What is, or are, the most significant common features of Hebrew years 264928H, 461920H, and 658912H?
Some of the common features of Hebrew years 264928H (Thu 8 Oct 261170g), 461920H (Mon 4 Feb 458165g), and 658912H (Sat 6 Jun 655159g) are:
1. They are all the 11th year of the GUChADZaT cycle.
2. They are all leap years.
3. All of their Tishrei moladot arrive at 21h 68p on the day of their molad.
4. All of their Shevat moladot are on the same day as their Tishrei moladot.
5. All of their Shevat moladot arrive at 0h 0p on the day of their molad.
6. All of their immediately subsequent Tishrei moladot arrive at 18h 657p on the day of their molad.
7. However, the most significant of their common features is that, absent Dehiyah Molad Zaqen, the first day of Shevat in each of these years would precede their molad of Shevat.
This Hebrew calendar phenomenon is fully explained in The Overpost Problem which shows that the function of Dehiyah Molad Zaqen is to prevent the first day of any Hebrew month from preceding the calculated time of the molad for that month.
Why are 352 day years not possible in the fixed Hebrew calendar?
Equation 4.5, in Properties of Hebrew Year Periods  Part 1, shows that the length of any 12 month Hebrew year is given by
where p' = the number of days Rosh Hashannah is postponed for the 1st year p" = the number of days Rosh Hashannah is postponed for the 2nd year f = the time on the day of the molad of Tishrei for the 1st year m = 8h 876p [f+m] = to 1d if f + 8h 876p => 1d, or else 0d.
So that the length of the 12 month year be equal to 352 days, it is necessary that f + m < 1d, p" = 0d, and p' = 2d.
p' = 2d implies that f => 18h Hence, f + 8h 876p => 18h + 8h 876p = 1d 2h 876p which contradicts the requirement that f + m < 1d.
Equation 4.5 above also shows that any other circumstance forcing the value of p" to become nonzero,
such as f => 9h 204p when the molad of Tishrei falls on a Tuesday, will make the 352 day year impossible.
Therefore, it is not possible to generate a 352 day year in the fixed Hebrew calendar.
Are people who become 32,896 days old on their Hebrew birthday born in a 12 month or a 13 month Hebrew year?
Anyone who is 32,896 days old on their Hebrew birthday is born in a 13 month Hebrew year.
247 Hebrew Year Periods tabulates the possible lengths for periods of Hebrew years that are from 1 to 247 years long.
A quick search of the tables shows that 32,896 days is the shortest length possible for periods of 90 years which contain the extra leap year.
For periods of 90 Hebrew years, the extra leap year is possible only if the period begins with a leap year.
If the 90 year period, as measured from Tishrei 1, does contain the extra leap year, then the last year of that period can be shown to always be a leap year.
Consequently, as looked upon from years beginning on Tishrei 1, if someone is 32,896 days old on their Hebrew birthday, and the 90th birthday is in a leap year, then that person was born in a 12 month year. Similarly, if the 32,896th day of life occurs in a 12 month year, then that person was born in a 13 month year.
However, if the time is measured from the person's 1st day of life, then it must be noted that their 1st birthday will be 13 months later, because the first year of that specific period of time is always 13 months long. So from this perspective, anyone who is 32,896 days old on their Hebrew birthday is born in a 13 month year.
As a side note, their 90th birthday will always be 13 months after their 89th birthday.
Correspondents Winfried Gerum and Robert H. Douglass both supplied excellent answers.
Winfried Gerum observed that:
Robert H. Douglass shared these results:If someone is exactly 32896 days old in his 90th (hebrew) birthday, he may be either born in the first half of a 13monthyear or in the second half of a 12month year. The next such years are 353: never! 354: 5789 (Nissan/Iyyar/Sivan/Tymmuz/Av/Elul) 354: 5890 (Nissan/Iyyar/Sivan/Tymmuz/Av/Elul) 383: 5790 (Tishrei/Cheshvan/Kislev) 384: never! 385: 5945 (Tevet/Shevat/Adar I) It the birth is in a leap year, the 90th anniversary is not and vice versa!
Thank you Winfried Gerum and Robert H. Douglass for your intriguing insights into this question.Here are my empirical results for the requirements if a Hebrew date is to by followed by the same date 90 years later with an offset of 32896 days: The first Adar following the first date must be followed by an Intercalary Month (II Adar) which is part of a 383day year. The last Adar preceding the final date must be followed by an Intercalary Month (II Adar) which is part of a 383day year. Since deficient years cannot follow one another, the year following this final 383day year may contain either 354 or 355 days. If it contains 354 days, the "Birthday Match" occurs through the end of Kislev. If it contains 355 days, the "Birthday Match" occurs through the end of Heshvan. In both cases, the "Birthday Match" also works prior to Tishrei, as far back as 1 Nissan (ie, immediately following the last Intercalary Month). Example of first case: HY 5668  5758 (or late 5667  5757). Example of second case: HY 5695  5785 (or late 5694  5784).
