WQ Archive 51 - 60

Weekly Question Archive 51 - 60

by Remy Landau

Question 51

How often do Hebrew years of 383 days go missing from all of the 19 year cycles in the full Hebrew calendar cycle?

Besides the regular leap years of 384 days, the deficient leap years of 383 days are the only other kind of year to skip some of the 19 year cycles.

As shown in the Additional Notes under the topic of The Keviyyot 383 day years occur 106,677 times in the full Hebrew calendar cycle of 689,472 years or 15.47% of all the years.

Consequently, it is hard to imagine that any 19 year cycle can be found without any 383 day year in it. However, over the full Hebrew calendar cycle exactly 9 of the 19 year cycles do not have a 383 day year.

The first such 19 year cycle to exclude a 383 day year will not begin until
Rosh Hashannah 114,780H corresponding to Tuesday 16 January 111,021g.

Correspondent Ari Z. Zivotofsky did a random sampling of a number of Rosh Hashannah dates mapped onto the Gregorian calendar. He then calculated the Gregorian equivalent of these Rosh Hashannah dates offset by 247 years. He was surprised to discover that all of the new Gregorian dates were exactly ONE day later than their earlier counterparts.

Ari Z. Zivotofsky showed these examples

```            Rosh Hashannah 5665H occurred on Sat 10 Sep 1904g
5912H             Sat 11 Sep 2151g

Rosh Hashannah 5669H occurred on Sat 26 Sep 1908g
5916H             Sat 27 Sep 2155g

Rosh Hashannah 5724H occurred on Thu 19 Sep 1963g
5971H             Thu 20 Sep 2210g

Rosh Hashannah 5726H occurred on Mon 27 Sep 1965g
5973H             Mon 28 Sep 2212g

```

Question 52

Are all Rosh Hashannah's separated by 247 years exactly one day apart in their Gregorian calendar equivalents?

NO! While many examples of the one day date difference easily can be found, a number of additional examples can be shown which do not conform to that difference.

Correspondent Ari Z. Zivotofsky did a random sampling of a number of Rosh Hashannah dates mapped onto the Gregorian calendar. He then calculated the Gregorian equivalent of these Rosh Hashannah dates offset by 247 years. He was surprised to discover that all of the new Gregorian dates were exactly ONE day later than their earlier counterparts.

Ari Z. Zivotofsky showed these examples

```            Rosh Hashannah 5665H occurred on Sat 10 Sep 1904g
5912H             Sat 11 Sep 2151g

Rosh Hashannah 5669H occurred on Sat 26 Sep 1908g
5916H             Sat 27 Sep 2155g

Rosh Hashannah 5724H occurred on Thu 19 Sep 1963g
5971H             Thu 20 Sep 2210g

Rosh Hashannah 5726H occurred on Mon 27 Sep 1965g
5973H             Mon 28 Sep 2212g

Ari Z. Zivotofsky then also found these examples

Rosh Hashannah 5662H occurred on Sat 10 Sep 1901g
5909H             Sat 10 Sep 2148g

Rosh Hashannah 5663H occurred on Thu  2 Oct 1902g
5910H             Thu  2 Oct 2149g

Rosh Hashannah 5664H occurred on Tue 22 Sep 1903g
5911H             Tue 22 Sep 2150g

The years 5725H (1964g) and 5850H (2089g) were also mentioned as showing
a 2 day Gregorian date difference after 247 years.
```

Correspondent Winfried Gerum made the following excellent analysis

```
The conjecture is, that 247 Hebrew years differ from 247 Gregorian
years by one day.

The length of the Gregorian year is     146,097/   400
= 365.2425 days.
The length of the Hebrew    year is 251,827,457/689472
= 365.246822.. days.

Summing 247 "average" years, the difference between 247 Hebrew and
Gregorian years is about 1.067... days.

So it is clear, that the difference between two Gregorian dates
247 Hebrew years apart cannot always be 1 day!
```
Thank you very much Winfried Gerum for this remarkably simple and elegant mathematical demonstration of the correct answer.

