In what proportion does any long and short period of

HHebrew years occur over the full Hebrew calendar cycle of689,472 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

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

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 discarding any remainder. 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

How frequently does

Rosh Hodeshoccur onShabbat?

Noting that ** Rosh Hodesh Shevat 5763H (Sat 4 Jan 2003g)** coincided with

The answer to this question is documented in The Roshei Hadashim.

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

This occurs as follows

YearShabbatas Length 2nd Day RH Month ====== =========== ===== 353 29853 Adar 354 124416 Heshvan 354 43081 Iyar 355 22839 Heshvan 355 22839 Adar 355 81335 Iyar 383 26677 Heshvan 383 40000 Adar 383 26677 v'Adar 383 40000 Iyar 385 45899 Heshvan 385 45899 Adar 385 32576 Iyar ===== ====== TOTAL 582091

Consequently, the number of times that ** Rosh Hodesh** occurs on

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.

In any given Hebrew year, how frequently does

Rosh Hodeshoccur onShabbat?

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

Consequently, the answer to this question requires finding all of the cases in which ** Shabbat** is either the

If ** Shabbat** is to be the

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 | F | 0 | 2 | F | 3 | 5 | F | 0 | 2 | 5 | F | 0 | 2 | 5 | F | 3 | F | 0 | 2 | 5 |

Month | Month's Weekday Start | |||||||||||||||||||

Tishrei | * | 0 | 2 | * | 3 | 5 | * | 0 | 2 | 5 | * | 0 | 2 | 5 | * | 3 | * | 0 | 2 | 5 |

Heshvan | . | 2 | 4 | . | 5 | 0 |
* | 2 | 4 | 0 |
. | 2 | 4 | 0 |
. | 5 | * | 2 | 4 | 0 |

Kislev | . | 3 | 5 | * | 6 | 1 | * | 4 | 6 | 2 | . | 3 | 5 | 1 | * | 6 | * | 4 | 6 | 2 |

Tevet | . | 4 | 6 | . | 1 | 3 | . | 6 | 1 | 4 | . | 4 | 6 | 2 | . | 1 |
. | 6 | 1 | 4 |

Shevat | * | 5 | 0 |
* | 2 | 4 | * | 0 | 2 | 5 | * | 5 | 0 | 3 | * | 2 | * | 0 | 2 | 5 |

Adar | * | 0 | 2 | 5 | * | 4 | * | 2 | 4 | 0 |
||||||||||

(v')Adar | . | 0 | 2 | . | 4 | 6 | . | 2 | 4 | 0 |
. | 2 | 4 | 0 |
. | 6 | . | 4 | 6 | 2 |

Nisan | * | 1 | 3 | * | 5 | 0 |
* | 3 | 5 | 1 | * | 3 | 5 | 1 | * | 0 |
* | 5 | 0 | 3 |

Iyar | . | 3 | 5 | . | 0 | 2 | . | 5 | 0 | 3 | . | 5 | 0 | 3 | . | 2 | . | 0 | 2 | 5 |

Sivan | * | 4 | 6 | * | 1 | 3 | * | 6 | 1 | 4 | * | 6 | 1 | 4 | * | 3 | * | 1 | 3 | 6 |

Tamuz | . | 6 | 1 |
. | 3 | 5 | . | 1 | 3 | 6 | . | 1 | 3 | 6 | . | 5 | . | 3 | 5 | 1 |

Av | * | 0 | 2 | * | 4 | 6 | * | 2 | 4 | 0 |
* | 2 | 4 | 0 |
* | 6 | * | 4 | 6 | 2 |

Elul | . | 2 | 4 | . | 6 | 1 |
. | 4 | 6 | 2 | . | 4 | 6 | 2 | . | 1 |
. | 6 | 1 | 4 |

Shabbatot |
. | 3 | 2 |
. | 2 | 3 |
. | 3 | 2 | 3 |
. | 3 | 2 | 3 |
. | 3 |
. | 3 | 3 | 3 |

The ** Shabbatot** and the

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

Correspondents **Jerrold Landau** and **Larry Padwa** provided excellent answers.

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 Adar I: Sun Mon Adar II Tue Wed 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

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

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

** Weekly Question 181** noted that in the Hebrew calendar the frequency of the shorter spans of

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

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

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

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

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

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

The Moladot For 4596H | ||
---|---|---|

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 |

Adar | Sat 14h 16m 17hl |
*Mon 28 Jan 836g |

v'Adar | 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

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

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

The Moladot For 4596H | ||
---|---|---|

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 |

Adar | Sat 14h 16m 17hl |
*Sat 26 Jan 836g |

v'Adar | 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

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 can answer (well, possibly), you must have something up your sleeve. 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**.