Weekly Question 153 introduces the fact that periods of 90 years which are 32,896 days long, not only contain the extra leap year possible, but also begin with a 13 month year and end with a 13 month year.
Which periods of Hebrew years beginning with 13 month years end with 13 month years?
Weekly Question 153 introduces the fact that periods of 90 years which are 32,896 days long, not only contain the extra leap year possible, but also begin with a 13 month year and end with a 13 month year.
Let Y = the number of years in some period of Hebrew years.
As measured from Tishrei 1, in order that a period of Y Hebrew years begin with a 13 month year, the period must begin in any one of years 3, 6, 8, 11, 14, 17, 19 of the mahzor qatan (19 year cycle) identified by leap year distribution GUChADZaT.
However, in order that the period of Y Hebrew years also always end with a 13 month year, the length of the period must be such that Y MOD 19 is either 1, 3, 6, 9, 11, 14, or 17, and that period must contain the extra month possible for that number of years.
This phenomenon can be demonstrated more easily than explained using the following two tables also shown in Properties of Hebrew Year Periods  Part 2.
Some of the entries have been highlighted to better show the relationship to the leap years, as numbered from GUChADZaT, of the 13 month counts in spans 1, 3, 6, 9, 11, 14, and 17.
It is to be noted that for the spans Y MOD 19 = {1, 3, 6, 9, 11, 14, 17} the extra month count, applying to M" + m", occurs only when the span begins in a leap year.
Year i of GUChADZaT  

SPAN  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19 
1  0  0  1  0  0  1  0  1  0  0  1  0  0  1  0  0  1  0  1 
2  0  1  1  0  1  1  1  1  0  1  1  0  1  1  0  1  1  1  1 
3  1  1  1  1  1  2  1  1  1  1  1  1  1  1  1  1  2  1  1 
4  1  1  2  1  2  2  1  2  1  1  2  1  1  2  1  2  2  1  2 
5  1  2  2  2  2  2  2  2  1  2  2  1  2  2  2  2  2  2  2 
6  2  2  3  2  2  3  2  2  2  2  2  2  2  3  2  2  3  2  2 
7  2  3  3  2  3  3  2  3  2  2  3  2  3  3  2  3  3  2  3 
8  3  3  3  3  3  3  3  3  2  3  3  3  3  3  3  3  3  3  3 
9  3  3  4  3  3  4  3  3  3  3  4  3  3  4  3  3  4  3  4 
10  3  4  4  3  4  4  3  4  3  4  4  3  4  4  3  4  4  4  4 
11  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4  5  4  4 
12  4  4  5  4  4  5  4  5  4  4  5  4  4  5  4  5  5  4  5 
13  4  5  5  4  5  5  5  5  4  5  5  4  5  5  5  5  5  5  5 
14  5  5  5  5  5  6  5  5  5  5  5  5  5  6  5  5  6  5  5 
15  5  5  6  5  6  6  5  6  5  5  6  5  6  6  5  6  6  5  6 
16  5  6  6  6  6  6  6  6  5  6  6  6  6  6  6  6  6  6  6 
17  6  6  7  6  6  7  6  6  6  6  7  6  6  7  6  6  7  6  6 
18  6  7  7  6  7  7  6  7  6  7  7  6  7  7  6  7  7  6  7 
19  7  7  7  7  7  7  7  7  7  7  7  7  7  7  7  7  7  7  7 
Some of the entries have been highlighted to better show the relationship to the leap years, as numbered from GUChADZaT, of the extra month counts in spans 1, 3, 6, 9, 11, 14, and 17.
It is to be noted that for the spans Y MOD 19 = {1, 3, 6, 9, 11, 14, 17} the extra month count, applying to M" + m", occurs only when the span ends in a leap year.
