WQ Archive 81 - 90

Weekly Question Archive 81 - 90

by Remy Landau

Question 81

Tu B'Shevat cannot occur on which day, or days, of the week?

Correspondent Larry Padwa suggested this question, which is very timely because
Tu B'Shevat 5760H (the 15th day of Shevat 5760H) occurs on Sat 22 Jan 2000g.

Tu B'Shevat is a minor celebration also known as The New Year of Trees.

Since Tu B'Shevat is the 15th day of Shevat, it must always fall on the same day of the week as the 1st day of Shevat, because it is 2 complete weeks away from the 1st of the month.

The simplest way of answering the question is to look at the topic of The First Day of the Month found in the Additional Notes. The first table in that item shows the The Start of Month Distribution By Day of Week.

That table indicates no occurrences of Shevat beginning on either Sunday or Friday.

Correspondent Benjamin W Dreyfus correctly stated as follows

```Tu Bishvat cannot fall on Friday or Sunday.

(It can be on Monday, Tuesday, Thursday, or Saturday in a leap year
and falls on the same day as the following Rosh Hashana, and can be on
Monday, Wednesday, Thursday, or Saturday in a common year.)
```
Correspondent Winfried Gerum provided this very thorough answer
```A Tu B' Shevat never occurs on a Friday or on a Sunday:

In a deficient year (353 days) Tu B' Shevat is the 131st day.
As this type of year begins either on a Shabbat or on a Monday,
Tu B'Shevat can be either a Thursday or a Shabbat.

In a regular year (354 days) Tu B' Shevat is the 132nd day.
As this type of year begins either on a Thursday or on a Thuesday,
Tu B'Shevat can be either a Wednesday or a Monday.

In an abundant year (355 days) Tu B' Shevat is the 133rd day.
As this type of year begins either on a Shabbat, a Thursday or on a
Monday, Tu B'Shevat can be either a Shabbat, a Thursday or a Monday.

In a deficient leap year (383 days) Tu B' Shevat is the 131st day.
As this type of year begins either on a Shabbat or on a Monday,
Tu B'Shevat can be either a Thursday or a Shabbat.
As this type of year begins either on a Shabbat, a Thursday or on a
Monday, Tu B'Shevat can be either a Thursday, Wednesday or a Shabbat.

In a regular year (384 days) Tu B' Shevat is the 132nd day.
As this type of year begins on a Tuesday
Tu B'Shevat can be only a Monday.

As this type of year begins either on a Shabbat, a Thursday or on a
Monday, Tu B'Shevat can be either a Shabbat, a Thursday or a Monday.
In an abundant year (385 days) Tu B' Shevat is the 133rd day.
```
Correspondent Larry Padwa provided the following analysis
```
This is most directly approached by considering the Pesach following
the Shevat in question.

The first day of Pesach cannot fall on Monday, Wednesday or Friday.

Now in a common year, Tu B'Shevat is 59 days before Pesach
(30 day Shevat and 29 day Adar). Thus in a common year Tu B'Shevat
cannot occur on the days of the week that are 59 days
(or 3 days when multiples of 7 are eliminated) before {Mon,Wed,Fri},
namely Friday, Sunday, and Tuesday.

In a leap year, Tu B'Shevat is 89 days before Pesach
(30 day Shevat, 30 day Adar I, and 29 day Adar II). Thus in a leap year
Tu B'Shevat cannot occur on the days of the week that are 89 days
(or 5 days when multiples of 7 are eliminated) before {Mon,Wed,Fri},
namely Wednesday, Friday, and Sunday.

The only days that are on both of the above lists are Sunday and Friday.

NOTE
----
I stated at the outset that this is the most direct approach. This is
because there are only two possibilities to consider--common or leap
year.

Another approach is to use the allowable days from the RH preceding the
Shevat in question; however this is slightly less direct because there
are three possibilities to consider corresponding to the lengths of the
intervening Cheshvan and Kislev
(i.e., deficient, normal or abundant year).
```
Thank you correspondents Benjamin W Dreyfus, Winfried Gerum, and Larry Padwa for your excellent work!
Correspondent Dwight Blevins asked about the frequency of each of the Hebrew calendar's different 19 year period lengths.
Question 82

What is the frequency of occurrence of each of the Hebrew calendar's different 19 year period lengths?

Correspondent Dwight Blevins asked about the frequency of each of the Hebrew calendar's different 19 year period lengths.

