Referring to Properties of Hebrew Year Periods - Part 1, the answer can be developed analytically as shown below.

Let w = the weekday disallowed for 1 Tishrei.
Let M + m = 354d 8h 876p and L = 356d.

Then L = D" - D' = 354d + [f + m] + p" - p' = 356d. (Equation 4.5)
Hence, [f + m] + p" - p' = 356d - 354d = 2d.
Consequently, it is necessary that  p" = 1d.

Leading to  D" = w + p" = w + 1d.
Hence,      w + 1d - D' = 356d
            D' = w - 355d 
            D' MOD 7d = (w - 355d) MOD 7d = (w - 5d) MOD 7d = (w + 2d) MOD 7d

Therefore, given the single non-allowable weekday w, the 356 day year will begin on D' = w + 2d and end on D" = w + 1d.

In the above example, with Friday removed, the 356 day year is seen to begin on Friday + 2d = Sunday. It will end on Friday + 1d = Shabbat since (Sunday + 356d) MOD 7d = Shabbat.

The same analysis can be applied in the case of the 382 day year.

Let w = the weekday disallowed for 1 Tishrei.
Let M + m = 383d 21h 589p and L = 382d.

Then L = D" - D' = 383d + [f + m] + p" - p' = 382d. (Equation 4.5)
Hence, [f + m] + p" - p' = 382d - 383d = -1d.
Consequently, it is necessary that  p' =  1d.

Leading to  D' = w + p' = w + 1d.
Hence,      D" - w - 1d = 382d
            D" = w + 383d 
            D" MOD 7d = (w + 383d) MOD 7d = (w + 5d) MOD 7d = (w - 2d) MOD 7d

Therefore, given the single non-allowable weekday w, the 382 day year will begin on D' = w + 1d and end on D" = w - 2d.

In the above example, with Friday removed, the 382 day year is seen to begin on Friday + 1d = Shabbat. It will end on Friday - 2d = Wednesday since (Friday + 382d) MOD 7d = Wednesday.

The keviyot for a calendar, missing only Friday for 1 Tishrei, is shown in the folowing table. The statistics are for the full Hebrew calendar cycle of 689,472 years.

Part of the answer to Weekly Question 163 may be derived from Weekly Question 162.

Which day, or days, of the week would have begun 356 and 382 day years had Tuesday, Thursday, and Shabbat been disallowed for 1 Tishrei?

Had Tuesday, Thursday, and Shabbat been disallowed for 1 Tishrei,then the 356 day year would have begun on Monday, while the 382 day year would have begun on Wednesday.

Referring to Properties of Hebrew Year Periods - Part 1, the answer can be developed analytically as shown below.

Let w = the weekday disallowed for 1 Tishrei.
Let M + m = 354d 8h 876p and L = 356d.

Then L = D" - D' = 354d + [f + m] + p" - p' = 356d. (Equation 4.5)
Hence, [f + m] + p" - p' = 356d - 354d = 2d.
Consequently, it is necessary that  p" = 1d.

Leading to  D" = w + p" = w + 1d.
Hence,      w + 1d - D' = 356d
            D' = w - 355d 
            D' MOD 7d = (w - 355d) MOD 7d = (w - 5d) MOD 7d = (w + 2d) MOD 7d

Therefore, given the single non-allowable weekday w, the 356 day year will begin on D' = w + 2d and end on D" = w + 1d.

In the above example, with Friday removed, the 356 day year is seen to begin on Friday + 2d = Sunday. It will end on Friday + 1d = Shabbat since (Sunday + 356d) MOD 7d = Shabbat.

The same analysis can be applied in the case of the 382 day year.

Let w = the weekday disallowed for 1 Tishrei.
Let M + m = 383d 21h 589p and L = 382d.

Then L = D" - D' = 383d + [f + m] + p" - p' = 382d. (Equation 4.5)
Hence, [f + m] + p" - p' = 382d - 383d = -1d.
Consequently, it is necessary that  p' =  1d.

Leading to  D' = w + p' = w + 1d.
Hence,      D" - w - 1d = 382d
            D" = w + 383d 
            D" MOD 7d = (w + 383d) MOD 7d = (w + 5d) MOD 7d = (w - 2d) MOD 7d

Therefore, given the single non-allowable weekday w, the 382 day year will begin on D' = w + 1d and end on D" = w - 2d.

Given that some weekday w must be bypassed for the weekday of 1 Tishrei, it is possible to relate to w the weekdays on which the 356 day and 382 day years potentially begin and end. This may be summarized as shown in the following table.

If the postponement days are given as w, w + 2d, and w + 4d then, according to the relationships tabulated above, the starts and ends of the 356 and 382 day years may be tabulated as follows:

The tabulation shows that in this situation, only w + 6d can start a 356 day year, because w + 2d and w + 4d have been disallowed for year starts.

