Does the fast of the 17th day of Tammuz occur most often on a Sunday?
YES!
Since 17 Tammuz 5761H occurred Sun 8 Jul 2001g, correspondent Ira Walfish asked if the fast of the 17th day of Tammuz occurred most often on a Sunday.
The First Day of The Month shows the statistical occurences of the first day of Tammuz over the full Hebrew calendar cycle of 689,472 years.
Start of Month Distribution by Week Day  

Sun  Mon  Tue  Wed  Thu  Fri  Sat  Totals  
Tishrei  0  193280  79369  0  219831  0  196992  689472 
Heshvan  0  196992  0  193280  79369  0  219831  689472 
Kislev  151093  68738  69853  127139  79369  193280  0  689472 
Tevet  193280  26677  124416  138591  0  206508  0  689472 
Shevat  0  193280  26677  124416  138591  0  206508  689472 
Adar  0  206508  0  193280  26677  124416  138591  689472 
v'Adar  0  85899  0  72576  0  68864  26677  254016 
Nisan  79369  0  219831  0  196992  0  193280  689472 
Iyar  0  193280  79369  0  219831  0  196992  689472 
Sivan  196992  0  193280  79369  0  219831  0  689472 
Tammuz  219831  0  196992  0  193280  79369  0  689472 
Av  0  219831  0  196992  0  193280  79369  689472 
Elul  193280  79369  0  219831  0  196992  0  689472 
Totals  1033845  1463854  989787  1345474  1153940  1282540  1258240  8527680 
Leap Adar  0  72576  0  68864  26677  0  85899  254016 
Since Tammuz 17 is exactly 2 weeks and 2 days later than the start of the month, it always falls on either Tuesday, Thursday, Saturday, or Sunday.
Also to be realized, in order to answer the question, is that rabbinically ordained fasts coinciding with Shabbat are postponed to Sunday.
Consequently, over the full Hebrew calendar cycle of 689,472 years, the fast of the 17th day of Tammuz occurs 193,280 + 79369 = 272,6549 times on Sunday, which is more often than any other day of the week.
Correspondents Winfried Gerum, Larry Padwa, and Ram Sinclair all noted that Tammuz 17 always occurred on the same week day as the first day of Pesach, and therefore derived the correct answer using the statistical distribution for the first day of Pesach.
Correspondent Larry Padwa answered on the basis of the meaning of Weekly Question 131:
This can be viewed as a trick question, depending on how one interprets it. In the complete cycle of 689,472 years, the 17th of Tammuz is distributed as follows: Sunday: 79,369 Tuesday: 219,831 Thursday: 196,992 Saturday: 193,280 From the above, it appears that the answer to the question is "no". HOWEVER, when the 17th of Tammuz occurs on Saturday, then the FAST of the 17th of Tammuz is observed on Sunday. Thus the FAST of the 17th of Tammuz occurs on Sunday 272,649 times in the full cycle (sum of the Saturday and Sunday values). The fast is therefore on Sunday more than on any other day of the week and it APPEARS that the answer is "Yes" **HOWEVER** since the fast occurs on Sunday 272,649 times, and does not occur on Sunday 416,823 times (sum of the Tuesday and Thursday) values, the answer to the question is NO! In summary:  17th Tammuz occurs on Sunday less frequently than on any other of the (nonzero) days.  Fast of 17th of Tammuz occurs on Sunday more frequently than on any other day.  Fast of 17th of Tammuz does not occur on a Sunday more than it does. Note: The reasoning is exactly the same for Tisha B'Av
Ram Sinclair provided the following correct logic:
If Saturday is not an allowed day, and in case of a Saturday the fast is postponed to Sunday Tammuz 18, then Sunday is the most likely day, I am looking at the RoshHashannah table in you site: Weekday  No. of Times  Frequency  Thursday 219,831 31.9% Saturday 196,992 28.6% Monday 193,280 28.0% Tuesday 79,369 11.5% Subtract 2 days of the week to get Tammuz 17: Weekday  No. of Times  Frequency  Tuesday 219,831 31.9% Thursday 196,992 28.6% Saturday 193,280 28.0% Sunday 79,369 11.5% Suppose you do not allow the fast to take place on Saturday and you will postponed the fast to Sunday Tammuz 18 in those years. Then let's add the odds: 11.5% + 28.0% = 39.5 . That make Sunday the most likely day.
