 WQ Archive 201 - 210

Question 201

Is the leap year distribution known as GUChADZaT an astronomical or arithmetical derivation?

The scholars Richard A. Parker and Waldo H. Dubberstein, in their paper Babylonian Chronology 626 B.C.-A.D. 45, (The University of Chicago Press 1942), indicated that the Babylonians had been using a calendar system which used a cycle of 235 lunar months in a period of 19 years. They also indicated that the extra month lunar years were distributed in a cycle that is familiar to us today. That calendar knowledge apparently preceded Meton's discoveries by a number of years.

The ancient Greek astronomer Meton (c. 5th cent. b.c.e.) observed that 235 lunation periods practically equalled 19 solar years. He therefore suggested a cyclical method of distributing 7 extra lunar months into every period of 19 lunar years. It is not known if he borrowed the idea from the ancient Babylonians or determined that independently. Also, it is not really known how Meton actually distributed the extra month lunar years within this synchronized period of time.

The fixed Hebrew calendar uses only two atronomical parameters. These parameters are the period of the molad which was set to 29d 12h 793p and the observation that 235 moladot equal 19 solar years.

What is rarely noticed or discussed in the Hebrew calendar literature is the fact that the ancient mathematicians chose to distribute the years of the 19 year cycles so that no calendar year could theoretically start more than one lunar month from its corresponding solar year.

The leap year distribution known as GUChADZaT is just one of the 19 ways in which the 12-month and 13-month years can be distributed so as to make possible the less than one month difference.

Consequently, the leap year distribution known as GUChADZaT is an arithmetical derivation, rather than an astronomical fact.

Question 202

Which are the fundamental arithmetic rules governing the leap year distribution known as GUChADZaT?

The two fundamental arithmetic rules governing the leap year distribution known as GUChADZaT can be either
0 < R(7 * Y + 1, 19) < 7 or 0 < R(12 * Y + 5, 19) < 7.

It is a well known fact that the 12-month Hebrew year is a few days shorter than the solar year and that the 13-month Hebrew year is a few days longer than the solar year. The exact amounts by which these years are either shorter or longer than the solar year have no particular relevance to the determination of the eventual leap year distribution in the 19 year cycle.

Of major importance is the fact that 235 lunar months are traditionally accepted as equalling 19 solar years.

And of even greater importance in this derivation is the fact that, at some point in the calendar's history, it was decided to limit the start of each lunar year to less than one lunar month from its corresponding solar year.

These considerations define a very simple algorithmic process for distributing the leap years within the mahzor qatan.

```
Let  S = the length of a solar year
Let  m = the length of a lunar month

By assumption, 235 * m = 19 * S

Hence, 13 * m - S = (13 * m * 19 - 235 * m) / 19 = 12 * m / 19

and    12 * m - S = (12 * m * 19 - 235 * m) / 19 = -7 * m / 19
```
The above indicates the existence of 2 algorithms for identifying leap years.

```
For Y years, when we proceed to add 13-month years,

13 * m * Y = S * Y + 12 * m * Y / 19

Let x = the number of months which must be sutracted from 12 * m * Y / 19 such that

0 < 12 * m * Y / 19 - x * m < m

Then, 0 < 12 * Y - 19 * x  < 19

which inequality represents the remainder of 12 * Y / 19.

Let R(12 * Y, 19) denote the remainder of 12 * Y / 19.

Now, when R(12 * Y, 19) > 6, then a lunar month must be subtracted.
Since, we are proceeding in this algorithm by adding 13-month years,
Y is a 13-month year whenever R(12 * Y, 19) < 7.
```
```
For Y years, when we proceed to add 12-month years,

12 * m * Y = S * Y - 7 * m * Y / 19

Let x = the number of months which must be added to -7 * m * Y / 19 such that

m > -7 * m * Y / 19 + x * m > 0

Then, 19 > -7 * Y + x * 19 > 0

which inequality represents -R(7 * Y, 19).

Now, when R(7 * Y, 19) > 11, then a lunar month must be added.
Since, we are proceeding in this algorithm by adding 12-month years,
Y is a 13-month year whenever R(7 * Y, 19) > 11.
```
With a bit of additional effort, it is possible to note that GUChADZaT emerges whenever either
0 < R(7 * Y + 1, 19) < 7 or 0 < R(12 * Y + 5, 19) < 7.

Correspondent Dwight Blevins shared an interesting Hanukah observation.

Question 203

Of what significance is Tuesday to Hanukah?