The table also shows that the only allowable start day for the 382 day year is w + 1d, because days w and w + 2d, the end days of the 382 day year, are not permitted to start any year in this example.

Thus, when w = 3 (Tue), then w + 2d = 5d (Thu), and w + 4d = 7d (Shabbat).

Hence, the start of the 356 day year, given by w + 6d MOD 7d = 2d, is Monday.

Similarly, the start of the 382 day year, given by w + 1d = 4d, is Wednesday.

The following table shows the distibution of the year lengths over the full Hebrew calendar cycle of 689,472 years when Tuesday, Thursday, and Shabbat are not permitted week days for 1 Tishrei.

It may be noted that when w = 4d (Wed), then w + 2d = 6d (Fri), and w + 4d MOD 7d = 1d (Sun), which actually are the days specified for Dehiyah Lo ADU Rosh. Not too surprisingly, when w = 4d then the 356 day year begins on w + 6d MOD7d = 3d = Tuesday, and the 382 day year ends on w - 2d = 2d = Monday.

What would have been the single year lengths had Shabbat been the only day allowed to start Rosh Hashannah?

When the postponements allowed are only one day, then the single year lengths can be either 353, 354, 355, 356, 382, 383, 384, or 385 days. The following tables, derived for the full Hebrew calendar cycle of 689472 years, represent the smallest number of weekday combinations needed to show all of the possible ways in which one day postponements can be designated.

In each of the above tables, it is to be noted that the postponements called for are at most one day. Also to be noted, is that the effect of prohibiting Rosh Hashannah starts on Sunday, Tuesday, and Thursday, is mathemathetically equivalent to the effect of using Dehiyah Lo ADU Rosh in the construction of the fixed Hebrew calendar. By just adding 3 to each day of the week, in the above table, the same statistical distribution of the single year lengths can be seen as in the tabulation under Dehiyah Lo ADU Rosh.

When Shabbat is the only day allowed to start Rosh Hashannah, then the holiday can be postponed by up to 6 days. Logically, all of the single year lengths would then have to be even multiples of 7 days, since that is the only way that each year could start on Shabbat.

Consequently, under this restriction, the single year lengths can only be either 350, 357, 378, or 385 days long. When Shabbat is the only day allowed to start Rosh Hashannah, it is possible to tabulate the distribution of the single year lengths over the full Hebrew calendar cycle of 689,472 years.

Weekly Question 164 introduced the idea that the effect of prohibiting Rosh Hashannah starts on Sunday, Tuesday, and Thursday, is mathemathetically equivalent to the effect of using Dehiyah Lo ADU Rosh in the construction of the fixed Hebrew calendar. That is because the single year statistical distributions remain the same, even though they may be ordered differently. Weekly Question 165 extends this idea.

What is the smallest number of weekday combinations needed in order to view all of the possible single year statistical distributions when up to 6 weekdays may be disallowed for 1 Tishrei?

Suppose that the days disallowed for 1 Tishrei are Sunday, Tuesday, and Thursday.

These weekdays are expressed numerically as 1, 3, and 5 respectively. The statistical effect, over the full Hebrew calendar cycle of 689,472 years, of these implied postponements may be tabulated as follows:-

Having developed these statistical distributions, it is no longer necessary to work out any other combination of 3 days whose selection is derived by adding exactly the same number to each of the originally forbidden 3 days.

The 7 combinations which are derived from 1, 3, and 5 are

All of these combinations will produce the statistical distributions that are mathematically equivalent to the effects of forbidding 1 Tishrei on week days 1, 3, and 5.

For example, combination 4, 6, 1 (+3 above) represents the week days that are used in Dehiyah Lo ADU Rosh. Adding 3 to each day in the above Keviyyot statistical table actually produces the Dehiyah Lo ADU Rosh ditribution table. That table is shown below.

The number of ways that 1 up to 6 days may be chosen from the week, is 126. Since 6 out of 7 choices are not required, because of they produce mathematically equivalent statistical distributions, the smallest number of combinations that need be considered is 18. These 18 combinations are:

One of the weekly scriptural readings during this time of year is the double portion known as Matot-Masei. These portions are Numbers 30:2 to 32:42 and Numbers 33:1 to 36:13 respectively. Occasionaly, the Parshiot Matot and Masei are each read on their own.

The following question first appeared as Weekly Question 132.

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.

Given this question, a number of correspondents wanted to know the most recent separate readings of these two Parshiyot.

The following question first appeared as Weekly Question 133.

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, Winfried Gerum, and Robert H. Douglass 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 Rosh-Hashannah on Thursday are September 8, 1983g; and September 16, 2004g respectively.