And Winfried Gerum stated the facts correctly:
Running a statistics, on which days of the week the 17th of Tammuz will fall over a full calendar cycle, one gets: Shabbat : 193279 times = 28.03 % Sunday : 79369 times = 11.51 % Tuesday : 219831 times = 31.88 % Thursday: 196992 times = 28.57 % However, a fast will not be observed on a Shabbat. If the 17th of Tammuz falls on a Shabbat, the fast is postponed by one day. Therefore the fast of the 17th Tammuz will be distributed over a full calendar cycle with the following numbers: Sunday : 272648 times = 39.54% Tuesday : 219831 times = 31.88 % Thursday: 196992 times = 28.57 % So the strange answer to Q131 is: while Sunday is the least frequent day of the week of a 17th Tammuz, the day of the week for the fast of the 17th Tammuz will be observed most frequently on a Sunday!
Thank you Ira Walfish, Winfried Gerum, Larry Padwa, and Ram Sinclair for your very thoughtful contributions to Weekly Question 131.
One of the more prevalent practices, among the Jewish people, is that of reading the entire Mosaic text of their scriptures (Torah) over the course of one Hebrew year. At Simchat Torah, the last few verses are read, and then the entire cycle is repeated once again from Bereshit (Genesis).
The scriptural readings are divided into contiguous weekly portions, which are read in their entirety each Shabbat morning. Each division is known as a Parshah or Sedrah. These portions are arranged so as to be completely read over the course of one Hebrew year.
Each portion is given a special name. Two portions that are normally read together are Matot and Masei.
These portions are Numbers 30:2 to 32:42 and Numbers 33:1 to 36:13 respectively.
Occasionally, these two portions are read separately, rather than together.
How often are the Parshiot Matot and Masei read separately?
The Parshiot Matot and Masei are read separately only in leap years that begin on Thursday.
This happens 72,576 times over the full Hebrew calendar cycle of
689,472 years or
exactly 2 out of 19 times.
One of the more prevalent practices, among the Jewish people, is that of reading the entire Mosaic text of their scriptures (Torah) over the course of one Hebrew year. At Simchat Torah, the last few verses are read, and then the entire cycle is repeated once again from Bereshit (Genesis).
The scriptural readings are divided into contiguous weekly portions, which are read in their entirety each Shabbat morning. Each division is known as a Parshah or Sedrah. These portions are arranged so as to be completely read over the course of one Hebrew year.
Each portion is given a special name. Two portions that are normally read together are Matot and Masei.
These portions are Numbers 30:2 to 32:42 and Numbers 33:1 to 36:13 respectively.
Occasionally, these two portions are read separately, rather than together.
Since there are 14 ways of laying out the Hebrew years (14 keviyyot), there exist only 14 ways of dividing the annual Torah reading cycle. As a result, the 14 different divisions can be easily tabulated in very compact form. One such tabulation may be found at the back of certain editions of the Chumash (Pentateuch) as translated by Alexander Harkavy, and published by the Hebrew Publishing Co. in New York (1928).
That source shows that the portions Matot and Masei are read separately only in leap years that begin on Thursday.
The Keviyyot shows the statistical distribution of all the leap years that begin on Thursday in the full Hebrew calendar cycle of 689,472 years.
YEAR LENGTH IN DAYS  