Hanukah 5674H begins on Shabbat 20 December 2003g.
Hag Urim Sameach!

The First Day of The Month in the Additional Notes shows the statistical weekday distribution of the first day of each month over the full Hebrew calendar cycle of 689,472 years.

Start of Month Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 193280 79369 0 219831 0 196992 689472
Heshvan 0 196992 0 193280 79369 0 219831 689472
Kislev 151093 68738 69853 127139 79369 193280 0 689472
Tevet 193280 26677 124416 138591 0 206508 0 689472
Shevat 0 193280 26677 124416 138591 0 206508 689472
Adar 0 206508 0 193280 26677 124416 138591 689472
v'Adar 0 85899 0 72576 0 68864 26677 254016
Nisan 79369 0 219831 0 196992 0 193280 689472
Iyar 0 193280 79369 0 219831 0 196992 689472
Sivan 196992 0 193280 79369 0 219831 0 689472
Tammuz 219831 0 196992 0 193280 79369 0 689472
Av 0 219831 0 196992 0 193280 79369 689472
Elul 193280 79369 0 219831 0 196992 0 689472
Totals 1033845 1463854 989787 1345474 1153940 1282540 1258240 8527680
Leap Adar 0 72576 0 68864 26677 0 85899 254016

Amazingly, the month of Kislev can begin on every day of the week other than Shabbat.

Since Hanukah begins on the 25th day of Kislev, it cannot begin 3 days after Shabbat. In other words, Hanukah can never begin on Tuesday. This therefore is the significance of Tuesday to Hanukah.

Thank you correspondent Dwight Blevins for having shared this very intriguing Hanukah fact!

Question 204

How many leap years would have been in the mahzor qatan if 240 lunar months had equalled 19 solar years?

```
Let C = the number of 12-month years in a mahzor qatan, i.e., a 19-year cycle.
Let L = the number of 13-month years in the same mahzor qatan.

Then C + L = 19; Hence, C = 19 - L.
Now, 12 * C + 13 * L = 240 (by hypothesis)
Hence, 12 * (19 - L) + 13 * L = 240
and,                        L = 240 - 12 * 19 = 12
```
Therefore, if 240 lunar months had equalled 19 solar years, the mahzor qatan would have contained 12 leap years.

Correspondent Glenn Leider sent the following response and comments:

```Were there NO leap years in a 19-year cycle, there would be 228 lunar
months, as is the case with the Islamic calendar. With 240 lunar months,
there would be 240-228 or 12 leap years in the 19-year cycle in
question. (Since there in reality 235 lunar months, there are instead 235-228
or 7 leap years in a 19-year cycle.)

Just as note: I am one who endorses W.M. Feldman's suggestion for
combining 17 cycles of 19 years (each containing 7 leap years) with one
cycle of 11 years (containing 4 leap years) for a total of 17x19 + 11 or
334 years. There would be 17x7 + 4 or 123 leap years in this cycle. 334
lunar (Islamic) years total 4,008 lunar months, so when figuring in the
leap years 334 solar years would total 4,008+123 or 4,131 lunar months.
I propose the following lengths, the first two in the ratio of 334 to
4,131:

1 lunar month = 2,551,442.70" = 29d 12h 792.810p (44' 2.70");
1 solar year = 31,556,915.55" = 365d 5h 874.665p (48' 35.55");
1 cycle = 10,540,009,793.7" = 121990d 20h 538.11p (29' 53.7").

My lunar month is only about 1/10" shorter than the astronomical lunar
month. My solar year is 10" shorter, but the year is getting shorter
and will eventually (in a couple of thousand years or so) be the length I
have assigned. What do you think?
```

Thank you Glenn Leider for sharing these magnificent observations!

Correspondent Avi Veisz noted that the answer to Weekly Question 203 had not included Dwight Blevins' logic. It is customary for the Weekly Question to quote its contributors and/or its successful respondents. Here is how Dwight Blevins phrased the observations which led to Weekly Question 203.

```Kislev 25, it appears, never (as in not ever) falls on Tuesday, which I
never before stopped to observe.  This fact, I assume would have to
imply that Heshvan is NEVER elognated to 30 days in a 384 day leap
year.  Otherwise, the Festival of Lights ALWAYS falls on either the
same day of the week as Tishrei 1 or the day before.  Therefore, when
Tishrei 1 falls on Tuesday, Kislev 25 ALWAYS falls on Monday and can never fall
on Tuesday, making that (ie, Tue) the only day of the week which does
not host Kislev 25.
```
The next Weekly Question was suggested by correspondent Glenn Leider.