Since the question did not specify the method of measurement, two entirely different answers are possible.

The first answer is that all periods of 19 consecutive Hebrew years have the mean lunar length of 6939d 16h 595p.

A 19 Hebrew year cycle (mahzor katan) consists of 235 months. Since one month has the lunation period of 29d 12h 793p, 235 months have

```235 * (29d 12h 793p) = 6815d 2820h  186,355p
= 6815d 2820h+172h 595p
= 6815d 2992h      595p
= 6815d+124d   16h 595p
= 6939d        16h 595p
```
The second answer is based on the number of whole days that exist in any period of 19 consecutive Hebrew years.

Within the entire 689,472 year cycle of the Hebrew calendar, the possible number of days for any period of 19 consecutive years, and their frequency of occurrence within the calendar cycle are as follows

```             number of days= 6,938 number of times=  11,263
6,939                  311,544
6,940                  250,123
6,941                  113,011
6,942                    3,531
```

Question 83

Which year, or years, of the 19 year cycle most frequently begin(s) the 6,938 day span of a 19 year period?

Within the entire 689,472 year cycle of the Hebrew calendar, the possible number of days for any period of 19 consecutive years, and their frequency of occurrence within the calendar cycle are as follows

```             number of days= 6,938 number of times=  11,263
6,939                  311,544
6,940                  250,123
6,941                  113,011
6,942                    3,531
```
Question 79 noted that the shortest of the 19 year periods, 6938 days, only can occur if the first year of the 19 year period is a LEAP year AND begins on a Monday.

Each of the shortest periods is distributed equally among each of the leap years in the 19 year cycle. That is, the shortest period of 19 years can occur 1,609 times starting from any of the leap years in a mahzor katan.

Question 84

What, today, is the value of the year of the Aera Alexandri?

The Hebrew calendar was subject to several different ways of counting the years.

Some year counts would start at the time a new king of ancient Israel ascended the throne. Other year counts would begin on the basis of biblical events, such as the first year in the life of Adam.

The current year count is based on an interpretation of the biblical text as regards the number of years since Bereshit, i.e., Creation, or Yetsirah. This count is now valued at 5760. However, different schools of biblical studies establish different counts.

It is not known at what point in the history of the Hebrew calendar the Yetsirah count was first adopted as the universal count.

In the medieval work The Chronology of Ancient Nations (4760H/1000g) by the Muslim author Al-Biruni, the Yetsirah count is not reported as being used by any of the Jewish communities then known to the author.

Some 175 years later, in the work Hilchot Kiddush HaHodesh, Maimonides uses the Yetsirah count in relation to details of the Hebrew calendar rules.

However, both of these medieval scholars do mention the Aera Alexandri, and do leave behind some definite details which help to correlate that particular year count to the Yetsirah count.

In Hilchot Kiddush HaHodesh, at 11:16, Maimonides indicates that year 4938H is the year 1489 li'shtorot (of contracts). That year count, according to rabbinic tradition, was the number of years that had elapsed since Alexander the Great ascended the throne, thus beginning the
Aera Alexandri year count. This particular year count is also referred as the epoch of the Seleucids.

Since the difference between the Yetsirah count and the Aera Alexandri count is 3449, the current year of the Aera Alexandri is 5760 - 3449 = 2311.

Question 85

In the Hebrew calendar, what is the proportion of 29 day months to 30 day months?

The proportion of 29 day months to 30 day months is 12,167 / 13,753 = 0.8847

Correspondents Ben Dreyfus, Larry Padwa, and Winfried Gerum all sent the correct answers, showing two different ways of approaching the problem. All three had one solution in common.

