WQ Archive 191 - 200

Question 191

In any period of HY Hebrew years, what are the minimum and maximum number of months?

In 1802g, the German mathematician Carl Friedrich Gauss published a formula which gave the Julian date of Pesach for any Hebrew year A.

The Gauss Pesach formula appeared without proof of its derivation in “Berechnung des jüdischen Osterfestes”, Monatliche Correspondenz zur Beförderung der Erd- und Himmels-Kunde, 5 (1802), 435-437 – reprinted in: Carl Friedrich Gauss Werke (Königlichen Gesellschaft der Wissenschaften, Göttingen, 1874), vol. 6, pp. 80-81.

With a bit of patience, it is actually quite possible to derive the fundamental elements of the Gauss formula.

One of the amazing results is that there are actually 2 formulas that come out of this effort.

The first formula, which was apparently favoured, but not shown, by Gauss, and provides for the maximum number of months in any period of HY years is

MAXSUM(HY) = [235 * HY + R(12 * Y, 19)] / 19

where R(12 * Y, 19) is the remainder of 12 * HY divided by 19.

The second formula which can be derived from the method used to derive the Gauss formula giving the minimum number of months is

MINSUM(HY) = [235 * HY - R(7 * HY, 19)] / 19

That second formula can actually be reduced to

MINSUM(HY) = INT(235 * HY / 19)

When HY MOD 19 = 0 then MINSUM(HY) = MAXSUM(HY)

When HY MOD 19 is not equal to 0 then it can be shown easily that

MAXSUM(HY) = MINSUM(HY) + 1 = INT(235 * HY / 19) + 1

```Divide the number of years by 19, yielding a quotient Q and remainder R.

Then the minimum and maximum number of years is 235*Q +12*R + X, where X
is obtained from the following table:

R (remainder)    X (minimum)        X (maximum)
-------------    -----------        -----------
0              0                   0
1              0                   1
2              0                   1
3              1                   2
4              1                   2
5              1                   2
6              2                   3
7              2                   3
8              2                   3
9              3                   4
10              3                   4
11              4                   5
12              4                   5
13              4                   5
14              5                   6
15              5                   6
16              5                   6
17              6                   7
18              6                   7

The first term (235*Q) comes from the fact that every 19 year cycle has
exactly 235 months.

The second term (12*R) comes from the fact that every year has at least
12 months.

The third term (X) reflects the minimum and maximum number of leap years
posssible in the span of years since the last multiple of 19.

Shabbat Shalom,
-Larry
```

Correspondent Dwight Blevins made a rather intriguing observation leading to the following Weekly Question.

Question 192

Which are two of the most important arithmetical properties of the 9th year of the mahzor qatan (19 year cycle) known as GUChADZaT?

In 1802g, the German mathematician Carl Friedrich Gauss published a formula which gave the Julian date of Pesach for any Hebrew year A.

The Gauss Pesach formula appeared without proof of its derivation in “Berechnung des jüdischen Osterfestes”, Monatliche Correspondenz zur Beförderung der Erd- und Himmels-Kunde, 5 (1802), 435-437 – reprinted in: Carl Friedrich Gauss Werke (Königlichen Gesellschaft der Wissenschaften, Göttingen, 1874), vol. 6, pp. 80-81.

With a bit of patience, it is actually quite possible to derive the fundamental elements of the Gauss formula.

Gauss either established or noticed that when the base formula 12 * A + 5 was divided by 19, the remainder would actually represent the multiple of 19th's of a single lunar month by which the start of the lunar year would exceed the start of the average solar year.

Thus, 12 * A + 5 leaves a remainder of 18 when divided by 19.

Since 18 is the maximum possible remainder after division by 19, that helps to explain exactly why the latest possible Rosh HaShannah's in any given period of time always coincide with the 9th year of GUChADZaT.

For example, Rosh HaShannah 5804H will begin on Mon 5 Oct 2043g. This date, October 5, is in the Gregorian calendar, the latest possible date today for the first day of Tishrei.

Note also that 12 * 5804 + 5 = 69,653 which leaves 18 when divided by 19.