Question 205

On average, about how many years does it take for the Hebrew calendar to drift by one full lunar month?

Correspondent Glenn Leider sent the following response and comments:

```As stated in The Rosh Hashannah Drift:

"The Hebrew calendar moves more slowly in time than does the Gregorian
calendar. As a result, the earliest possible start day for Rosh
Hashannah is moving later and later into the Gregorian year.

The average Hebrew year length is 365.246822... days

The average Gregorian year length is 365.2425 days

Hence, the Hebrew calendar is drifting through the Gregorian calendar
at an average rate of about 1 day in every 231.374 years."

How long will it take for this drift to reach a full lunar month? Using
your figures, it should take 29.5306 x 231.374 or 6832.61 years, ...
```

Once again, thank you Glenn Leider for sharing these magnificent observations!

Question 206

In the Gauss Pesach formula, which year will first produce zero for the March offset?

The Gauss Pesach Formula was published without commentary in 1802g. Given a Hebrew year, the formula automatically calculates the corresponding Julian date for Pesach in that year.

In the computer language QBASIC, the March offset may be calculated as follows:-

```
da = (12 * dyear + 17) MOD 19
db = dyear MOD 4
dm = 32 + 4343 / 98496 + da + da * (272953 / 492480) + db / 4
dm = dm - dyear * (313 / 98496)
Marchoffset = INT(dm)
```
Because the average Hebrew year is shorter than the average Julian year, the date of Pesach occurs earlier and earlier in the Julian year. This in turn implies that the March offset will decrease over time and eventually become zero, and as time progresses, negative.

The March offset first becomes zero for the Hebrew year 9877H (6117j).

Since March 0 can represent either February 28 or February 29, the Gauss formula needs additional work so as to provide a correct Julian date for Pesach in that year.

Question 207

For the Gauss Pesach formula, what would be the simplest way to avoid a zero March offset as early as year 9877H (6117j)?

The Gauss Pesach Formula was published without commentary in 1802g. Given a Hebrew year, the formula automatically calculates the corresponding Julian date for Pesach in that year.

In the computer language QBASIC, the March offset may be calculated as follows:-

```
da = (12 * dyear + 17) MOD 19
db = dyear MOD 4
dm = 32 + 4343 / 98496 + da + da * (272953 / 492480) + db / 4
dm = dm - dyear * (313 / 98496)
Marchoffset = INT(dm)
```
Because the average Hebrew year is shorter than the average Julian year, the date of Pesach occurs earlier and earlier in the Julian year. This in turn implies that the March offset will decrease over time and eventually become zero, and as time progresses, negative.

The March offset first becomes zero for the Hebrew year 9877H (6117j).

The simplest way to extend the life of the Gauss formula appears to be to add 304 days to dm. This appears to yield consistently Julian dates for the 24th day of Shevat up the year 88,370H (Wed 21 May 84,609j).

Subtracting 304 days from this particular date provides us the Pesach date which in the Julian calendar is Sun 21 Jul 84,608j.

Since this year marks Maimonides' 800th yahrzeit, the following question is derived from his 1175g work on the Hebrew calendar Hilkhot Qiddush HaHodesh.

Question 208

In what year will the molad of Nisan be on Sunday at 17h 107p, as shown in chapter 6:7 of Mamonides' Hilkhot Qiddush HaHodesh?

Often cited as the repetition of the law, Maimonides' 14 volume work called Mishneh Torah was compiled over a 10 year period towards the latter end of the 12th century. The 3rd volume of this religious legal compendium includes Hilkhot Qiddush HaHodesh, which is a documentation of the rules and regulations governing the organization of the Hebrew calendar.

In order to explain the actual calculation of the moladot, chapter 6:7 of Hilkhot Qiddush HaHodesh begins with an example of a molad of Nisan set to 1d 17h 107p. Such a molad, of course, leads to a molad of Iyar that is 3d 5h 900p.

Theoretically, this particular molad of Nisan will first occur as Nisan 38,058H on Sun 14 Aug 34,298g.

The following originally appeared as Weekly Question 72. The Persian born Muslim scholar Al-Biruni documented the fixed Hebrew calendar 175 years before Maimonides.

Question 72

Was Al-Biruni correct in stating that 2 deficient years cannot follow each other because "the perfect months among the months of the cycle (Enneadecateris) preponderate over the imperfect ones"?