Year i of GUChADZaT  

SPAN  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19 
1  0  0  1  0  0  1  0  1  0  0  1  0  0  1  0  0  1  0  1 
2  1  0  1  1  0  1  1  1  1  0  1  1  0  1  1  0  1  1  1 
3  1  1  1  1  1  1  1  2  1  1  1  1  1  1  1  1  1  1  2 
4  2  1  2  1  1  2  1  2  2  1  2  1  1  2  1  1  2  1  2 
5  2  2  2  2  1  2  2  2  2  2  2  2  1  2  2  1  2  2  2 
6  2  2  3  2  2  2  2  3  2  2  3  2  2  2  2  2  2  2  3 
7  3  2  3  3  2  3  2  3  3  2  3  3  2  3  2  2  3  2  3 
8  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3  2  3  3  3 
9  3  3  4  3  3  4  3  4  3  3  4  3  3  4  3  3  3  3  4 
10  4  3  4  4  3  4  4  4  4  3  4  4  3  4  4  3  4  3  4 
11  4  4  4  4  4  4  4  5  4  4  4  4  4  4  4  4  4  4  4 
12  4  4  5  4  4  5  4  5  5  4  5  4  4  5  4  4  5  4  5 
13  5  4  5  5  4  5  5  5  5  5  5  5  4  5  5  4  5  5  5 
14  5  5  5  5  5  5  5  6  5  5  6  5  5  5  5  5  5  5  6 
15  6  5  6  5  5  6  5  6  6  5  6  6  5  6  5  5  6  5  6 
16  6  6  6  6  5  6  6  6  6  6  6  6  6  6  6  5  6  6  6 
17  6  6  7  6  6  6  6  7  6  6  7  6  6  7  6  6  6  6  7 
18  7  6  7  7  6  7  6  7  7  6  7  7  6  7  7  6  7  6  7 
19  7  7  7  7  7  7  7  7  7  7  7  7  7  7  7  7  7  7  7 
Therefore, periods of Y Hebrew years beginning with 13 month years will always end with 13 month years as long as
Y MOD 19 = either 1, 3, 6, 9, 11, 14, or 17 and the period also contains the extra leap month possible for Y years.
The next question first appeared as Weekly Question 37.
When did the least frequently occurring 247 Hebrew year period last begin?
As shown by the 247 year spans table in 247 Hebrew Year Periods, reproduced below, periods of 247 Hebrew years can be either 90214, 90215, or 90216 days long.
247 YEAR SPANS  

3,055 months = 90,215d 23h 175p 

M'+/  DAYS  MOD 7d  OCCURS 
2  0d  0  0 
1  90,214d  5  10,317 
0  90,215d  6  3,439 
1  90,216d  0  675,716 
2  0d  0  0 
3  0d  0  0 
The maximum variance is 2 days 
These periods occur at the rate of 1.5%, 0.5%, and 98% of the time respectively. Hence, the 90,215day period is the least frequently occurring period of 247 years.
The 90,215day period of 247 Hebrew years occurs only 3,439 times in the full Hebrew calendar cycle of 689,472 years.
This period of time last began on Tue 27 Sep 1927g (5688H).
What would be the dehiyot (Hebrew calendar postponement rules) were Dehiyah Molad Zaqen applied only to the Tishrei moladot that immediately follow Hebrew leap years?
Dehiyah Molad Zaqen calls for a postponement to the next allowable day for 1 Tishrei whenever the molad of Tishrei is on an allowable day on or past 18h.
This rule prevents the first day of any month from completing prior to the occurrence of its corresponding molad. A full explanation of this calendar arithmetic phenomenon is found in The Overpost Problem.
Absent Dehiyah Molad Zaqen, the molad could occur as late as 5h 422p on the second day of Shevat.
The analysis in The Overpost Problem indicates that the calendar problem only occurs in leap years. Consequently, it seems possible to control the phenomenon by applying the necessary postponement only to the Tishrei moladot that immediately follow leap years.
Assuming that Dehiyah Molad Zaqen is to be applied only to post leap year moladot on or past 18h then the dehiyah that eliminates 356 day years may be changed so that the postponement takes place whenever the molad of Tishrei for a 12 month year occurs on a Tuesday on or past 15h 204p.
Dehiyah Lo ADU Rosh, forbidding 1 Tishrei on either Sunday, Wednesday, or Friday, remains unchanged.
When these 3 rules are substituted for the currently known set of rules, single years are either 353, 354, 355, 383, 384, or 385 days long, and all of the moladot in the full Hebrew calendar cycle of 689,472 years occur prior to the closing of the first day of their corresponding months.
At first sight, this solution does appear to be reasonably complete.
What additional problem, or problems, might have be solved when Dehiyah Molad Zaqen is applied as suggested in Weekly Question 155?
Weekly Question 156 required that Dehiyah Molad Zaqen be applied exclusively to the years immediately following 13 month years.
The analysis in The Overpost Problem indicates that the calendar problem only occurs in leap years. Consequently, it seems possible to control the phenomenon by applying the necessary postponement only to the Tishrei moladot that immediately follow leap years as follows:
1.Apply Dehiyah Molad Zaqen only to post leap year moladot past 17h 1079p.
2. Dehiyah Lo ADU Rosh, forbidding 1 Tishrei on either Sunday, Wednesday, or Friday, remains unchanged;
3. Whenever the molad of Tishrei of a 12 month year is on Tuesday past 15h 203p, postpone 1 Tishrei to Thursday.
When these 3 rules are substituted for the currently known set of rules, single years are either 353, 354, 355, 383, 384, or 385 days long, and all of the moladot in the full Hebrew calendar cycle of 689,472 years occur prior to the closing of the first day of their corresponding months.
