Tu B'Shevatcannot 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

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

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

Correspondent **Benjamin W Dreyfus** correctly stated as follows

CorrespondentTu 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.)

CorrespondentA 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.

Thank you correspondentsThis 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).

Correspondent

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

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

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

Which year, or years, of the 19 year cycle

most frequentlybegin(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

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**.

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.,

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

Some 175 years later, in the work * Hilchot Kiddush HaHodesh*,
Maimonides uses the

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

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**.

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%.

I see two ways to approach this, and both (happily) yield the same result.First Method--Quick but possibly impreciseThe 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 preciseI 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 theanswer to the question (barring typo's or calculation errors) is 4002943/4524737 which also rounds to 88.47%.EXACT

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

It is very difficult to quarrel with successful results. However, the
* fraction of day method* presented by

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**

leading to

**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

Since the value ofx=(1-f)*(x+y)=(1-f)y f*(x+y) f

**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

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

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 |

6,939 | 17,099 | 111 | 124 | 1,897,989 | 2,120,276 |

6,940 | 13,648 | 110 | 125 | 1,501,280 | 1,706,000 |

6,941 | 5,246 | 109 | 126 | 571,814 | 660,996 |

6,942 | 295 | 108 | 127 | 31,860 | 37,465 |

Grand Sum | 36,288 | - | - | 4,002,943 | 4,524,737 |

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

Correspondent

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.

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.

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**.

What do

Shabbat, Monday, and Wednesdayhave 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 First Paged 5 Nov 2004 Next Revised 5 Nov 2004