The Persian born Muslim scholar Al-Biruni documented the fixed Hebrew calendar 175 years before Maimonides.

The medieval scholar Al-Biruni claimed that 2 deficient years could not follow each other because there are more 30 day months than 29 day months in the Hebrew calendar's 19 year cycle.

As stated on page 153 lines 15-18, in Dr. E.C. Sachau's 1879g translation of Al-Biruni's year 1000g work The Chronology of Ancient Nations

The reason why two imperfect years cannot follow each other is this, that the perfect months among the months of the cycle (Enneadecateris) preponderate over the imperfect ones. For the small cycle comprises 6,940 days, i.e. 125 perfect months and only 110 imperfect ones.

There is no real connection between the number of months in a 19 year cycle and the inability of 2 deficient months to follow each other.

353 day years can begin only on Mondays or Saturdays. Two such years together would cause the 3rd year to begin either on Sunday or on Friday. So that cannot happen.

383 day years can begin on Mondays, Thursdays, and Saturdays. If such years are followed by a 353 day year, then the 3rd year could begin on either Tuesday, Friday, or Sunday.

The need exists to examine the possibility of 383+353 or 353+383 day years beginning on Monday.

The span of time of the moladot for 2 successive years, of which one is leap, is
25*(29d 12h 793p) = 738d 16h 385p (= 3d 16h 385p).

Hence, the earliest possible molad for the beginning of the 3rd year would be
(0d 18h 0p) + (3d 16h 385p) = 4d 12h 385p.

On the other hand, a 383 day year beginning on Monday followed by a 353 day year
would necessarily end on [2 + 383 + 353] remainder 7 = 3d,
at least 1d 12h 385p too short of the earliest possible molad of Tishrei for that 3rd year.

Therefore two imperfect years cannot follow each other.

This is also explained on pages 11-12 of Remy Landau's recently published article, Al-Biruni's Hebrew Calendar Enigmas in the journal Mo'ed Annual for Jewish Studies, Vol 13, 2003.

The following originally appeared as Weekly Question 76.

Question 72

Which year, or years, of the mahzor qatan (19 year cycle) cannot begin a 19 year period of 6,942 days?

Measured from the first day of Tishrei, 19 year periods can be either 6938, 6939, 6940, 6941, or 6942 days long.

The longest of the 19 year periods, 6942 days, cannot occur if the first year of the 19 year period is a LEAP year

Correspondent Larry Padwa not only sent the correct answer, but he also backed it up with a mathematical proof, which will be shown next week. Larry Padwa's proof also touches on the next Weekly Question.

The following originally appeared as Weekly Question 77.

Question 72

Measured from the 1st day of Tishrei, on which day, or days, of the week can the longest period of 19 years begin?

The longest possible periods of 19 years are launched from the closing 1h 21m 12p of Shabbat's molad of Tishrei.

Correspondent Winfried Gerum provided the correct answer.

The answer to Q77 is, that the longest 19-year periods (6942 days), as measured from Tishrei 1 to Tishrei 1, always commence on a shabbat.

```1) Since 6942 is congruent to 5 (mod 7), then if year x+19 begins 6942
days later than year x, then the day of the week that begins year x+19
must be 5 days later (or 2 days earlier) than that of year x.

Consider the four cases:

a) Year x begins on Monday.   Then year x+19 must begin on Saturday.
b) Year x begins on Tuesday.  Then year x+19 must begin on Sunday.
This is impossible.
c) Year x begins on Thursday. Then year x+19 must begin on Tuesday.
d) Year x begins on Saturday. Then year x+19 must begin on Thursday.

At this point, we are left with cases a, c, and d.

2) The molad of year x+19 is 2d 16h 595p later than molad of year x.
(This is always the case).

Case a: If year x begins on Monday, then molad year x is no later than
2d 17h 1079p. (else Dehiyyah Molad Zaken would postpone the beginning
of year x to Tuesday). Therefore molad of year x+19 would be no later
than 5d 10h 595p which would mean RH of year x+19 would be Thursday.
But case a) requires RH of x+19 to be Saturday,
so case a) is impossible.

Case c: By reasoning exactly similar to case a), if year x begins on
Thursday, then year x+19 would begin on Monday (not Tuesday as required
by case c). Thus case c is impossible.

Case d: If year x begins on Saturday, then molad year x is no later
than 0d 17h 1079p (else Dehiyyah Molad Zaken would postpone the
beginning of year x to Monday).

Now consider a year x whose molad is on Saturday after 16h 689p and
before 18h. This would leave RH for year x on Saturday, and
the molad of year x+19 will be on Tuesday after 9h 204p.

If years x and x+19 are leap years, then RH for year x+19 will be on
Tuesday, and the requirement for case d) fails.

However, if years x and x+19 are common years, then Dehiyyah GaTaRad
kicks in and RH for year x+19 will be on Thursday, satisfying the
requirement of case d.

Thus the only time that a 19 year interval has a number of days which
is congruent to 5 (mod 7) is when the starting year is a common year
whose molad is between 0d 16h 689p, and 0d 18h--a period of about an
hour and twenty-two minutes!

Finally, of the possible lengths of 19 year intervals (6939-6942 days),
only 6942 is congruent to 5 (mod 7). Thus when case d) is satisfied,
the number of days in the interval is in fact 6942.

QED
```