Correspondent Jerrold Landau attracted my attention to the fact that, beginning with
Rosh Hashannah 5760H (Sat 11 Sep 1999g), 4 out of the next 5 consecutive Rosh Hashannah's will begin on Saturday.

Question 53

After Rosh Hashannah 5764H (Shabbat 27 September 2003g) when again will 4 out of 5 consecutive Rosh Hashannah's begin on Shabbat?

The next pattern of of 5 consecutive Hebrew years in which 4 out of 5 begin on Shabbat will not occur until Rosh Hashannah 5936H (Saturday 16 Sep 2175g).

The 5 Rosh Hashannah's are

```
5936H corresponding to Sat 16 Sep 2175g
5937H                  Sat  5 Oct 2176g
5938H                  Tue 23 Sep 2177g
5939H                  Sat 12 Sep 2178g
5940H                  Sat  2 Oct 2179g
```

Correspondent Winfried Gerum sent the following correct answer:

The next period with 4 out of 5 years beginning on Shabbat are the years 5936 through 5940.

Three consecutive years cannot all commence on Shabbat. Therefore, periods with 4 out of 5 years starting on Shabbat must have their 3rd year starting on some day other than Shabbat.

Thank you very much Winfried Gerum for sharing these observations with us.

The next occurence of this will be in about 175 years. RH 5936 (Saturday 16-September 2175G) begins the next cycle.

It is interesting to note that this pattern occurs with other days of the week much sooner.

Thank you also Larry Padwa for not only sharing your observations with us but actually suggesting the next Weekly Question.

Question 54

After Rosh Hashannah 5764H (Shabbat 27 September 2003g) when will 4 out of 5 consecutive Rosh Hashannah's begin on the same day other than Shabbat?

Correspondent Winfried Gerum sent the following correct answer:

The next period of five years with four of them beginning on the same day of the week with Rosh Hashannah not falling on a Sabbath are the years 5771H through 5775H (2010g thru 2014g).

The four Rosh haShannas will all begin on Thursday while Rosh haShanna 5773H (2012g) will begin on Monday. As with all such five year periods the third year starts on a different day of the week.

Thank you very much Winfried Gerum for sharing these observations with us.

It is interesting to note that this pattern occurs with other days of the week much sooner. RH 5771H (Thurs 9 Sep 2010g) is the first of four out of five consecutive RH's beginning on Thursday.

Thank you also Larry Padwa for not only sharing your observations with us but once again suggesting the next Weekly Question.

Question 55

After Rosh Hashannah 5775H (Thu 25 Sep 2014g) when next will 4 out of 5 Rosh Hashannah's begin on the same day?

RH 5771H (Thurs 9-Sept 2010G) is the first of four out of five consecutive RH's beginning on Thursday, and those five years are immediately followed by RH 5776H (Mon 14-Sept 2015G) which is the first of four out of five consecutive RH's beginning on Monday.

A further curiosity is that in the string of Thursday RH's the non-Thursday RH is Monday, and in the string of Monday RH's the non-Monday RH is Thursday.

Thus, in the ten years beginning 5771H (2010G), all RH's begin on Monday or Thursday.

The pattern is: Th,Th,Mo,Th,Th,Mo,Mo,Th,Mo,Mo.

Thank you Larry Padwa for once again having shared your amazing observations with us.

Correspondent Winfried Gerum also sent the following thought provoking comments:

The next such period is from 5776H through 5780H.

Of these "Thursday"-Periods 14 overlap during a full calendar cycle. I.e., there are 14 periods of eight years with six out of eight Rosh Hashanot falling on a Thursday.

The most recent such period was the years 3173H .... 3180H.

The next such period will be the years 35883H through 35890H.