DAY  353  354  355  383  384  385  TOTALS 
Mon  39369  0  81335  40000  0  32576  193280 
Tue  0  43081  0  0  36288  0  79369 
Thu  0  124416  22839  26677  0  45899  219831 
Sat  29853  0  94563  40000  0  32576  196992 
TOTALS  69222  167497  198737  106677  36288  111051  689472 
Given this question, a number of correspondents wanted to know the most recent separate readings of these two Parshiyot.
[Relative to the year 5760H] When was the most recent separate readings of Parshiot Matot and Masei and when will this next happen?
One of the more prevalent practices, among the Jewish people, is that of reading the entire Mosaic text of their scriptures (Torah) over the course of one Hebrew year. At Simchat Torah, the last few verses are read, and then the entire cycle is repeated once again from Bereshit (Genesis).
The scriptural readings are divided into contiguous weekly portions, which are read in their entirety each Shabbat morning. Each division is known as a Parshah or Sedrah. These portions are arranged so as to be completely read over the course of one Hebrew year.
Each portion is given a special name. Two portions that are normally read together are Matot and Masei.
These portions are Numbers 30:2 to 32:42 and Numbers 33:1 to 36:13 respectively.
The Parshiot Matot and Masei are read separately only in leap years that begin on Thursday.
This happens 72,576 times over the full Hebrew calendar cycle of
689,472 years or
exactly 2 out of 19 times.
The last time that each of these portions was read by themselves on
2 separate Shabbatot occurred on the
21st and 28th Tammuz 5744H corresponding coincidentally
to the 21st and 28th July 1984g.
The next time that each of these portions will be read entirely on
2 separate Shabbatot is the
23 Tammuz 5765H and 1 Av 5765H corresponding
to 30 July 2005g and 6 August 2005g respectively.
Correspondents Larry Padwa, Ram Sinclair, and Winfried Gerum found the correct Hebrew years in answer to this question. Correspondent Larry Padwa remarked that
The last leap year that began on Thursday was 5744H, so the last time that they were read separately was in 1984g. The next occurrence will be in 5765H (2005g).
Correspondent Ram Sinclair stated that
The last and next occurrences of a leap year's RoshHashannah on Thursday are September 8, 1983g; and September 16, 2004g respectively.Correspondent Winfried Gerum made these interesting observations:
Q133 is really simple: The last leap year commencing on a Thursday was 5744H (19830908) The next such years are 5765H (20040916) 5768H (20070913) 5771H (20100909) 5774H (20130905) One might ask, how these years are distributed. Such years recur after 3,10,11,13,14,21 or 24 years. Surprisingly the three year interval is quite frequent (48.72%)! The next short span is 57655768. But occassionaly one has to wait 24 years between two such years. The next long wait is 59235947
Thank you correspondents Larry Padwa, Ram Sinclair, and Winfried Gerum for sharing your answers with us.
Israel's political comic strip Dry Bones recently wondered why there were no Haggim (biblically ordained festivals) in the month of August.
Can the Haggim (biblically ordained Festivals) ever take place in August?
YES!
The festival of Rosh Hodesh always takes place in August.
However, outside of Rosh Hodesh, no other Hag (biblically ordained festival) presently occurs in August.
This was not always the case. The Rosh Hashannah Drift shows that Rosh Hashannah last occurred in August in the Gregorian year 873g.
Today, Rosh Hashannah cannot be celebrated any earlier than the Gregorian date September 5. And this won't happen until the Gregorian year 2083g.
Consequently, all of the other Haggim are moving later and later into the Gregorian year so that one day, barring any changes to either the Hebrew and Gregorian calendars, these biblically ordained festivals theoretically will be observed in the month of August.
Other than Rosh Hodesh, which Hag (biblically ordained Festival) will be first to next take place in August, and when?
The Rosh Hashannah Drift tables help to solve this problem without too much calculation effort.
The Hag (biblically ordained festival) that comes closest to the subsequent Rosh Hashannah is Shavuot (also known as Pentecost).
Since Shavuot is exactly 113 days from the subsequent Rosh Hashannah, it is necessary to determine which Rosh Hashannah comes exactly 113 days after August 1.
This is the Rosh Hashannah corresponding to November 22, since August 1 + 113 days = November 22.
According to The Rosh Hashannah Drift tables,
that coincidence will first occur for
Rosh Hashannah 16,767H on Sat 22 Nov 13,006g.
Hence, other than Rosh Hodesh, the first Hag that will take place in August is Shavuot 16,766H, corresponding to Fri 1 Aug 13,006g.
Since the second day of Shavuot is not biblically ordained, the answer is complete.
Correspondent Larry Padwa once again sent in the correct answer.
Thank you Larry Padwa for sharing with us your excellent work!The number of days between the first day of Shavuout and the first day of Rosh Hashannah (RH) is 113. Counting forward 113 days from 1Aug takes us to 22Nov, and by the RH drift tables, we see that RH next occurs on 22Nov in 16767H (13006g). This would also be the first time that 6Sivan falls in Augustin fact it would be August 1, 13006.
Why do the longest spans of 120 Hebrew years begin and end with abundant leap years starting on Thursday?
The tables shown in 247 Hebrew Year Periods for periods of 119 and 120 Hebrew years answer part of this question.
119 YEAR SPANS  