These observations lead to the second major arithmetic property of the 9th year of GUChADZaT. This will be discussed in the next Weekly Question.

Question 193

What is the second major arithmetic property of the 9th year of the mahzor qatan (19 year cycle) known as GUChADZaT?

The most important part of the Gauss Pesach formula is that it leads to a very simple summation for the number of months that have elapsed up to any year HY in the Hebrew calendar.

Let R(x, 19) = the non-negative remainder after x is divided by 19.

Then,

19 * SUM(HY) = (HY + 2) * 235 + R(12 * HY + 5, 19)

Let XY = the number of years elapsed from some HY

Then,

19 * SUM(HY + XY) = (HY + 2 + XY) * 235 + R(12 * HY + 5 + 12 * XY, 19)

The number of months between HY and HY + XY

= 19 * SUM(HY + XY) - 19 * SUM(XY)
= (HY + 2 + XY) * 235 + R(12 * HY + 5 + XY, 19) - (HY + 2) * 235 - R(12 * HY + 5, 19)
= XY * 235 + R(12 * HY + 5 + 12 * XY) - R(12 * HY + 5, 19)

When HY = 9, then R(12 * HY + 5, 19) = 18

Hence, the number of months in XY years beginning at the 9th year of GUChADZaT

= (XY * 235 + R(12 * XY + 18, 19)- 18) / 19

This single valued expression also represents the minimum number of months that any period of XY years may have.

Therefore, a second property of the 9th year of GUChADZaT is that any period of years beginning in the 9th year will always have the minimum number of months possible for that period of years.

Rabbi Erwin Schild, Rabbi Emeritus of Congregation Adath Israel in Downsview, Ontario, asked the following question.

The following terms need explaining before asking the question.

A parshah or sedrah is the portion of the Mosaic scriptures (Torah) selected for reading during a particular week.

Shabbat Shuvah is the Shabbat preceding the day of Yom Kippur.

Parshah or Sedrah Ha'azinu is found in D'vorim (Deuteronomy) 32:1-52.

Question 194

How often is parshah Ha'azinu read on Shabbat Shuvah?

Rabbi Erwin Schild, Rabbi Emeritus of Congregation Adath Israel in Downsview, Ontario, asked this week's question.

The following terms need explaining.

A parshah or sedrah is the portion of the Mosaic scriptures (Torah) selected for reading during a particular week.

Shabbat Shuvah is the Shabbat preceding the day of Yom Kippur.

Parshah or Sedrah Ha'azinu is found in D'vorim (Deuteronomy) 32:1-52.

One of the more prevalent practices, among the Jewish people, is that of reading the entire Mosaic text 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. The two portions that can each be read on Shabbat Shuvah are Vayelech and Ha'azinu.

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 Humash (Pentateuch) as translated by Alexander Harkavy, and published by the Hebrew Publishing Co. in New York (1928).

That source shows that the portion Ha'azinu is read whenever Rosh Hashannah begins on either Thursday or Shabbat.

The Keviyyot shows the statistical distribution of all the years that begin on Thursday or Shabbat in the full Hebrew calendar cycle of 689,472 years.

Keviyyot - Under ALL Postponement Rules
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

The statistics show that parshah Ha'azinu is read on about 61% of all of the Shabbat Shuvah's.

Question 195

What is an unusual feature of Hebrew year 5764H?

An unusual feature of the year 5764H, which began on Shabbat 27 September 2003g, is that the Hebrew dates of its first few months also coincide with the same Gregorian dates of the years 1927g, 1965g, 1973g, and 1984g.

This made a number of people observe that the tragic Yom Kippur War in 1973g occurred on the same Gregorian date as that of Yom Kippur 5764H. That however, as seen from the table below, was mere coincidence, and not due to arithmetic phenomena involving calendar cycles.