Except for these 14 periods there are no other 8 year periods with six Rosh Hashanot on the same day of the week.

Thank you very much Winfried Gerum for sharing these observations with us.

Question 56

Do the numbers 2 4 7 10 12 15 18 constitute a valid leap year distribution for the calculation of the Hebrew calendar?

YES!

The traditional literature of the Hebrew calendar, such as the 8th century Seder Olam, and the 11th century work The Chronology of Ancient Nations by the near Eastern scholar Albiruni, show a number of variants as regards the leap year distribution inside a mahzor katan (the nineteen year Hebrew calendar cycle).

The Encyclopedia Judaica, in its article on the Calendar very correctly notes that

Apparent variations in the ordo intercalationis ... are but variants of the selfsame order.

The leap year order that is used will depend entirely on the year of the 19 year cycle that is selected to be the first year of the year counts, ie, the epochal year.

Since the Hebrew calendar uses only one pattern in which to arrange the 7 leap years of a 19 year cycle, there can only be at most 19 leap year distributions resulting from the choice of a specific epochal year within the mahzor katan. Assuming that the epochal year 1H is the first year of the 19 year cycle, then these distributions are:-

```
YEAR Leap Year Distribution
==== ======================
1   3  6  8 11 14 17 19
2   2  5  7 10 13 16 18
3   1  4  6  9 12 15 17
4   3  5  8 11 14 16 19
5   2  4  7 10 13 15 18
6   1  3  6  9 12 14 17
7   2  5  8 11 13 16 19
8   1  4  7 10 12 15 18
9   3  6  9 11 14 17 19
10   2  5  8 10 13 16 18
11   1  4  7  9 12 15 17
12   3  6  8 11 14 16 19
13   2  5  7 10 13 15 18
14   1  4  6  9 12 14 17
15   3  5  8 11 13 16 19
16   2  4  7 10 12 15 18
17   1  3  6  9 11 14 17
18   2  5  8 10 13 16 19
19   1  4  7  9 12 15 18
```

The numbers 2 4 7 10 12 15 18 can be seen against the 16th year of the 19 year cycle whose first year is also year 1H. Consequently, any Hebrew calendar system whose year count begins relative to year 16H will be required to use the leap year distribution 2 4 7 10 12 15 18 if it is to maintain synchronization with the fixed Hebrew calendar.

Correspondent Winfried Gerum gave the following correct answer

... if one does not start counting19-year cycles in the year 1 but instead, the year 16, then leap years fall into the stated sequence of numbers.

Good work Winfried Gerum!

Question 57

The Encyclopedia Judaica, in its article on the Calendar cites the following leap year distributions as having been used

```       2  5  7 10 13 16 18
1  4  6  9 12 15 17
3  5  8 11 14 16 19
3  6  8 11 14 17 19
```

Additionally, the EJ suggests that the following values were used at one time as the epochal moladot

```      4d 20h 408p
2d  5h 204p
6d 14h   0p
3d 22h 876p
```

Is the EJ entirely correct?

NO!

In the Encyclopedia Judaica, the epochal molad given as 4d 20h 408p should be 2d 20h 385p .

The traditional literature of the Hebrew calendar, such as the 8th century Seder Olam, and the 11th century work The Chronology of Ancient Nations by the near Eastern scholar Albiruni, show a number of variants as regards the leap year distribution inside a mahzor katan (the nineteen year Hebrew calendar cycle).

The Encyclopedia Judaica, in its article on the Calendar very correctly notes that

Apparent variations in the ordo intercalationis ... are but variants of the selfsame order.

The leap year order that is used will depend entirely on the year of the 19 year cycle that is selected to be the first year of the year counts, ie, the epochal year.

Since the Hebrew calendar uses only one pattern in which to arrange the 7 leap years of a 19 year cycle, there can only be at most 19 leap year distributions resulting from the choice of a specific epochal year within the mahzor katan.