1,471 months = 43,439d 12h 103p 
1,472 months = 43,469d 0h 896p 

M'+/  DAYS  MOD 7d  OCCURS  M"+/  DAYS  MOD 7d  OCCURS  
2  0d  0  0  2  43,467d  4  11,804 

1  43,438d  3  15,428 
1  43,468d  5  222,717 

0  43,439d  4  38,281 
0  43,469d  6  66,682 

1  43,440d  5  53,027 
1  43,470d  0  279,405 

2  43,441d  6  2,128 
2  0d  0  0  
3  0d  0  0  3  0d  0  0  
The maximum variance is 32 days 
120 YEAR SPANS  

1,484 months = 43,823d 9h 692p 
1,485 months = 43,852d 22h 405p 

M'+/  DAYS  MOD 7d  OCCURS  M"+/  DAYS  MOD 7d  OCCURS  
2  0d  0  0  2  0d  0  0  
1  43,822d  2  114,296 
1  43,851d  3  1,404 

0  43,823d  3  188,409 
0  43,852d  4  21,423 

1  43,824d  4  167,629 
1  43,853d  5  96,784 

2  43,825d  5  73,986 
2  43,854d  6  19,332 

3  0d  0  0  3  43,855d  0  6,209 

The maximum variance is 33 days 
These two tables show that the longest period of 120 Hebrew years is 43,855 days, and that the longest period of 119 Hebrew years is 43,470 days.
Since the difference between these two longest periods is 385 days both the first year and the last year of the longest 120 Hebrew year period must be abundant leap years.
Otherwise, all other sums of 1 year + 119 years would be less than 43,855 days.
The table for the 120 year period also shows that the
43,855 days can only result because of a
2 day postponement from Tuesday to Thursday.
This is fully explained in Properties of Hebrew Year Periods  Part 1.
Since 385 days and 43,855 days are whole numbers of weeks, the longest spans of 120 Hebrew years must begin and end with abundant leap years starting on Thursday.
Properties of Hebrew Year Periods  Introduction notes that the period of 137 Hebrew years is the very first span to have the maximum number of 10 possible lengths in days.
The gematria of the Hebrew word kabalah is 137. This, of course, is an interesting coincidence.
So it is eminently appropriate to reserve Weekly Question 137 to the subject of such coincidences.
Which biblical coincidence involves 137 years?
Properties of Hebrew Year Periods  Introduction notes that the period of 137 Hebrew years is the very first span to have the maximum number of 10 possible lengths in days.
137 YEAR SPANS  

1,694 months = 50,024d 19h 902p 
1,695 months = 50,054d 8h 615p 

M'+/  DAYS  MOD 7d  OCCURS  M"+/  DAYS  MOD 7d  OCCURS  
2  0d  0  0  2  50,052d  2  105  
1  50,023d  1  6,297 
1  50,053d  3  42,865 

