew years whose year lengths are not multiples of 19 exhibit 2 different lengths in months. These two monthly lengths are one month apart.

The reason for this particular Hebrew calendar phenomenon does not come from the fact that Hebrew leap years have an extra month, or that the leap years are distributed in a particular manner. Those reasons do not explain the one month difference in the periods of time which are not multiples of 19 Hebrew years.

It is possible to demonstrate the phenomenon using simple arithmetic techniques, as evident from a reading of
Properties of Hebrew Year Periods - Part 2.

The one month differences arise from a quirk in the remaindering arithmetic that is used to calculate the Hebrew calendar. The same quirk applies also to both the Julian and Gregorian calendars. However, in those calendars, the differences in the lengths of equal numbers of years is only one day.

Weekly Question 181 noted that in the Hebrew calendar the frequency of the shorter spans of H Hebrew years occurred 12 * H MOD 19 times in any cycle of 19 Hebrew years.

The same modulus arithmetic uncovers the amazing fact that when H and Hebrew year HY are such that

(12 * H MOD 19) + [(12 * HY + 5) MOD 19] > 18

then Hebrew year HY will start the shorter period of H Hebrew years.

Example

Let H = 120 Hebrew years

Then, 12 * H MOD 19 = 12 * 120 MOD 19 = 15.

Consequently, all of the shorter periods of time will occur whenever

[(12 * HY + 5) MOD 19] > 3

This will be true whenever HY is either year 1, 2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 15, 16, 18, or 19 of the mahzor qatan GUChADZaT.

By implication, the longer periods of 120 will be started whenever

[(12 * HY + 5) MOD 19] < 4

This will be true whenever HY is either year 3, 6, 14, or 17 of the mahzor qatan GUChADZaT.

It is interesting to note that Rosh Hashanah 5676H (Thu 9 Sep 1915g) began the longest possible period of 120 Hebrew years.

Question 185

In days, what was the length of year 4596H?

The modern calculation of the moladot for the year 4596H is shown in the following table. The proleptic Gregorian calendar is used to map the Hebrew dates.

The modern calculation shows that year 4596H has a length of 385 days. This value is easily spotted by the fact that the months of Heshvan and Kislev are both 30 day months, and that the year also contains the leap month of Adar.

Month Moment of Molad First Day Chodesh
Tishrei
`Fri 22h 36m 12hl`
`  Sat  1 Sep  835g `
Heshvan
`Sun 11h 20m 13hl`
` *Mon  1 Oct  835g `
Kislev
`Tue  0h  4m 14hl`
` *Wed 31 Oct  835g `
Tevet
`Wed 12h 48m 15hl`
` *Fri 30 Nov  835g `
Shevat
`Fri  1h 32m 16hl`
`  Sat 29 Dec  835g `
`Sat 14h 16m 17hl`
` *Mon 28 Jan  836g `
`Mon  3h  1m  0hl`
` *Wed 27 Feb  836g `
Nisan
`Tue 15h 45m  1hl`
`  Thu 27 Mar  836g `
Iyar
`Thu  4h 29m  2hl`
` *Sat 26 Apr  836g `
Sivan
`Fri 17h 13m  3hl`
`  Sun 25 May  836g `
Tammuz
`Sun  5h 57m  4hl`
` *Tue 24 Jun  836g `
Av
`Mon 18h 41m  5hl`
`  Wed 23 Jul  836g `
Elul
`Wed  7h 25m  6hl`
` *Fri 22 Aug  836g `

However, it appears that the calendar calculations at about the year 4596H did not include Dehiyyah Molad Zaqen. This idea appears to be a reasonable conclusion in light of a particular document found in the Cairo Geniza.

It appears that sometimes in 835g, the Resh Galuta (Exilarch) composed a letter which indicated a particular problem with the calendar calculations as then known. The problem seemed to be an inability to cope with new moons that arrived a bit too early. A further analysis of the letter indicated that this particular year was only 384 days long, that is, one day short of the result that would have been achieved if Dehiyyah Molad Zaqen had been included in the calculations.

There is a very good presentation of the letter in Sacha Stern's Calendar and Community, Oxford Press 2001g.

Therefore, from what appears to be valid archeological evidence, year 4596H was calculated to be 384 days long, and not the anticipated 385 days that today's calculations would have given.

The calendar calculation of the moladot for the year 4596H, absent Dehiyyah Molad Zaqen, is shown in the following table. The proleptic Gregorian calendar is used to map the Hebrew dates.