The EJ article Calendar mentions only the first 4 possible leap year variations assuming that the epochal year 1H is the first year of the 19 year cycle. These distributions are:-

```
YEAR Leap Year Distribution
==== ======================
1   3  6  8 11 14 17 19
2   2  5  7 10 13 16 18
3   1  4  6  9 12 15 17
4   3  5  8 11 14 16 19
```

Consequently, the epochal moladot should be

```       2d  5h 204p  (for the 1st year)
6d 14h   0p  (for the 2nd year)
3d 22h 876p  (for the 3rd year)
and 2d 20h 385p  (for the 4th year) instead of the value 4d 20h 408p
```

Question 58

Which Hebrew year could most reasonably be represented as an epochal year with the presence of the molad 4d 20h 408p?

The Encyclopedia Judaica article Calendar mentions only the first 4 possible leap year variations assuming that the epochal year 1H is the first year of the 19 year cycle. These distributions are:-

```
YEAR Leap Year Distribution
==== ======================
1   3  6  8 11 14 17 19
2   2  5  7 10 13 16 18
3   1  4  6  9 12 15 17
4   3  5  8 11 14 16 19
```

Consequently, the epochal moladot should be

```       2d  5h 204p  (for the 1st year)
6d 14h   0p  (for the 2nd year)
3d 22h 876p  (for the 3rd year)
and 2d 20h 385p  (for the 4th year) instead of the value 4d 20h 408p
```

The moladot of Tishrei which correspond to the value 4d 20h 408p are for the Hebrew years

```       117357H (Thu 15 Jan 113598g)
308062H (Thu 27 Apr 304305g)
616124H (Thu  3 Dec 612370g)
```

These Hebrew years are respectively, the 13th, 15th, and 11th years of the mahzor katan using the leap year distribution GUChADZaT.

Clearly, the above possibilities are not sensible in terms of epochal years. However, the value
4d 20h 408p also comes as the molad of Heshvan for the year 0H. Hence, it would be very reasonable to use the year 0H as an epochal year. Its molad of Tishrei is 3d 7h 695p.

Consequently, if the Encyclopedia Judaica were to correct this particular passage in their well written article on the Calendar the two corrections that could be recommended would be that:-

1. The epochal molad of 4d 20h 408p be replaced by the value 3d 7h 695p which represents the molad of Tishrei 0H;

2. The leap year distribution 3 5 8 11 14 16 19 be replaced by 1 4 7 9 12 15 18 which represents the leap year distribution for an epochal year beginning at year 0H.

Question 59

Which Hebrew month least frequently has the molad of 0d 0h 0p?

In the full Hebrew calendar cycle of 689472 years each and every month will experience the molad 0d 0h 0p at least 2 times.

Of all the months, the leap month Adar will experience the molad 0d 0h 0p exactly two times in the full Hebrew calendar cycle. All of the other months will see this molad either 3 or 4 times.

Therefore, the leap month Adar least frequently has the molad 0d 0h 0p.

Question 60

Is the Hebrew 5760H a leap year because it begins on the Gregorian date of September 11?

YES!

The last Rosh Hashannah to coincide with the Gregorian date September 11 will begin on Tuesday 7142H (3381g).

Until then, all of the Rosh Hashannah's coinciding with the Gregorian date September 11 will mark the start of a Hebrew leap year.

Correspondent Winfried Gerum had another quite correct opinion.

```As the two calendars are not in any way related there is no reason
why a Rosh Hashanna falling on the 11th day of September is forced
to be a leap year.

However, during the years 5350H through 7142H there are 63
Rosh Hashannah's falling on September 11th and all of them happen
to be leap years.

Prior to 5350H, there are some Rosh Hashannah's falling on
September 11, but none of them is a leap year.
```

Good work Winfried Gerum!

The first time that a Rosh Hashannah began a leap year on the Gregorian date September 11 was Monday 4511H (750g).

``` First  Begun 21 Jun 1998