Nice work Larry and Winfried!

Correspondents Winfried Gerum and Larry Padwa both noticed and shared additional facts governing the 19 year periods as related to the shortest periods of 6,938 days.

The following originally appeared as Weekly Question 78.

Question 78

Measured from the first day of Tishrei, which year, or years, of the mahzor katan (19 year cycle) can begin a 19 year period of 6,938 days?

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.

Correspondents Larry Padwa, Winfried Gerum, and Dwight Blevins provided correct answers to this question.

Correspondent Larry Padwa not only sent the correct answer, but he also backed it up with a mathematical proof, which will be shown next week. Larry Padwa's proof also touches on the next Weekly Question.

The following originally appeared as Weekly Question 79.

Question 79

Measured from the 1st day of Tishrei, on which day, or days, of the week can the shortest period of 19 years begin?

Measured from the first day of Tishrei, 19 year periods can be either 6938, 6939, 6940, 6941, or 6942 days long.

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.

Correspondents Larry Padwa, Winfried Gerum, and Dwight Blevins provided correct answers to this question.

Correspondent Larry Padwa not only sent the correct answer, but he also backed it up with a mathematical proof, which is as follows:

Since 6938 is congruent to 1 (mod 7), then if year x+19 begins 6938 days later than year x, then the day of the week that begins year x+19 must be one day later than that of year x. Since the only days on which a year can begin are (Mon, Tue, Thu, Sat), the only possibility of two consecutive days are that year x begins on a Monday and year x+19 begins on a Tuesday. Furthermore, since 6938 is the only number in the set {6938,6939,6940,6941,6942} of allowable days for 19 year intervals that is congruent to 1 (mod 7), it follows that x beginning on Monday and x+19 beginning on Tuesday is necessary and sufficient for the 19 year interval to contain 6938 days.

The molad of year x+19 is always 2d 16h 595p later than the molad of year x.

For RH of year x to be on Monday, its molad can be as early as 0d 18h 0p, and as late as 2d 15h 588p (if x-1 is a leap year), or as late as 2d 17h 1079p (if x-1 is not a leap year).

This means that if RH for year x is on Monday, then the earliest that the molad of x+19 could be is 3d 10h 595p. Now, if x (and x+19) are common years, then Dehiyyah GaTaRad would cause RH of x+19 to be on Thursday, thus failing our requirement of a Tuesday RH. However if x (and x+19) are leap years, then GaTaRad would not apply, and RH for x+19 would be on Tuesday if the Molad of x+19 is no later than 3d 17h 1079p. This would occur if the Molad of x is no later than 1d 1h 484p (which would of course still leave RH of year x on Monday).

To summarize: If the molad for a leap year is between 0d 18h 0p and 1d 1h 484p, then the 19 years beginning with year x will contain 6938 days.

Correspondent Winfried Gerum made the following observations:

The answer to Q77 is, that the longest 19-year periods (6942 days) always commence on a shabbat.

```Looking at a full calender cycle, one finds, that if one counts just
cycles starting at year 1H one gets

cycles of  length 6939 commencing on Shabbat or Tuesday or
Thursday
length 6940 commencing on Shabbat or Monday
length 6941 commencing on Thursday or Tuesday
length 6942 commencing always on a Shabbat

If one considers 19-year periods starting with any year there are also
periods of length 6938 days commencing always on a Monday
(see 5790..5808)
```

Correspondent Dwight Blevins sent in the following answer:

In the upcoming discussions of the 6938 period question on the web--the only examples I've found occur in the 19th year of the cycle, and re-occur only at intervals of 247 years. Examples are 1464 - 1483 ce, 1711 - 1730, 1958 - 1977, 2205 - 2224, etc. The set-up limits are Monday for the first year of the period, and Tueday for the declaration of Tishrie 1, day one of the next period. Thus a 6938/7 = 991.14285 week, or a 0.14285 x 7 = 1 day rotation or advance from the first day of the period.
Thank you correspondents Larry Padwa, Winfried Gerum, and Dwight Blevins for these most intriguing answers.