At first sight, this solution does appear to be reasonably complete.
However, this particular combination of rules introduces a new keviah, namely, 385 day years beginning on Tuesday.
In order to maintain the calendar's existing keviyot, as shown in The Keviyyot, 385 day years beginning on Tuesday need to be eliminated. This elimination is made possible by the introduction of the following dehiyah:
4. Whenever the molad of Tishrei of a 13 month year is on Tuesday past 20h 490p, postpone 1 Tishrei to Thursday.
The 3rd and 4th dehiyot above will now help to maintain the existing calendar's 14 keviyot.
The following first appeared as Weekly Question 38.
What's the smallest number of consecutive Hebrew years that cannot add up to a whole number of weeks?
The smallest number of Hebrew years that cannot add to a whole number of weeks is 8 years.
1 Hebrew year can have 385 days which is 55 weeks. 2 consecutive Hebrew years can have 707 days which is 101 weeks. 3 consecutive Hebrew years can have 1092 days which is 156 weeks. 4 consecutive Hebrew years can have 1449 days which is 207 weeks. 4 consecutive Hebrew years also can have 1477 days which is 211 weeks. 5 consecutive Hebrew years can have 1799 days which is 257 weeks. 6 consecutive Hebrew years can have 2184 days which is 312 weeks. 7 consecutive Hebrew years can have 2541 days which is 363 weeks. 7 consecutive Hebrew years also can have 2569 days which is 367 weeks.
However, 8 consecutive Hebrew years do not have a single length that is formed from a whole number of weeks.
247 Hebrew Year Periods shows these lengths statistics for spans of 8 Hebrew years.
8 YEAR SPANS  

98 months = 2,893d 23h 1034p 
99 months = 2,923d 12h 747p 

M'+/  DAYS  MOD 7d  OCCURS  M"+/  DAYS  MOD 7d  OCCURS  
2  0d  0  0  2  0d  0  0  
1  2,892d  1  9  1  2,922d  3  65,095 

0  2,893d  2  12,828 
0  2,923d  4  201,652 

1  2,894d  3  15,004 
1  2,924d  5  358,028 

2  2,895d  4  8,447 
2  2,925d  6  28,409 

3  0d  0  0  3  0d  0  0  
The maximum variance is 33 days 
Of all of the lengths possible for 8 consecutive Hebrew years, the shortest of these is 2,892 days long and it only occurs 9 times in the full 689,472 year cycle of the Hebrew calendar.
HAG SAMEACH!
In our times, is March 28 the earliest possible date for Pesach?
No!
The earliest possible date for the start of Pesach in our times is March 26.
Correspondents Larry Padwa and Robert H. Douglass provided these wonderful observations.
Larry Padwa's research found the following
The first day of Pesach will occur on March 26 in eleven years (2013G). This is the earliest date in our lifetimes. It last occurred on March 26 in 1899. In order to get an earlier date than March 26, we have to go all the way back to March 25, 1766. Also the first day of Pesach occurred on March 27 several times, most recently in 1994, and will again in 2032.
In addition to the Gregorian calendar, Robert H. Douglass mapped his findings onto the Julian calendar which moves more slowly than the Hebrew calendar, producing a number of startling and delightfule results!
In 1994 Pesach was March 27, and in 2013 Pesach will be March 26. On the Julian calendar, Pesach comes progressively earlier, while on the Gregorian calendar, Pesach comes progressively later. In the year 360 CE, Pesach was on March 18 Julian for the first time (according to the current rules for the Hebrew Calendar). 664 was the first occurrence on March 17. 1101 was the first time for March 16. 1348 was the first time for March 15. On the Gregorian calendar, which began in 1582, the earliest Pesach ever was on March 24 in the year 1652 (a single occurrence on this date). This year was also the first time for March 14 Julian. 1766 was the last time for March 25 Gregorian. For March 26 Gregorian, the most recent occurrence was 1899, and the final two occurrences will be in 2013 and 2089. 2013 also marks the first time for March 13 Julian. The last occurrence for March 27 Gregorian will be in 2260, which will also be the first occurrence for March 12 Julian. The final occurrence for March 28 Gregorian will be in 2488. HAG SAMEACH!  Robert H. Douglass
Thank you Larry Padwa and Robert H. Douglass for your fabulous answers!
HAG SAMEACH!
The next question first appeared as Weekly Question 40.
Since the period of the molad is less than 30 days, is it possible for the molad to occur beyond the 30th day of a Hebrew month?
The answer raises a number of apparent paradoxes.
Although the period of the molad is less than 30 days, it is possible for a molad to make its appearance beyond the 30th day of some Hebrew months.