Ben Dreyfus expressed his solution as follows

```In 353-day years there are 7 29-day months and 5 30-day months.
Likewise, 354 = 6,6;  355 = 5,7;  383 = 7,6;  384 = 6,7;  385 = 5,8.

Multiply these by the number of years of each type in the full cycle to
find the frequency of each month in the full cycle:

29: 69222*7+167497*6+198737*5+106677*7+36288*6+111051*5 = 4002943
30: 69222*5+167497*6+198737*7+106677*6+36288*7+111051*8 = 4524737

Therefore, 29-day months are 46.9% of the total,
and        30-day months are 53.1%.
```
Larry Padwa showed two different methods of finding the same answer
```I see two ways to approach this, and both (happily) yield the same
result.

First Method--Quick but possibly imprecise

The length of a month is 29d 12h 793p. Since one complete day consists
of 25920 parts, and the amount that a true month exceeds 29 days is
12h and 793p (or 13753 parts) it follows that the proportion of
30 day months to 29 day months is 13753/25920.

Therefore the proportion of 29 day months to 30 day months is
(25920-13753)/25920 or 12167/25920 which rounds to 46.94%.

Second Method--Brute Force but absolutely precise

I will count the number of 29 day months and the number of 30 months
in the complete 689472 year cycle.

Year    # of such   # of 29-day  # of 30-day  # of 29-day  # of 30-day
Length  yrs in      months in 1  months in 1  months in    months in
full cycle  such yr      such yr      full cycle   full cycle
------  ----------  -----------  -----------  -----------  -----------
353       69222       7            5            484554       346110
354      167497       6            6           1004982      1004982
355      198737       5            7            993685      1391159
383      106677       7            6            746739       640062
384       36288       6            7            217728       254016
385      111051       5            8            555255       888408

TOTAL    689472                                4002943      4524737

Thus the EXACT answer to the question (barring typo's or calculation
errors) is 4002943/4524737 which also rounds to 88.47%.
```
Winfried Gerum helped to confirm everybody's brute force methods.
```    The calendar uses as the time between two moladot the value of
765433 halakim which is 29+(13753/25920) days.

For the average month to match this value 13753 out of 25920 months
must have 30 days, which leaves 12167 (25920-13753) 29-day months.

This gives a ratio of 12167/13753 = .884679706246

If you dont trust this analysis, you may look at the number
of months in the full calendar cycle of 689472 years:

Let A  = length of year
M1 = number of 29-day month
C  = number of years of length A
M2 = number of 29-day months in years of length A
N1 = number of 30-day months
N2 = number of 30-day months in years of length A

A      M1 C      M2        N1 C     N2
353 29 7* 69222= 484554 30 5* 69222= 346110
354 29 6*167497=1004982 30 6*167497=1004982
355 29 5*198737= 993685 30 7*198737=1391159
383 29 7*106677= 746739 30 6*106677= 640062
384 29 6* 36288= 217728 30 7* 36288= 254016
385 29 5*111051= 555255 30 8*111051= 888408
-------             -------
29          4002943 30          4524737
=======             =======

So a full calendar cycle has 4002943 30-day month and 4524737 30-day
months. The greatest common divisor of the two numbers is 329. If you
divide the numbers by 329 then 4002943/4524737 becomes
12167/13753 = .884679706246
```

Thank you Ben Dreyfus, Larry Padwa, and Winfried Gerum for sharing your solutions with us.

It is very difficult to quarrel with successful results. However, the fraction of day method presented by Larry Padwa and Winfried Gerum is so elegant that it deserves a more rigorous mathematical approach to prove that it really does work.

Question 86

For the full Hebrew calendar cycle of 689472 years, is it possible to show mathematically the actual proportion of 29 day months to 30 day months?

Let Cycle mean the full Hebrew calendar cycle.

The period of the molad is given by the constant 29d 12h 793p.

Let f stand for the fraction of the day represented by (12h 793p)/(24*1080) = 13753/25920.
Then the period of the molad can be expressed as 29 + f days.

Let x and y = number of 29 and 30 day months in the Cycle.

Then x*29 + y*30 = total number of days in the Cycle.

Since x + y = total number of months in the Cycle,

(29+f)*(x+y) = total number of days in the Cycle.

Consequently,

(29+f)*(x+y) = x*29 + y*30

y = f * (x + y)

Adding x to both sides of the equation,

```        (x+y) - f*(x+y) = x
(1-f) *   (x+y) = x
```

The proportion of 29 day months to 30 day months is then

```        x = (1-f)*(x+y) = (1-f)
y       f*(x+y)     f
```
Since the value of f = 13753/25920,

x/y = (25920-13753)/13753 = 12167/13753 = 0.88468

Several observations can be immediately made.

```1. The number of 30 day months is directly related to the fraction f.
2. Because f > 0.5 there are more 30 day months than 29 day months.
3. The proportion of 29 day months to 30 day months is independent
of the number of days or months in the Cycle.
4. When the two month lengths are 1 day apart, then no matter what the
length of the months, the same proportion x/y will result.
```

All of the brute force arithmetical solutions to Question 85 relied on statististics and facts that related to the full Hebrew calendar cycle of 689,472 years.