Question 209

What would have happened to the Hebrew calendar if at least 2 consecutive weekdays had been specified as postponement (dehiyot) days for Rosh HaShannah?

Properties of Hebrew Year Periods - Part 1 shows that the length L of any period of Hebrew years is given by

```4.5 L  = D" - D' = M + INT(f+m) + p" - p'
```
```where

M  = the integer    portion of the molad period for the given period of Hebrew years
m  = the fractional portion of the molad period for the given period of Hebrew years
f  = any fractional part of a day such that 0 <= f <= d'
p' = the number of days that Tishrei 1 is postponed at the start of the year
p" = the number of days that Tishrei 1 is postponed at the end   of the year
D' = the day of 1 Tishrei H'
D" = the day of 1 Tishrei H"
```

Assuming any 2 consecutive days as forbidden days for 1 Tishrei, then, whenever the day of the molad of the subsequent year is the first of the proscribed 2 days, p" = 2d.

When p' = 0, INT(f+m) = 1, and p" = 2, then the single year length becomes L = 354 + 1 + 2 = 357 days.

Thus, the 357-day single Hebrew year would be created were 2 consecutive weekdays allowed for the postponement of 1 Tishrei.

Correspondent Dennis Kluk made an observation which helped to suggest the next Weekly Question.

Question 210

When again, in the next 1000 years, will the 30th day of Adar coincide with the 29th day of February in the Gregorian calendar?

The answer to this question is actually found in First Day Hebrew-Gregorian Coincidences.

The 30th day of Adar is really only possible in 13-month Hebrew years. That particular day is today considered to be part of the first month of Adar. Consequently, it is only necessary to search the data presented for a coincidence of v'Adar 1 with March 1 in a Gregorian leap year.

Since 5260H (1500g), these coincidences are seen to have occured as follows

```

```
Due to the The Rosh Hashannah Drift, the year 5584H (1824g) was the last time, for many thousands of years to come, that February 29 coincided with Adar 30.

Correspondent Dennis Kluk asked the question and correspondent Glenn Leider sent a correct answer.

Dennis Kluk made the following remarks

```We had a civil and a Jewish leap day recently but they weren't the same
day. However, I noticed that on Sunday 29 February 1824 it was Adar 30
5584. When will this coincedence happen again?
```
Glenn Leider submitted the following answer
```Note: I inserted a Roman numeral one (I) after Adar since it's only
Adar I of a Jewish leap year has 30 days. The last time the above
condition happened was on the 30th of Adar I 5584H or the 29th of February
1824g. To determine the next possible occurrence, the following rules
apply:

1. The earliest Gregorian date occurs in the 17th year of a 19-year
Jewish cycle. In the 294th such cycle the 17th year was 5548H.

2. Gregorian leap years usually occur every 4 years, such as in 1824g.
So for the next possible occurrence, weneed to proceed 19 x 4 or 76
years, bringing us to 5660H or 1900g.

Unfortunately, 1900g is a century year which isn't divisible by 400 so
is NOT a Gregorian leap year! So let's go another 76 years to 5736H or
1976g. Alas, that year the 30th of Adar I fell on March 2. 76 years
later, the 30th of Adar I 5812H will fall on March 1, 2052g. The 30th of
Adar I 5888H will fall on March 2, 2128g. And the 30th of Adar I 5964H
will fall on March 4, 2204g.

It only gets worse after that! Ten more 76-year cycles later the 30th
of Adar I 6724H will fall on March 6, 2964g. That means that the 30th of
Adar I NEVER falls on February 29 during the next 1000 years. Sorry!
```

The only comment that should be made with regards to Glenn Leider's response is that Hebrew and Gregorian calendar dates do not necessarily match up in 19 year cycles.

Both correspondents Dennis Kluk and Glenn Leider are to be congratulated for making these excellent observations!

Thank you!

``` First  Begun 21 Jun 1998