Month Moment of Molad First Day Chodesh
Tishrei
`Fri 22h 36m 12hl`
`  Sat  1 Sep  835g `
Heshvan
`Sun 11h 20m 13hl`
` *Mon  1 Oct  835g `
Kislev
`Tue  0h  4m 14hl`
`  Tue 30 Oct  835g `
Tevet
`Wed 12h 48m 15hl`
`  Wed 28 Nov  835g `
Shevat
`Fri  1h 32m 16hl`
`  Thu 27 Dec  835g `
`Sat 14h 16m 17hl`
` *Sat 26 Jan  836g `
`Mon  3h  1m  0hl`
` *Mon 25 Feb  836g `
Nisan
`Tue 15h 45m  1hl`
`  Tue 25 Mar  836g `
Iyar
`Thu  4h 29m  2hl`
` *Thu 24 Apr  836g `
Sivan
`Fri 17h 13m  3hl`
`  Fri 23 May  836g `
Tammuz
`Sun  5h 57m  4hl`
` *Sun 22 Jun  836g `
Av
`Mon 18h 41m  5hl`
`  Mon 21 Jul  836g `
Elul
`Wed  7h 25m  6hl`
` *Wed 20 Aug  836g `

The above table indicates that the molad of Tishrei 4597H was 5d 20h 569p.

As explained in The Overpost Problem, failure to apply Dehiyyah Molad Zaqen in this instance is the reason why the molad of Shevat 4596H occurred on the 2nd day of that month.

Correspondent Dwight Blevins correctly calculated the length of year 4596H according to current knowledge of the calendar rules using a rather intriguing method.

```
Hi Remy,

I always suspect that when you have a question on a level that even I
But, I'll bite on this one anyway.  Who knows, I might learn something
new in the process.  I cannot do the finite calculation of the molad
math by formula, so here's my rough, quick version.

Year 4596H = 835 - 836 CE (leap year)

The molad of 835 fell about 6:03 AM, Saturday, Aug. 28 (Julian), thus
Tishri 1 was declared that day.  The molad of 4597 fell about 1.5305941
x 13 mos. later = 19.897723 days later in the week, which, by the
nearest seven is 19.897723 - 14.0 = 5.897723 days west of 0.5023,
4596H.  The molad time of 4597H is then, 5.897723 + 0.5023 = 6.4 days,
or 6d 9h 648p.

So, as is always the case, a 13 month year by molad advance is 383.8977
days long.  In this case, being the difference from about 3:36 to 6:03
AM, Friday Sept. 15, 836 CE, which would have been the 384 day mark from
6:03 AM, Saturday, Aug. 28, 4596H.

However, at this point, postponement rule 1 kicks in, pushing the
declaration to Sat. Sept. 16, 836 CE, for a total length, by sevens, of
385 days.  My molad time of reference for 835 CE is likely only ball
park, since I collected it by projecting forward from a molad time I
found in my archives, from 56 CE.  I do this by simply multiplying the
number of 19 year cycles by 2.6895 days, and, in this case adding the
fraction beyond the nearest seven to the reference molad.  Yes, I know.
Why I just learn to use the formula?  I would never have been the one to
find the northwest passage, since I always insist on taking the long way
around and making it 10 times more difficult than need be.

I'm not expecting this answer to show up on your website, even if it is
fairly close to correct.  That is, unless you are looking for an
anecdote.  One of the other correspondents will supply the exact scoop
on the math, which I shall be interested to read.

Warm Regards,

Dwight Blevins
```

Thank you correspondent Dwight Blevins for sharing your thoughts and methods to this week's question.

The following originally appeared as Weekly Question 56.

Question 56

Do the numbers 2 4 7 10 12 15 18 constitute a valid leap year distribution for the calculation of the Hebrew calendar?

YES!

The traditional literature of the Hebrew calendar, such as the 8th century Seder Olam, and the 11th century work The Chronology of Ancient Nations by the near Eastern scholar Al-Biruni, show a number of variants as regards the leap year distribution inside a mahzor qatan (the nineteen year Hebrew calendar cycle).

The Encyclopedia Judaica, in its article on the Calendar very correctly notes that

Apparent variations in the ordo intercalationis ... are but variants of the selfsame order.

The leap year order that is used will depend entirely on the year of the 19 year cycle that is selected to be the first year of the year counts, ie, the epochal year.