Tishrei 1HYEARLENGTH
`Tue 27 Sep  1927g`
`  5688H`
`   354`
`Mon 27 Sep  1965g`
`  5726H`
`   353`
`Thu 27 Sep  1973g`
`  5734H`
`   355`
`Thu 27 Sep  1984g`
`  5745H`
`   354`
`Sat 27 Sep  2003g`
`  5764H`
`   355`

Question 196

What effect, or effects, would be realized if postponement to Tuesday occurred whenever the molad of Tishrei occurred on a Monday following a 13-month Hebrew year?

The number of possible Hebrew single year lengths would remain the same as is currently known. These lengths in days are 353, 354, 355, 383, 384, and 385 days.

However, postponing to Tuesday, any post-leap year whose molad of Tishrei is on Monday, introduces another qeviyyah, namely, a 385-day year beginning on Tuesday.

The Qeviyyot shows the effects of the dehiyyot on the calendar.
Properties of Hebrew Year Periods explains the effects of the dehiyyot on the calendar .

The following statistical table, using the rule of a post-leap year Monday postponement shows the effect of such modification over the full Hebrew calendar cycle of 689,472 years.

Keviyyot - post leap mondays out
YEAR LENGTH IN DAYS
DAY 353 354 355 383 384 385 TOTALS
Mon 39369 0 57832 40000 0 32576 169777
Tue 0 66584 0 0 12784 23504 102872
Thu 0 124416 22839 26677 0 45899 219831
Sat 29853 0 94563 40000 0 32576 196992
TOTALS 69222 191000 175234 106677 12784 134555 689472

Question 197

What major change, or changes, if any, would have to be made to the Gauss Pesach formula to accommodate a post 13-month year postponement from Monday to Tuesday?

Initially, the 382-day year test would have to remove the additional test for the fraction m > 23,269 / 25,920.

A postponement taking place from a post 13-month year Monday to Tuesday negates the possibility of eliminating the test for Dehiyyah Molad Zaqen by simply adding 6 hours to the resulting time of the Tishrei molad.

As a result, the Gauss Pesach Formula also would have to

test separately for Dehiyyah Molad Zaqen;

alter the constant 32 + 4,343 / 98,496 to 31 + 78,215 / 98,496;

alter the 356-day year test fraction to m > 827 / 2,160.

A QBasic program which demonstrates one form of the Gauss Pesach formula is found in the Additional Notes under The Gauss Pesach Formula.

Question 198

What minimum Tishrei molad time would create a 385-day year beginning on Tuesday if postponement took place from a post 13-month year Monday to Tuesday?

Properties of Hebrew Year Periods - Part 1 indicates that the length in days of a 385-day year may be given by

385d = 383d + INT(f + 21h 589p) + p" - p'

```
where f  = the fractional part of the starting molad of Tishrei
p' = the number of days Tishrei 1 is postponed at the start of the period
p" = the number of days Tishrei 1 is postponed at the end of the period
```

In the above instance it is necessary that p" = 1d.

Consequently, p' = 0; INT(f + 21h 589p) = 1; leading to the minimum f = 2h 491p.

Since the year must be 385d, that year will start on Tuesday if the next year is to end on Tuesday.

Therefore, the minimum Tishrei molad time required to create a 385-day year beginning on Tuesday, if postponement takes place from a post 13-month year Monday to Tuesday, is 3d 2h 491p.

Question 199

In how many ways can 7 leap years be distributed in 19 year cycles, such that the 13-month years have between each other either one or two 12-month years?

Last Week's Question

In how many ways can 7 leap years be distributed in 19 year cycles, such that the 13-month years have between each other either one or two 12-month years?

There are 57 ways in which 7 leap years can be distributed in 19 year cycles, such that the 13-month years have between each other either one or two 12-month years. All 57 possibilities are listed below. The number 0 is really the equivalent of the number HY MOD 19, where HY is a Hebrew year whose value is a multiple of 19.