0  50,024d  2  153,412 
0  50,054d  4  106,535 

1  50,025d  3  133,504 
1  50,055d  5  164,130 

2  50,026d  4  66,656 
2  50,056d  6  12,957 

3  50,027d  5  3,011 
3  0d  0  0  
The maximum variance is 33 days 
The maximum variance is the difference between the longest and the shortest possible lengths in days for a given period of Hebrew years. In the case of 137 years, the maximum variance is 33 days.
The gematria of the Hebrew word kabalah (meaning tradition) is 137. This, of course, is an interesting coincidence.
The coincidence becomes more intriguing when it is noticed that the 10 lengths in days are arranged in 2 groupings consisting of 5 lengths on the 1,694 months side, and 5 lengths on the 1,695 months side. This grouping is highly reminiscent of the almost universal symbolic representation of the Shnei Luchot (the two tablets), which shows a grouping of 5 items on one tablet, and 5 items on the other.
However, the biblical coincidence comes from Exodus 6:16 and 6:20.
Exodus 6:16 states that the life of Levi was 137 years
long.
Exodus 6:20 states that the life of Amram
(the father of Mosheh) also was 137 years long.
It appears that the biblical text does not explicitly state as equal the length of the lives of any other two biblical personalities.
So, this quite an interesting biblical coincidence subject to considerable
speculation.
The Weekly Question will not engage in any
such speculation.
Rosh Hashannah 5762H (Tue 18 Sep 2001g) inaugurates a 354 day Hebrew year.
The molad of Tishrei will occur on Tuesday at 4h 5m 16hl. Consequently, Rosh Hashannah 5762H begins on the day of the molad of Tishrei.
None of these calendar features are common to both Hebrew years 5761H and 5762H.
At least which two calendar features does the new Hebrew year 5762H have in common with the previous Hebrew year 5761H?
Rosh Hashannah 5762H (Tue 18 Sep 2001g) inaugurated a 354 day Hebrew year.
The molad of Tishrei occurred on that Tuesday at 4h 5m 16hl. Consequently, Rosh Hashannah 5762H began on the day of the molad of Tishrei.
None of these calendar features are common to both Hebrew years 5761H and 5762H.
Here are some of the common features.
Both 5761H and 5762H
1. are 12 month years
2. are in the 304th mahzor katan (19 year cycle)
counting from Yetsirah (1H)
(i.e. Creation or Genesis)
3. began following the Gregorian date September 16
(See September 16)
4. lack a Hebrew month whose first day coincides with the first day of a
Gregorian month
(See The First Day of The Month)
5. have the Nisan Descent property
(See The Nisan Descent)
6. begin the relative leap year cycle given as
gimelbettetbetgimel (or, 3 2 3 3 3 2 3)
(See Cycles and Moladot)
7. etc...
Correspondent Dr. John Stockton matched Remy Landau's Gregorian Pesach dates against a number of his calculated Gregorian Easter dates. The result was intriguing.
When last did the first day of Pesach coincide with Gregorian Easter?
Sunday Nisan 5741H (19 April 1981g).
Correspondent Dr. John Stockton matched Remy Landau's Gregorian Pesach dates against a number of his calculated Gregorian Easter dates. The result was so intriguing that it inspired this Weekly Question.
Correspondents Larry Padwa and Robert H. Douglass also provided the correct answers.
Larry Padwa shared all of the Pesach/Gregorian Easter coincidences of the 20th Gregorian century as follows:
5741. April 19, 1981. Before that  April 18, 1954 April 17, 1927 April 1, 1923 April 12, 1903
Robert H. Douglass provided this answer:
Sunday, April 19, 1981.
Correspondents Dr. John Stockton, Larry Padwa, and Robert H. Douglass all made the same fascinating observation which is the subject of the next Weekly Question.
When next will the first day of Pesach coincide with Gregorian Easter?
Correspondents Dr. John Stockton, Larry Padwa, and Robert H. Douglass all gave the answer which pointed to Pesach 5883H corresponding to Sunday 11 April 2123g.
This will happen 142 years less 8 days from the last time Gregorian Easter coincided with Pesach on Sunday Nisan 5741H (19 April 1981g), thus skipping the whole of the 21st Gregorian century.
Thank you correspondents Dr. John Stockton, Larry Padwa, and Robert H. Douglass for sharing your most fascinating results proving most undoubtedly that Gregorian Easter and Pesach can and do coincide at times, even if we will have to wait just a little bit longer for their next meeting.
Calendar arithmetic reveals that any 120 Hebrew years as measured from Rosh Hashannah to Rosh Hashannah can have either:
43,822 days; 43,823 days; 43,824 days; 43,825 days;
43,851 days; 43,852 days; 43,853 days; 43,854 days; or 43,855 days.
Simple arithmetic shows that these lengths, after
division by 7 produce the remainders
2, 3, 4, 5, 3, 4, 5, 6, and 0.
The lengths do not give all of the possible remainders after division by 7 since the remainder 1 is missing.
First Begun 21 Jun 1998 First Paged 5 Nov 2004 Next Revised 12 Nov 2004