Instantly, as shown by the considerations of the The Overpost Problem, it is possible to realize that the molad can never occur 30 days after the first day of deficient months. If that were the case, then the molad would come after the first day of some months.
However, the molad can arrive 30 days after the first day of full months. Such days would be the first days of the subsequent new month.
Under no circumstance is it ever possible for the molad to arrive 31 days after the first day of any Hebrew month.
Calendar arithmetic shows that the molad can occur 30 days after the first day Tishrei, Kislev, Shevat, Adar I, Nisan, Sivan, and Av.
In abundant years, only the month of Heshvan can, in some years, have its molad occur on the 31st day of its predecessor month of Tishrei.
The phenomenon of a molad on the 31st day of its preceding month does not occur in every Hebrew year.
The next question first appeared as Weekly Question 41.
When was the most recent occurrence of a molad on the 31st day of its preceding month?
The molad can arrive 30 days after the first day of full months. Such days would be the first days of the subsequent new month.
Under no circumstance is it ever possible for the molad to arrive 31 days after the first day of any Hebrew month.
Calendar arithmetic shows that the molad can occur 30 days after the first day Tishrei, Kislev, Shevat, Adar I, Nisan, Sivan, and Av.
In abundant years, only the month of Heshvan can, in some years, have its molad occur on the 31st day of its predecessor month of Tishrei.
The phenomenon of a molad on the 31st day of its preceding month does not occur in every Hebrew year.
This last happened for the molad of Heshvan 5759H (1998g), which
occurred on
Wednesday 21 October 1998g at 1h 39m 16hl.
Correspondent Robert H. Douglass correctly answered as follows:
There was a molad on Wednesday, October 21, 1998 gregorian at 1 hr 718 parts. This day was the First of Heshvan... 30 days after September 21 which was the First of Tishrei in that year (HY 5759). This is the most recent occurrence of a molad on the "31st" of the previous month.
Correspondent Robert H. Douglass also provided the answer to the next question which first appeared as
Weekly Question 42.
When will be the next occurrence of a molad on the 31st day of its preceding month?
The molad can arrive 30 days after the first day of full months. Such days would be the first days of the subsequent new month.
Under no circumstance is it ever possible for the molad to arrive 31 days after the first day of any Hebrew month.
Calendar arithmetic shows that the molad can occur 30 days after the first day Tishrei, Kislev, Shevat, Adar I, Nisan, Sivan, and Av.
In abundant years, only the month of Heshvan can, in some years, have its molad occur on the 31st day of its predecessor month of Tishrei.
The phenomenon of a molad on the 31st day of its preceding month does not occur in every Hebrew year.
This last happened for the molad of Heshvan 5759H (1998g), which
occurred on
Wednesday 21 October 1998g at 1h 39m 16hl.
The next molad to occur on the 31st day of its preceding month will be the molad of Elul 5762H corresponding to Friday 9 August 2002g. The time of this molad will be 0h 10m 9hl.
Correspondent Robert H. Douglass correctly answered as follows:
Thank you Robert H. Douglass for sharing your very complete answer!The next such occurrrence will be on Friday, August 9,2002 gregorian at 0 hrs 189 parts, the First of Elul... 30 days after July 10 which will be the First of Av this year (HY 5762).
The next question which first appeared as Weekly Question 43.
The following 2 tables are found in Dr. C. E. Sachau's 1879 translation of Albiruni's 11th century work The Chronology of Ancient Nations.
The tables show the week days possible for both the first day of a Hebrew month (Arabic numerals) and the first day of Rosh Hodesh (Roman numerals) when it coincides with the last day of the old month.
These tables may be found on pages 155 and 156 of the cited work.
The terms imperfect, regular, and perfect respectively refer to Hebrew years that are 353 or 383 days long, 354 or 384 days long, 355 or 385 days long.
While there has been a slight modification to the labels so as to make their spelling and content more understandable, the actual numbers, both Roman and Arabic, have not changed in the values shown.