Q133 is really simple:
The last leap year commencing on a Thursday was
   5744H (1983-09-08)
The next such years are
   5765H (2004-09-16)
   5768H (2007-09-13)
   5771H (2010-09-09)
   5774H (2013-09-05)

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
5765-5768. But occassionaly one has to wait 24 years between two such
years. The next long wait is 5923-5947

Correspondent Robert H. Douglass stated that

This was explained in the current weekly question as: "the portions
Matot and Masei are read separately only in leap years that begin on
Thursday."

The most recent time this occurred was HY 5744, a leap year of 385 days
beginning on Thursday, September 8, 1983 CE.

The next time this will occur is HY 5765, a leap year of 383 days
beginning on Thursday, September 16, 2004 CE.

Why is it easy to count mentally the days from Rosh Hashannah 5663H to Rosh Hashannah 5763H?

The Gregorian calendar dates for these two year beginnings provides a clue.

Rosh Hashannah 5663H began on Thu 2 Oct 1902g.
Rosh Hashannah 5763H begins on Sat 7 Sep 2002g.

Since the year 2000g was a Gregorian leap year,
the 100 Gregorian years from 1902g to 2002g had 365 * 100 + 25 days = 36,525 days.

Since Sep 7 is 25 days earlier than Oct 2,
the span from Thu 2 Oct 1902g to Sat 7 Sep 2002g is 36,525 - 25 = 36,500 days long.

Consequently, the 100 Hebrew years beginning on Rosh Hashannah 5663H were exactly 36,500 days long.

When next, at Rosh Hashannah, will a 100th Hebrew anniversary be exactly 36,500 days long?

The next Hebrew centennial to have exactly 36,500 days at Rosh Hashannah will be marked at the start of 5771H, corresponding to Thu 9 Sep 2010g.

Rosh Hashannah 5671H began on Tue 4 Oct 1910g.
Rosh Hashannah 5771H begins on Thu 9 Sep 2010g.

Since the year 2000g was a Gregorian leap year,
the 100 Gregorian years from 1910g to 2010g had 365 * 100 + 25 days = 36,525 days.

Since Sep 9 is 25 days earlier than Oct 4,
the span from Tue 4 Oct 1910g to Thu 9 Sep 2010g is 36,525 - 25 = 36,500 days long.

Consequently, the 100 Hebrew years beginning on Rosh Hashannah 5671H will be exactly 36,500 days long.

Correspondent Robert H. Douglass wasted no time in responding with the correct answer.

That would be Hebrew Year 5771, beginning on Thursday, Sept. 9, 2010...
exactly 36500 days after the start of Hebrew Year 5671 on Tuesday, Oct.
4, 1910.

Regards,   Robert H. Douglass

Thank you Robert H. Douglass for your very prompt and correct answer to the question.

To get the greatest number possible of year lengths, which days of the week would have to be forbidden for 1 Tishrei?

The greatest number of year lengths that Dehiyot can produce is 12.

The 12 year lengths are generated whenever any 3 consecutive weekdays are forbidden from starting Rosh Hashannah.

This is illustrated over the complete Hebrew calendar cycle of 689472 years in the following statistical tabulation which eliminates Sunday, Monday, and Tuesday from starting Rosh Hashannah.

It is interesting to note that, when 3 consecutive weekdays are forbidden from starting Rosh Hashannah, the year lengths of 351, 352, 357, 380, and 381 days are generated, and the 382 day year length cannot be produced.

To get the shortest possible 13 month year length, which day, or days, of the week would have to be forbidden for 1 Tishrei?

To get the shortest possible 13 month year, at least 5 consecutive days would have to be forbidden for 1 Tishrei.

When at least 5 consecutive week days are forbidden for 1 Tishrei, then the shortest 13 month year length possible is 378 days. This length is also the shortest possible leap year length that can be generated whenever any of the weekdays are forbidden for 1 Tishrei.

This is illustrated over the complete Hebrew calendar cycle of 689472 years in the following statistical tabulations which allow either only Friday and Shabbat, or Shabbat, to start Rosh Hashannah.

The shortest year length possible, whenever a certain number of week days are forbidden for 1 Tishrei, is 350 days. This length appears in the above 2 tables.

To generate the shortest possible year length, which day, or days, of the week would have to be forbidden for 1 Tishrei?

The shortest year length possible, whenever a certain number of week days are forbidden for 1 Tishrei, is 350 days.

To get the shortest possible year length, at least 4 consecutive weekdays would have to be forbidden for 1 Tishrei.

This is illustrated over the complete Hebrew calendar cycle of 689,472 years in the following statistical tabulations which forbid at least 4 consecutive weekdays to start Rosh Hashannah.

The next Hebrew year will be Hebrew year 5763H beginning on Shabbat 7 September 2002g. This year will be 385 days long, and therefore, 13 months long.

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