Since the Hebrew calendar uses only one pattern in which to arrange the 7 leap years of a 19 year cycle, there can only be at most 19 leap year distributions resulting from the choice of a specific epochal year within the mahzor qatan. Assuming that the epochal year 1H is the first year of the 19 year cycle, then these distributions are:-

```
YEAR Leap Year Distribution
==== ======================
1   3  6  8 11 14 17 19
2   2  5  7 10 13 16 18
3   1  4  6  9 12 15 17
4   3  5  8 11 14 16 19
5   2  4  7 10 13 15 18
6   1  3  6  9 12 14 17
7   2  5  8 11 13 16 19
8   1  4  7 10 12 15 18
9   3  6  9 11 14 17 19
10   2  5  8 10 13 16 18
11   1  4  7  9 12 15 17
12   3  6  8 11 14 16 19
13   2  5  7 10 13 15 18
14   1  4  6  9 12 14 17
15   3  5  8 11 13 16 19
16   2  4  7 10 12 15 18
17   1  3  6  9 11 14 17
18   2  5  8 10 13 16 19
19   1  4  7  9 12 15 18
```

The numbers 2 4 7 10 12 15 18 can be seen against the 16th year of the 19 year cycle whose first year is also year 1H. Consequently, any Hebrew calendar system whose year count begins relative to year 16H will be required to use the leap year distribution 2 4 7 10 12 15 18 if it is to maintain synchronization with the fixed Hebrew calendar.

Correspondent Winfried Gerum gave the following correct answer

... if one does not start counting19-year cycles in the year 1 but instead, the year 16, then leap years fall into the stated sequence of numbers.

Good work Winfried Gerum!

The following originally appeared as Weekly Question 57.

Question 57

The Encyclopedia Judaica, in its article on the Calendar cites the following leap year distributions as having been used

```       2  5  7 10 13 16 18
1  4  6  9 12 15 17
3  5  8 11 14 16 19
3  6  8 11 14 17 19
```

Additionally, the EJ suggests that the following values were used at one time as the epochal moladot

```      4d 20h 408p
2d  5h 204p
6d 14h   0p
3d 22h 876p
```

Is the EJ entirely correct?

NO!

In the Encyclopedia Judaica, the epochal molad given as 4d 20h 408p should be 2d 20h 385p .

The traditional literature of the Hebrew calendar, such as the 8th century Seder Olam, and the 11th century work The Chronology of Ancient Nations by the near Eastern scholar Al-biruni, show a number of variants as regards the leap year distribution inside a mahzor katan (the nineteen year Hebrew calendar cycle).

The Encyclopedia Judaica, in its article on the Calendar very correctly notes that

Apparent variations in the ordo intercalationis ... are but variants of the selfsame order.

The leap year order that is used will depend entirely on the year of the 19 year cycle that is selected to be the first year of the year counts, ie, the epochal year.

Since the Hebrew calendar uses only one pattern in which to arrange the 7 leap years of a 19 year cycle, there can only be at most 19 leap year distributions resulting from the choice of a specific epochal year within the mahzor katan.

The EJ article Calendar mentions only the first 4 possible leap year variations assuming that the epochal year 1H is the first year of the 19 year cycle. These distributions are:-

```
YEAR Leap Year Distribution
==== ======================
1   3  6  8 11 14 17 19
2   2  5  7 10 13 16 18
3   1  4  6  9 12 15 17
4   3  5  8 11 14 16 19
```

Consequently, the epochal moladot should be

```       2d  5h 204p  (for the 1st year)
6d 14h   0p  (for the 2nd year)
3d 22h 876p  (for the 3rd year)
and 2d 20h 385p  (for the 4th year) instead of the value 4d 20h 408p
```

The following originally appeared as Weekly Question 58.

Question 58

Which Hebrew year could most reasonably be represented as an epochal year with the presence of the molad 4d 20h 408p?

The Encyclopedia Judaica article Calendar mentions only the first 4 possible leap year variations assuming that the epochal year 1H is the first year of the 19 year cycle. These distributions are:-

```
YEAR Leap Year Distribution
==== ======================
1   3  6  8 11 14 17 19
2   2  5  7 10 13 16 18
3   1  4  6  9 12 15 17
4   3  5  8 11 14 16 19
```

Consequently, the epochal moladot should be

```       2d  5h 204p  (for the 1st year)
6d 14h   0p  (for the 2nd year)
3d 22h 876p  (for the 3rd year)
and 2d 20h 385p  (for the 4th year) instead of the value 4d 20h 408p
```

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.