 Table Showing On What Days of the Week the Beginning of the Months Falls Throughout the Year Table of Common Years ====================================================================== S M i e g n M n s a u i r m s h T S K e i T a S N h T i s n i E m i I i A e e s h i s l m v y s d v v l v Quality t h u A u a a a a a e e a of the i r l v z n r n r t t v n Year i i ===== = ===== = ==== = ===== = ===== ===== ===== ========= === 4 III 2 1 VI 6 5 IV 3 2 I 7 6 V 4 III 2 I Perfect 7 2 I 7 6 V 4 3 II 1 7 VI 5 4 3 2 I Imperfect 7 6 V 4 3 II 1 7 VI 5 4 III 2 1 VII 6 V 4 III Perfect 2 4 III 2 1 VII 6 5 IV 3 2 I 7 6 5 4 III Imperfect 2 6 V 4 3 II 1 7 VI 5 4 III 2 1 VII 6 5 IV Regular 3 2 I 7 6 V 4 3 II 1 7 VI 5 4 III 2 I 7 VI Perfect 5 1 VII 6 5 IV 3 2 I 7 6 V 4 3 II 1 7 VI Regular 5  Table Showing On What Days of the Week the Beginning of the Months Falls Throughout the Year Table of Leap Years ============================================================================ S M i e g n M n s a u i S r m s e h T c P S K e i T a S N u r h T i s n i E m i I i A n A i e e s h i s l m v y s d d d m v v l v Quality t h u A u a a a a u a u a e e a of the i r l v z n r n r s r s t t v n Year i i ===== = ===== = ==== = ===== ===== = ===== ===== ===== ========= === 6 V 4 3 II 1 7 VI 5 4 III 2 I 7 6 V 4 III 2 I Perfect 7 4 III 2 1 VII 6 5 IV 3 2 I 7 VI 5 4 3 2 I Imperfect 7 1 VII 6 5 IV 3 2 I 7 6 V 4 III 2 1 VII 6 V 4 III Perfect 2 6 V 4 3 II 1 7 VI 5 4 III 2 I 7 6 5 4 III Imperfect 2 1 VII 6 5 IV 3 2 I 7 6 V 4 III 2 1 VII 6 5 IV Regular 3 4 III 2 1 VII 6 5 IV 3 2 I 7 VI 5 4 III 2 I 7 VI Perfect 5 2 I 7 6 V 4 3 II 1 7 I 5 IV 3 2 1 7 VI Imperfect 5 
Are the Albiruni Rosh Hodesh tables as shown in the Sachau translation free of numeric typographical error?
The following 2 tables are found on pages 155 and 156 in Dr. C. E. Sachau's 1879 translation of Albiruni's 11th century work The Chronology of Ancient Nations.
The tables show the week days possible for both the first day of a Hebrew month (Arabic numerals) and the first day of Rosh Chodesh (Roman numerals) when it coincides with the last day of the old month.
The terms imperfect, regular, and perfect respectively refer to
Hebrew years that are
353 or 383 days long, 354 or 384 days long, 355 or 385 days long.
While there has been a slight modification to the labels so as to make their spelling and content more understandable, the actual numbers, both Roman and Arabic, have not changed in the values shown.
 Table Showing On What Days of the Week the Beginning of the Months Falls Throughout the Year Table of Common Years ====================================================================== S M i e g n M n s a u i r m s h T S K e i T a S N h T i s n i E m i I i A e e s h i s l m v y s d v v l v Quality t h u A u a a a a a e e a of the i r l v z n r n r t t v n Year i i ===== = ===== = ==== = ===== = ===== ===== ===== ========= === 4 III 2 1 VI 6 5 IV 3 2 I 7 6 V 4 III 2 I Perfect 7 2 I 7 6 V 4 3 II 1 7 VI 5 4 3 2 I Imperfect 7 6 V 4 3 II 1 7 VI 5 4 III 2 1 VII 6 V 4 III Perfect 2 4 III 2 1 VII 6 5 IV 3 2 I 7 6 5 4 III Imperfect 2 6 V 4 3 II 1 7 VI 5 4 III 2 1 VII 6 5 IV Regular 3 2 I 7 6 V 4 3 II 1 7 VI 5 4 III 2 I 7 VI Perfect 5 1 VII 6 5 IV 3 2 I 7 6 V 4 3 II 1 7 VI Regular 5  Table Showing On What Days of the Week the Beginning of the Months Falls Throughout the Year Table of Leap Years ============================================================================ S M i e g n M n s a u i S r m s e h T c P S K e i T a S N u r h T i s n i E m i I i A n A i e e s h i s l m v y s d d d m v v l v Quality t h u A u a a a a u a u a e e a of the i r l v z n r n r s r s t t v n Year i i ===== = ===== = ==== = ===== ===== = ===== ===== ===== ========= === 6 V 4 3 II 1 7 VI 5 4 III 2 I 7 6 V 4 III 2 I Perfect 7 4 III 2 1 VII 6 5 IV 3 2 I 7 VI 5 4 3 2 I Imperfect 7 1 VII 6 5 IV 3 2 I 7 6 V 4 III 2 1 VII 6 V 4 III Perfect 2 6 V 4 3 II 1 7 VI 5 4 III 2 I 7 6 5 4 III Imperfect 2 1 VII 6 5 IV 3 2 I 7 6 V 4 III 2 1 VII 6 5 IV Regular 3 4 III 2 1 VII 6 5 IV 3 2 I 7 VI 5 4 III 2 I 7 VI Perfect 5 2 I 7 6 V 4 3 II 1 7 I 5 IV 3 2 1 7 VI Imperfect 5 
The Albiruni tables are read right to left, and line by line.