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

The ** Encyclopedia Judaica**, in its article on the

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

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

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

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**.

The Encyclopedia Judaica, in its article on the

cites the following leap year distributions as having been usedCalendar2 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 19Additionally, 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 876pIs the EJ entirely correct?

**NO!**

In the ** Encyclopedia Judaica**, the epochal molad given as

The traditional literature of the Hebrew calendar, such as the 8th century
** Seder Olam**, and the 11th century work

The ** Encyclopedia Judaica**, in its article on the

Apparent variations in theordo 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

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) and2d 20h 385p(for the 4th year) instead of the value4d 20h 408p

The following originally appeared as ** Weekly Question 58**.

Which Hebrew year could most reasonably be represented as an

epochalyear 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) and2d 20h 385p(for the 4th year) instead of the value4d 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

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**.

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

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

The following originally appeared as ** Weekly Question 63**.

Is it nececessary that

Rosh Hashannahbe postponed so as to have amoladprecede theShabbat M'Vorchimon which its month is announced?

**NO!**

The ** Shabbat** which immediately precedes the observance of

A ** molad** can precede the

The most recent ** molad** to occur prior to

That ** molad** arrived at

The following originally appeared as ** Weekly Question 67**.

In the full Hebrew calendar cycle of 689472 years, which

molad of Tishreiwill be the first to be repeated as amolad of Tishrei?

The time of any ** molad** is usually given as

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

For example, the ** molad of Tishrei 1H**,

** BaHaRad** will not be repeated as a

By one of these delightful coincidences, in the full Hebrew calendar cycle
of **689472 years**, ** BaHaRad** is also the very first

The following originally appeared as ** Weekly Question 68**.

In the full Hebrew calendar cycle of 689472 years, how often do the

moladot of Tishreirepeat themselves?

The time of any ** molad** is usually given as

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

In the full Hebrew calendar cycle of **689472 years**, any ** molad of Tishrei** is repeated either

For example, ** BaHaRad**, the

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**.

What fraction of the

moladot of Tishreiare repeated exactly 3 times in the full Hebrew calendar cycle of 689,472 years?

The ** moladot of Tishrei** can be repeated either exactly

There are **181,440** possible values for the ** moladot of Tishrei** and all are represented at least

Therefore, the fraction of theLet 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 ofmoladot of Tishreiin the full cycle) Hence, T = 36,288 which = 1/19 of 689,472 And 3*T = 108864 which = 3/19 of 689,472

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

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

Although the ** Weekly Question** is intrigued by

[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

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

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.

Thank you **Larry Padwa** for sharing your Hebrew calendar researches.

[Relative to June 2003g] When last did a calculated

moladresult in a time prior to its announcement onShabbat?

The ** Shabbat** which immediately precedes the observance of

The timing of the ** molad** is permitted to precede the

The most recent ** molad** to occur prior to the

That ** molad** arrived at

The following originally appeared as ** Weekly 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

** Rosh Hashannah 4H** occurred on

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**.

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

**383 day years** can begin on ** Mondays, Thursdays, and Saturdays**. If such years are followed by a

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

Hence, the earliest possible ** molad** for the beginning of the

On the other hand, a **383 day year** beginning on ** Monday** followed by a

would necessarily end on

at least

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

The following originally appeared as ** Weekly 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

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.

The following originally appeared as ** Weekly 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.

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

The following originally appeared as ** Weekly 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**.

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

The following originally appeared as ** Weekly 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

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

Which are two of the most important arithmetical properties of the 17th year of the

mahzor qatan(19 year cycle) known asGUChADZaT?

In **1802g**, the German mathematician **Carl Friedrich Gauss** published a formula which gave the ** Julian date** of

The **Gauss Pesach** formula appeared without proof of its derivation in

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

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

The most interesting observation which can be made from the above, is that the12 * 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

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

For example, ** Rosh HaShannah 5774H** will begin on

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

What is the second major arithmetic property of the 17th year of the

mahzor qatan(19 year cycle) known asGUChADZaT?

In **1802g**, the German mathematician **Carl Friedrich Gauss** published a formula which gave the ** Julian date** of

The **Gauss Pesach** formula appeared without proof of its derivation in

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

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

Therefore, a second property of the **17th year** of ** GUChADZaT** is that any period of

First Begun 21 Jun 1998 First Paged 2 Jan 2005 Next Revised 2 Jan 2005