Clearly, the above possibilities are not sensible in terms of epochal years. However, the value
4d 20h 408p also comes as the molad of Heshvan for the year 0H. Hence, it would be very reasonable to use the year 0H as an epochal year. Its molad of Tishrei is 3d 7h 695p.

Consequently, if the Encyclopedia Judaica were to correct this particular passage in their well written article on the Calendar the two corrections that could be recommended would be that:-

1. The epochal molad of 4d 20h 408p be replaced by the value 3d 7h 695p which represents the molad of Tishrei 0H;

2. The leap year distribution 3 5 8 11 14 16 19 be replaced by 1 4 7 9 12 15 18 which represents the leap year distribution for an epochal year beginning at year 0H.

The following originally appeared as Weekly Question 59.

Question 59

Which Hebrew month least frequently has the molad of 0d 0h 0p?

In the full Hebrew calendar cycle of 689,472 years each and every month will experience the molad 0d 0h 0p at least 2 times.

Of all the months, the leap month Adar will experience the molad 0d 0h 0p exactly two times in the full Hebrew calendar cycle. All of the other months will see this molad either 3 or 4 times.

Therefore, the leap month Adar least frequently has the molad 0d 0h 0p.

The following originally appeared as Weekly Question 63.

Question 63

Is it nececessary that Rosh Hashannah be postponed so as to have a molad precede the Shabbat M'Vorchim on which its month is announced?

NO!

The Shabbat which immediately precedes the observance of Rosh Hodesh is known as Shabbat M'Vorchim. On that particular Shabbat, during the synagogue services, it is customary to announce not only the coming of the new month, but also the specific time of the new month's molad.

A molad can precede the Shabbat M'Vorchim on which its month is announced even though Rosh Hashannah is not postponed for that year. This particular phenomenon occurs in 1.18% of all the Hebrew years.

The most recent molad to occur prior to Shabbat M'Vorchim for a year in which Rosh Hashannah was not postponed was the molad of Sivan 5689H (Fri 7 Jun 1929g).

That molad arrived at 6d 19h 724p.

The following originally appeared as Weekly Question 67.

Question 67

In the full Hebrew calendar cycle of 689472 years, which molad of Tishrei will be the first to be repeated as a molad of Tishrei?

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*25920 = 181,440 parts in one week.

Since there are no common factors between 7 and 25920 we must go for 7*25920 = 181,440 months before the time of a given molad is repeated.

For example, the molad of Tishrei 1H, BaHaRad, will return as the molad of Sivan 14,670H (Mon 23 Jun 10,910g) and again 181,440 months later as the molad of Tevet 29,340H (Mon 7 Apr 25,580g).

BaHaRad will not be repeated as a molad of Tishrei until 117,358H (Mon 4 Jan 113,599g).

By one of these delightful coincidences, in the full Hebrew calendar cycle of 689472 years, BaHaRad is also the very first molad of Tishrei that will be repeated as a molad of Tishrei.

The following originally appeared as Weekly Question 68.

Question 68

In the full Hebrew calendar cycle of 689472 years, how often do the moladot of Tishrei repeat themselves?

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.

Since there are no common factors between 7 and 25,920 we must go for 7*25920 = 181,440 months before the time of a given molad is repeated.

Since there are only 181,440 possible values for the moladot, the average number of times that any molad of Tishrei can be repeated is 689,472 / 181,440 = 3.8 repetitions.

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, BaHaRad, the molad of Tishrei 1H (Mon 7 Sep -3760g), is repeated 4 times in the full Hebrew calendar cycle.

That molad will next occur for the following Hebrew years

 117,358H 04 Jan 113,599g 308,063H 16 Apr 304,306g 498,768H 26 Jul 495,013g 689,473H 04 Nov 685,720g

The following originally appeared as Weekly Question 69.

Question 69

What fraction of the moladot of Tishrei are repeated exactly 3 times in the full Hebrew calendar cycle of 689,472 years?

The moladot of Tishrei can be repeated either exactly 3 times or exactly 4 times in the full Hebrew calendar cycle of 689,472 years.

There are 181,440 possible values for the moladot of Tishrei and all are represented at least 3 times in the full Hebrew calendar cycle.

```
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
```
Therefore, the fraction of the moladot of Tishrei which are repeated exactly 3 times in the full Hebrew calendar cycle of 689,472 years is 3/19.

Question 186

From the Hebrew calendar perspective, what do the months of June and July 2003g have in common?