The first day of Rosh Hodesh is always 29 days following the first day of a Hebrew month. Since 29 is 1 more than an even multiple of 7, the weekday of the subsequent Rosh Hodesh always falls one weekday later than the first day of the current month.
As an example, if the first day of the month falls on Tuesday, then the first day of the subsequent Rosh Hodesh will be Wednesday.
When a given month is 29 days long, then both the first day of the next month and the subsequent Rosh Hodesh coincide. That is why the Albiruni tables only show a single Arabic numeral as the week day following a 29 day month.
When a month is 30 days long, then the first day of the subsequent month is one day later in the week than the first day of the subsequent Rosh Hodesh. For example, if the first day of a 30 day month is Friday, then the subsequent Rosh Hodesh is on a Saturday, and the first day of the next month is on Sunday.
The Albiruni tables show this distinction using Roman numerals for the first day of Rosh Hodesh followed immediately by an Arabic numeral that represents a value one higher than the Roman numeral.
When the numeric values of the week days for Rosh Hodesh and the first day of the months are listed side by side, as is done in the Albiruni tables, the numbers read continuously from some starting point on to 7, and recycle starting at 1.
This pattern becomes evident in the Albiruni tables when reading all the numbers on a line by line, right to left basis.
That's why the two typographical errors in these tables can be so easily spotted.
The first error is in the first table. On the first line, under the heading of Tammuz, the Roman numeral value VI must be VII.
The second error is in the second table. On the last line, under the heading of Adar Secundus, the Roman numeral value I must be VI.
In the year 4681H (920g) Aaron ben Meir proposed that 642 halakim be added to Dehiyah Molad Zaqen limit, thus causing Rosh HaShannah for the years 4683H (922g), 4684H (923g), and 4688H (927g) (as well as their preceding Pesach's) to arrive 2 days earlier than predicted in the currently fixed Hebrew calendar.
Because of this difference in dates, ben Meir's proposal created a major calendar controversy between the schools of Palestine and Babylon. Please see The Ben Meir Years for more detailed information.
In changing the Dehiyah Molad Zaqen limit, what could Aaron ben Meir also have done to produce the same dates for Rosh Hashannah, as found in the currently fixed Hebrew calendar?
Dehiyah Molad Zaqen is the Hebrew calendar mechanism which prevents the calculated time of the molad from occurring on the 2nd day of a new month.
For example, absent Dehiyah Molad Zaqen, 1 Shevat 5760 would have been on Thursday 6 January 2000g, while the molad of Shevat 5760H would have been on Friday 2 Shevat 5760H at 0h 733p.
To correct for this problem, Dehiyah Molad Zaqen calls for a postponement to the next allowable day for 1 Tishrei, whenever the molad of Tishrei is on or past 18h (12:00 noon) on an allowable day for 1 Tishrei.
Analysis shows that the maximum possible value for the limit to Dehiyah Molad Zaqen is 18h 656p.
In the year 4681H (920g) Aaron ben Meir proposed that 642 halakim be added to the Dehiyah Molad Zaqen limit, thus causing Rosh HaShannah for the years 4683H (922g), 4684H (923g), and 4688H (927g) (as well as their preceding Pesach's) to arrive 2 days earlier than predicted in the currently fixed Hebrew calendar.
Because of this difference in dates, ben Meir's proposal created a major calendar controversy between the schools of Palestine and Babylon. Please see The Ben Meir Years for more detailed information.
The following table compares the currently held Rosh Hashannah dates in relation to the dates that are generated under Aaron Ben Meir's proposal, after 642 halakim are added to the limits of both the 356 and 382 Day Rules so as to maintain the existing keviyot.
Rosh Hashannah Dates Under the Ben Meir Rules
Comparison of RH Dates 900g2300g Current RH Dates Meir's RH Dates ================ =============== Thu 922 Oct 1  Tue 922 Sep 29 Mon 923 Sep 20  Sat 923 Sep 18 Sat 927 Sep 6  Thu 927 Sep 4  Tue 1108 Sep 15  Mon 1108 Sep 14  Sat 1330 Sep 23  Thu 1330 Sep 21 Thu 1334 Sep 9  Tue 1334 Sep 7 Tue 1335 Sep 27  Mon 1335 Sep 26  Thu 1514 Oct 1  Tue 1514 Sep 29 Mon 1515 Sep 20  Sat 1515 Sep 18 Sat 1519 Sep 6  Thu 1519 Sep 4  Sat 1655 Oct 2  Thu 1655 Sep 30 Tue 1700 Sep 14  Mon 1700 Sep 13 Mon 1840 Sep 28  Sat 1840 Sep 26 Sat 1844 Sep 14  Thu 1844 Sep 12  Tue 2025 Sep 23  Mon 2025 Sep 22 Thu 2028 Sep 21  Tue 2028 Sep 19 Mon 2029 Sep 10  Sat 2029 Sep 8  Sat 2247 Oct 2  Thu 2247 Sep 30 Thu 2251 Sep 18  Tue 2251 Sep 16 Tue 2252 Oct 5  Mon 2252 Oct 4
The difference in dates arises because Ben Meir did not also add 642 halakim to the moladot themselves. If Ben Meir also had decided that the molad of the year 2H was on Friday at 14h 642p instead of the traditionally accepted Friday at 14h 0p, then no conflict whatever would have arisen over the dates of Rosh Hashannah and the Roshei Hadashim (new months) since these would all have remained the same.