The Arrangements
1
`0  2  4  7 10 13 16`
2
`0  2  5  7 10 13 16`
3
`0  2  5  8 10 13 16`
4
`0  2  5  8 11 13 16`
5
`0  2  5  8 11 14 16`
6
`0  2  5  8 11 14 17`
7
`0  3  5  7 10 13 16`
8
`0  3  5  8 10 13 16`
9
`0  3  5  8 11 13 16`
10
`0  3  5  8 11 14 16`
11
`0  3  5  8 11 14 17`
12
`0  3  6  8 10 13 16`
13
`0  3  6  8 11 13 16`
14
`0  3  6  8 11 14 16`
15
`0  3  6  8 11 14 17`
16
`0  3  6  9 11 13 16`
17
`0  3  6  9 11 14 16`
18
`0  3  6  9 11 14 17`
19
`0  3  6  9 12 14 16`
20
`0  3  6  9 12 14 17`
21
`0  3  6  9 12 15 17`
22
`1  3  5  8 11 14 17`
23
`1  3  6  8 11 14 17`
24
`1  3  6  9 11 14 17`
25
`1  3  6  9 12 14 17`
26
`1  3  6  9 12 15 17`
27
`1  3  6  9 12 15 18`
28
`1  4  6  8 11 14 17`
29
`1  4  6  9 11 14 17`
30
`1  4  6  9 12 14 17`
31
`1  4  6  9 12 15 17`
32
`1  4  6  9 12 15 18`
33
`1  4  7  9 11 14 17`
34
`1  4  7  9 12 14 17`
35
`1  4  7  9 12 15 17`
36
`1  4  7  9 12 15 18`
37
`1  4  7 10 12 14 17`
38
`1  4  7 10 12 15 17`
39
`1  4  7 10 12 15 18`
40
`1  4  7 10 13 15 17`
41
`1  4  7 10 13 15 18`
42
`1  4  7 10 13 16 18`
43
`2  4  6  9 12 15 18`
44
`2  4  7  9 12 15 18`
45
`2  4  7 10 12 15 18`
46
`2  4  7 10 13 15 18`
47
`2  4  7 10 13 16 18`
48
`2  5  7  9 12 15 18`
49
`2  5  7 10 12 15 18`
50
`2  5  7 10 13 15 18`
51
`2  5  7 10 13 16 18`
52
`2  5  8 10 12 15 18`
53
`2  5  8 10 13 15 18`
54
`2  5  8 10 13 16 18`
55
`2  5  8 11 13 15 18`
56
`2  5  8 11 13 16 18`
57
`2  5  8 11 14 16 18`

GUChADZaT is actually found as the 15th entry in the list.

The values in the 3rd row are such that 12 * HY MOD 19 < 7.

Correspondent Larry Padwa made an excellent try at the solution to this problem but was not able to identify all of the possible arrangements. It is impressive to note that Larry Padwa's approach to the answer led to a list of arrangements that is the exact reverse order of the first 21 entries in the above table. According to Larry Padwa,

```This seems more like a mathematical/enumeration problem than a calendar
question. The only way I could think of to solve it is to actually
enumerate all of the possibilities.
I've come up with 21 as the answer (but I very well may have made some
errors). My reasoning is as follows:

First, label the years 0 through 18. We can arbitrarily assume (without
loss of generality) that year 0 is one of the leap years. Then the
following twenty sequences are allowed:

1) 0,3,6,9,12,15,17
2) 0,3,6,9,12,14,17
3) 0,3,6,9,12,14,16
4) 0,3,6,9,11,14,17
5) 0,3,6,9,11,14,16
6) 0,3,6,9,11,13,16
7) 0,3,6,8,11,14,17    (THIS IS THE ONE WE ACTUALLY USE)
8) 0,3,6,8,11,14,16
9) 0,3,6,8,11,13,16
10) 0,3,6,8,10,13,16
11) 0,3,5,8,11,14,17
12) 0,3,5,8,11,14,16
13) 0,3,5,8,11,13,16
14) 0,3,5,8,10,13,16
15) 0,3,5,7,10,13,16
16) 0,2,5,8,11,14,17
17) 0,2,5,8,11,14,16
18) 0,2,5,8,11,13,16
19) 0,2,5,8,10,13,16
20) 0,2,5,7,10,13,16
21) 0,2,4,7,10,13,16

Please let me know if this is correct, or if I have missed some or
duplicated some.
Shabbat Shalom,
-Larry
```

Larry Padwa's solution correctly provides the first 21 patterns that satisfy the answer. Excellent work Larry and thanks for sharing your results!