About every 2 or 3 years, the first day of some Hebrew month will coincide with the first day of some Gregorian month.

This year, the month of Sivan 5763H begins on the first day of June 2003g. As a result of both months being 30 days, the subsequent month of Tammuz 5763H sees its first day fall on the first day of July 2003g.

The question was correctly answered by correspondents Larry Padwa and Dennis Kluk.

Larry Padwa sent this answer almost as soon as the question was posted.

```Since June 1, 2003 coincides with 1-Sivan 5763, and since June and Sivan
both have 30 days, the Gregorian dates in June and July this year
correspond with the dates in Sivan and Tamuz
(until July 29<-->Tamuz 29).
```

Dennis Kluk's correct answer also included a number of interesting questions.

```The day number of the Gregorian months of
June and July 2003 = the day number of the Jewish months Sivan and Tammuz 5763
but only through the 29th of July/Tammuz while also noting that this is only
true from midnight to sunset each of those days.

How often in the full Gregorian cycle of 400 years and the full
Jewish cycle of 689472 does this happen?

When was the most recent previous time this happened and
when will it next happen?
```

Thank you Larry Padwa and Dennis Kluk for sharing your calendar insights.

The Weekly Question would also like to thank correspondent Dwight Blevins who correctly pointed out that 3/19 of 689,472 = 108,864 and not 98,064 as originally shown in the answer to Weekly Question 186.

Although the Weekly Question is intrigued by Dennis Kluk's questions it is only prepared to answer one. The answer to the question involving the Gregorian and Hebrew calendar cycles is heavily dependent on the section of years that is being examined.

Question 187

[Relative to year 2003g] When last did the first day of Sivan coincide with the first day of the Gregorian month of June?

As shown in First Day Hebrew-Gregorian Coincidences of the Additional Notes, the coincidence of the first day of Sivan with the first day of the Gregorian month of June last occurred in the Hebrew year 5744H (1984).

About every 2 or 3 years, the first day of some Hebrew month will coincide with the first day of some Gregorian month.

This year, the month of Sivan 5763H began on the first day of June 2003g. As a result of both months being 30 days, the subsequent month of Tammuz 5763H has its first day fall on the first day of July 2003g.

The question was correctly answered by correspondent Larry Padwa who did some interesting research...

```Here is some empirical data (past and future) for years at
intervals of 19 years.

Year    GDate of 1-Sivan
----    ----------------
1908    May 31
1927    June 1
1946    May 31
1965    June 1
1984    June 1
2003    June 1
2022    May 31
2041    May 31
2060    May 30
2079    May 31

As you can see from this (admittedly small) sample, the variation of the
date between consecutive 19 year dates is at most one day. Of course
this variation accumulates with time, resulting in the eventual drift
apart of the two calendars.
```

Question 188

[Relative to June 2003g] When last did a calculated molad result in a time prior to its announcement on Shabbat?

The Shabbat which immediately precedes the observance of Rosh Hodesh is known as Shabbat M'Vorchim. On that particular Shabbat, during the synagogue services, it is customary to announce not only the coming of the new month, but also the specific time of the new month's molad.

The timing of the molad is permitted to precede the Shabbat M'Vorchim service during which it will be announced.

The most recent molad to occur prior to the Shabbat M'Vorchim service in which it was announced was the molad of Sivan 5763H (Sat 31 May 2003g).

That molad arrived at 0d 7h 559p.

The following originally appeared as Weekly Question 71.

Question 71

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

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.

The following originally appeared as Weekly Question 72.

Question 72

Was 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"?

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 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] remainder 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.

This is also explained on pages 11-12 of Remy Landau's recently published article, Al-Biruni's Hebrew Calendar Enigmas in the journal Mo'ed Annual for Jewish Studies, Vol 13, 2003.

The following originally appeared as Weekly Question 76.

Question 76

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

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.

The following originally appeared as Weekly Question 77.

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?