The most famous example of this type of adjustment can be found in The Gauss Pesach Formula. A close examination of its constants shows that in order to eliminate Dehiyah Molad Zaqen, Karl Friedrich Gauss added six hours to the limits of the dehiyot AND AS WELL added 6 hours to BaHaRaD, the molad of Tishrei 1H (Mon 7 Sep 3760g).
The answer suggested that absent Dehiyah Molad Zaqen, the molad of Shevat 5760H would have arrived on 2 Shevat 5760H (Fri 7 Jan 2000g).
Absent Dehiyah Molad Zaqen, when next will a molad occur on the 2nd day of a Hebrew month?
Dehiyah Molad Zaqen is the Hebrew calendar mechanism which prevents the calculated time of the molad from occurring on the 2nd day of a new month.
For example, absent Dehiyah Molad Zaqen, 1 Shevat 5760H would have been on Thursday 6 January 2000g, while the molad of Shevat 5760H would have been on Friday 2 Shevat 5760H at 0h 733p.
To correct for this problem, Dehiyah Molad Zaqen calls for a postponement to the next allowable day for 1 Tishrei, whenever the molad of Tishrei is on or past 18h (12:00 noon) on an allowable day for 1 Tishrei.
Analysis shows that the maximum possible value for the limit to Dehiyah Molad Zaqen is 18h 656p.
The molad of Tishrei 5777H is Shabbat at 20h 724p. Since 5777H is the year past a 13 month year, then absent Dehiyah Molad Zaqen, at least one month in the year 5776H will have its molad on the second day of that month.
Absent Dehiyah Molad Zaqen, the next molad to occur on the second day of a Hebrew month will be the molad of Shevat 5776H calculated to be on Sunday 10 January 2016g at 2h 67p, while the first day of Shevat 5776H would be Shabbat 9 January 2016g.
Correspondent Robert H. Douglass correctly confirmed this result.
Absent Dehiyah Molad Zaqen, when next will a molad occur on the 2nd day of a Hebrew month? This would be Shevat of HY 5776. There is a molad on Sunday, Jan 10, 2016 CE at 2h 67p. The first of Shevat is on Monday, Jan 11. If there were no Dehiyah Molad Zaqen, the first of Shevat would be on Saturday, Jan 9, 2016... which would place the Molad on the second day of the Hebrew month. The explanation is as follows. The subsequent Molad of Tishrei would be Saturday, Oct 1, 2016 at 20h 724p. Normally the first of Tishrei would defer to Sunday because of Molad Zaqen, and again to Monday because of Dehiyyah Lo ADU Rosh, thereby starting the Year on Monday, Oct 3, 2016. This gives HY 5776 a length of 385 days. If it were not for Molad Zaqen, then the next first of Tishrei would be on the day of the Molad, Saturday Oct 1, 2016, giving HY 5776 a length of 383 days. For this reason, the month of Shevat would commence two days earlier than on the actual calendar, placing the Molad of Shevat on the second day of that month. Regards, Robert H. Douglass
Correspondent Robert H. Douglass then went on to ask, and to answer, Weekly Question 160.
Is Birkat HaHamah (the blessing of the sun) always said in the Hebrew month of Nisan?
No!
Birkat HaHamah (the Blessing of the Sun) is recited once in every 28 years. This timing is based on one of the rabbinic ideas which estimated the length of the solar year to be exactly 365d 6h long, in spite of rabbinic knowledge of a better estimate for that year.
Correspondent Robert H. Douglass not only asked, but also answered, Weekly Question 160.
Noting that the sun traditionally was considered to have been at the vernal equinox on Wed 22 Adar 1H (24 Feb 3760g; 26 Mar 3759j), he calculated that the last time that Birkat HaHamah would not have been recited in Nisan was Wed 27 v'Adar 5461H (6 Apr 1701g; 26 Mar 1701j).
Thank you also correspondent Larry Padwa for sending in the correct answer!
Correspondent Robert H. Douglass then asked the next question which is now shown as Weekly Question 161.
First Begun 21 Jun 1998 First Paged 5 Dec 2004 Next Revised 5 Dec 2004