Question 200

Of the 57 arrangements given in answer to Weekly Question 199, which pattern, or patterns, can be used to generate all of the other arrangements?

There are 57 ways in which 7 leap years can be distributed in 19 year cycles, such that the 13-month years have between each other either one or two 12-month years. All 57 possibilities are listed below. The number 0 is really the equivalent of the number HY MOD 19, where HY is a Hebrew year whose value is a multiple of 19.

The Arrangements
1
`0  2  4  7 10 13 16`
2
`0  2  5  7 10 13 16`
3
`0  2  5  8 10 13 16`
4
`0  2  5  8 11 13 16`
5
`0  2  5  8 11 14 16`
6
`0  2  5  8 11 14 17`
7
`0  3  5  7 10 13 16`
8
`0  3  5  8 10 13 16`
9
`0  3  5  8 11 13 16`
10
`0  3  5  8 11 14 16`
11
`0  3  5  8 11 14 17`
12
`0  3  6  8 10 13 16`
13
`0  3  6  8 11 13 16`
14
`0  3  6  8 11 14 16`
15
`0  3  6  8 11 14 17`
16
`0  3  6  9 11 13 16`
17
`0  3  6  9 11 14 16`
18
`0  3  6  9 11 14 17`
19
`0  3  6  9 12 14 16`
20
`0  3  6  9 12 14 17`
21
`0  3  6  9 12 15 17`
22
`1  3  5  8 11 14 17`
23
`1  3  6  8 11 14 17`
24
`1  3  6  9 11 14 17`
25
`1  3  6  9 12 14 17`
26
`1  3  6  9 12 15 17`
27
`1  3  6  9 12 15 18`
28
`1  4  6  8 11 14 17`
29
`1  4  6  9 11 14 17`
30
`1  4  6  9 12 14 17`
31
`1  4  6  9 12 15 17`
32
`1  4  6  9 12 15 18`
33
`1  4  7  9 11 14 17`
34
`1  4  7  9 12 14 17`
35
`1  4  7  9 12 15 17`
36
`1  4  7  9 12 15 18`
37
`1  4  7 10 12 14 17`
38
`1  4  7 10 12 15 17`
39
`1  4  7 10 12 15 18`
40
`1  4  7 10 13 15 17`
41
`1  4  7 10 13 15 18`
42
`1  4  7 10 13 16 18`
43
`2  4  6  9 12 15 18`
44
`2  4  7  9 12 15 18`
45
`2  4  7 10 12 15 18`
46
`2  4  7 10 13 15 18`
47
`2  4  7 10 13 16 18`
48
`2  5  7  9 12 15 18`
49
`2  5  7 10 12 15 18`
50
`2  5  7 10 13 15 18`
51
`2  5  7 10 13 16 18`
52
`2  5  8 10 12 15 18`
53
`2  5  8 10 13 15 18`
54
`2  5  8 10 13 16 18`
55
`2  5  8 11 13 15 18`
56
`2  5  8 11 13 16 18`
57
`2  5  8 11 14 16 18`

GUChADZaT is actually found as the 15th entry in the list.

The values in the 3rd row are such that 12 * HY MOD 19 < 7.

However, only the first 3 entries in the above table are independent of each other.

The Arrangements
1
`0  2  4  7 10 13 16`
2
`0  2  5  7 10 13 16`
3
`0  2  5  8 10 13 16`

All of the other entries are actually derived by adding a constant value, ranging from 1 to 18, to each of the entries in the first 3 rows and then remaindering by 19.

For example, by adding 17 to each entry in the 3rd row, and then remaindering by 19, the resulting values are 17, 0, 3, 6, 8, 11, and 14. These values represent GUChADZaT. If this procedure is applied to the 3rd row with all of the constants ranging from 1 to 18, then all of the possible variations of GUChADZaT will be developed as shown in Cycles and Moladot.

``` First  Begun 21 Jun 1998