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.

```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.

The following originally appeared as Weekly Question 78.

Question 78

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

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

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.

The following originally appeared as Weekly Question 79.

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?

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 189

Which are two of the most important arithmetical properties of the 17th year of the mahzor qatan (19 year cycle) known as GUChADZaT?

In 1802g, the German mathematician Carl Friedrich Gauss published a formula which gave the Julian date of Pesach for any Hebrew year A.

The Gauss Pesach formula appeared without proof of its derivation in “Berechnung des jüdischen Osterfestes”, Monatliche Correspondenz zur Beförderung der Erd- und Himmels-Kunde, 5 (1802), 435-437 – reprinted in: Carl Friedrich Gauss Werke (Königlichen Gesellschaft der Wissenschaften, Göttingen, 1874), vol. 6, pp. 80-81.

The application of the formula showed that if the remainder of 12 * A + 17 when divided by 19 was greater than 11 then the Hebrew year A was a 13 month year.

It is very clear from simple experimentation, that underlying the Gauss formula was the mahzor qatan (19 year cycle) known as GUChADZaT.

What was camouflaged in this extraordinary formula, was the fact that the expression 12 * A + 17 was actually derived from the base formula 12 * A + 5.

In that form, all of the leap years are ordered in such a way that the remainder is less than 7 when the results of that formula are divided by 19.

This can be easily shown as follows

```
12 *  3 + 5 =  41 which leaves 3 when divided by 19
12 *  6 + 5 =  77 which leaves 1 when divided by 19
12 *  8 + 5 = 101 which leaves 6 when divided by 19
12 * 11 + 5 = 137 which leaves 4 when divided by 19
12 * 14 + 5 = 173 which leaves 2 when divided by 19
12 * 17 + 5 = 209 which leaves 0 when divided by 19
12 * 19 + 5 = 233 which leaves 5 when divided by 19
```
The most interesting observation which can be made from the above, is that the 17th year of GUChADZaT corresponds to a zero remainder in that formula.

It is that very property which helps to explain exactly why the earliest possible Rosh HaShannah's in any given period of time always coincide with the 17th year of GUChADZaT.

For example, Rosh HaShannah 5774H will begin on Thu 5 Sep 2013g. This date, September 5, is in the Gregorian calendar, the earliest possible date today for the first day of Tishrei.

Note also that 12 * 5774 + 5 = 69,293 which leaves 0 when divided by 19.

These observations lead to the second major arithmetic property of the 17th year of GUChADZaT. This will be discussed in the next Weekly Question.

Question 190

What is the second major arithmetic property of the 17th year of the mahzor qatan (19 year cycle) known as GUChADZaT?

In 1802g, the German mathematician Carl Friedrich Gauss published a formula which gave the Julian date of Pesach for any Hebrew year A.

The Gauss Pesach formula appeared without proof of its derivation in “Berechnung des jüdischen Osterfestes”, Monatliche Correspondenz zur Beförderung der Erd- und Himmels-Kunde, 5 (1802), 435-437 – reprinted in: Carl Friedrich Gauss Werke (Königlichen Gesellschaft der Wissenschaften, Göttingen, 1874), vol. 6, pp. 80-81.

The application of the formula showed that if the remainder of 12 * A + 17 when divided by 19 was greater than 11 then the Hebrew year A was a 13 month year.

It is very clear from simple experimentation, that underlying the Gauss formula was the mahzor qatan (19 year cycle) known as GUChADZaT.

What was camouflaged in this extraordinary formula, was the fact that the expression 12 * A + 17 was actually derived from the base formula 12 * A + 5.

In that form, all of the leap years are ordered in such a way that the remainder is less than 7 when the results of that formula are divided by 19.

The most important part of the formula is that it leads to a very simple summation for the number of months that have elapsed up any year HY in the Hebrew calendar.

Let R(x, k) = the non-negative remainder after x is divided by k.

Then,

19 * SUM(HY) = (HY + 2) * 235 + R(12 * HY + 5, 19)

Let XY = the number of years elapsed from some HY

Then,

19 * SUM(HY + XY) = (HY + 2 + XY) * 235 + R(12 * HY + 5 + XY, 19)

The number of months between HY and HY + XY

= 19 * SUM(HY + XY) - 19 * SUM(XY)
= (HY + 2 + XY) * 235 + R(12 * HY + 5 + XY, 19) - (HY + 2) * 235 - R(12 * HY + 5, 19)
= XY * 235 + R(12 * HY + 5 + 12 * XY) - R(12 * HY + 5, 19)

When HY = 17, then R(12 * HY + 5, 19) = 0

Hence, the number of months in XY years beginning at the 17th year of GUChADZaT

= (XY * 235 + R(12 * XY, 19)) / 19

This single valued expression also represents the maximum number of months that any period of XY years may have.

Therefore, a second property of the 17th year of GUChADZaT is that any period of years beginning in the 17th year will always have the maximum number of months possible for that period of years.

``` First  Begun 21 Jun 1998