Question 87

Using statistics relevant only to the 19 year periods, is it possible to find the proportion of 29 day months to 30 day months?

Last Week's Question

Using statistics relevant only to the 19 year periods, is it possible to find the proportion of 29 day months to 30 day months?

Let x and y respectively represent the number of 29 and 30 day months in any 19 year cycle.

Let d = the total number of days in any 19 year cycle.

Then x*29 + y*30 = d.

Now, x + y = the total number of months in any 19 year cycle = 235 so that y = 235 - x

Hence, x*29 + (235-x)*30 = d

leading to 7050 - d = x

The 19 year cycles can have either 6939, 6940, 6941, or 6942 days with an occurrence of either 17099, 13648, 5246, 295 times in the full Hebrew calendar cycle.

The following table can then be derived.

19 year Cycle Statistics
19 Year
Length
In Days
Total
Occurrences
In Full Cycle
29 Day
Months
In 19 Years
30 Days
Month
In 19 Years
Total
29 Day
Months
Total
30 Days
Month
6,93917,099111124 1,897,9892,120,276
6,94013,648110125 1,501,2801,706,000
6,941 5,246109126 571,814660,996
6,942 295108127 31,86037,465
Grand Sum36,288-- 4,002,9434,524,737

Therefore, the proportion of 29 day months to 30 day months is 4,002,943 / 4,524,737 = 0.8847

Correspondent Winfried Gerum made a study of the time period during which the Hebrew and Gregorian year counts are equal.

This topic was first touched upon in Question 29. But it seems that Winfried Gerum has the definitive answers.

It is assumed that both the Hebrew and Gregorian calendars keep their rules fixed over the intended period of time.

Question 88

On what day will both the Hebrew and Gregorian calendars first show the exact same year?

Correspondent Winfried Gerum made a study of the time period during which the Hebrew and Gregorian year counts are equal.

This topic was first touched upon in Question 29. But it seems that Winfried Gerum has the definitive answers.

It is assumed that both the Hebrew and Gregorian calendars keep their rules fixed over the intended period of time.

The accuracy of the Hebrew calendar is fixed by the value of the mean lunation period coupled to the 19 year cycle of 235 lunar months.

That leads to an average Hebrew year length of 365.2468 days.

The mean Gregorian year is 365.2425 days.

Hence, the average Hebrew year is slower than the mean Gregorian year by about one day in every 230 years.

Consequently, the Gregorian year count, while still distant from the Hebrew year count, will eventually catch up to and exceed the Hebrew year count.

Correspondent Winfried Gerum predicts that the catchup will first occur on 28 Elul 317,675,828 corresponding to 1 January 317,675,828.

Winfried Gerum made another interesting observation.

Question 89

On what day will both the Hebrew and Gregorian calendars first end the exact same year?

Correspondent Winfried Gerum made a study of the time period during which the Hebrew and Gregorian year counts are equal.

This topic was first touched upon in Question 29. But it seems that Winfried Gerum has the definitive answers.

It is assumed that both the Hebrew and Gregorian calendars keep their rules fixed over the intended period of time.

The accuracy of the Hebrew calendar is fixed by the value of the mean lunation period coupled to the 19 year cycle of 235 lunar months.

That leads to an average Hebrew year length of 365.2468 days.

The mean Gregorian year is 365.2425 days.

Hence, the average Hebrew year is slower than the mean Gregorian year by about one day in every 230 years.

Consequently, the Gregorian year count, while still distant from the Hebrew year count, will eventually catch up to and exceed the Hebrew year count.

Correspondent Winfried Gerum predicts that the catchup will first occur on 28 Elul 317,675,828 corresponding to 1 January 317,675,828.

Winfried Gerum also made the interesting observation that the Hebrew and the Gregorian calendars, if they do not change rules, will both first end the same year on the same day of
29 Elul 317,760,226 corresponding to 31 December 317,760,226.

Question 90

What do Shabbat, Monday, and Wednesday have in common?

Correspondent Avi Veisz noted that

Shabbat, Monday, and Wednesday are excluded from Purim day.

The Talmudic sages prevented Purim from from being observed either on Monday, Wednesday, or Saturday.

``` First  Begun 21 Jun 1998