Which event is more infrequent?

Question

This year, we have a couple of infrequent events, and I'm trying to figure out which is less frequent, considering a 19 year cycle (which, I think, is the best time unit to use). Some of the data might be at this site, but I haven't found specifically what I am looking for.

There are 3 possible Shabbatot (outside of Israel) on which we can take out 3 sifrei Torah (as far as I know, but if there are more, please let me know): Shabbat Rosh Chodesh Tevet, Channukah; Shabbat Rosh Chodesh Adar, Parshat Sh'kalim; and Shabbat Rosh Chodesh Nisan, Parshat Hachodesh.

I do realize that on Simchat Torah outside of Israel we also take out 3 (or more) but that isn't part of the grouping I want to consider.

In some years, some of these happen, though I don't know if there are years in which all or none happen.

We also have a case in which we take out 2 sifrei Torah on a "weekday" (defined however you want, to exclude Shabbat, Yom Tov and Chol Hamo'ed Pesach). Rosh Chodesh Channukah is the one weekday when this happens (as far as I remember - tell me others if I am wrong), and it happens every year.

My question is "which happens less frequently over the cycle: a 3 sefer Torah shabbat or a 2 sefer Torah weekday?"

Answer 1

tl;dr

Over a span of 689,472 years (the complete Hebrew calendar cycle), a three-Torah shabbat happens 465,929 times, and a two-Torah weekday happens 1,009,765 times.

For technical details, see below the line.

---

First, let's consider the frequency of three-Torah shabbatot:

• Shabbat Rosh Chodesh Tevet occurs in year types בשה, גכה, בשז ,גכז.

• Shabbat Rosh Chodesh Adar (II) occurs in year types השא, זחא, החא.

• Shabbat Rosh Chodesh Nisan occurs in year types הכז, בשז, גכז.

Using the frequencies from the 689,472-year cycle (taken from here) and summing, I get a total of 67.6% i.e. an 'average' year contains approximately two-thirds of a three-torah shabbat.

Now, as noted in the question, every year contains at least one weekday Rosh Chodesh Tevet, so we could stop here and conclude that a three-Torah shabbat occurs more infrequently than a two-Torah weekday.

However, for completeness' sake, we note that there are two Rosh Chodesh Tevet weekdays in the following year types: הכז, השא, זשג, השג, זשה.

Summing over the frequencies from the same source as before gives a total of 146.5% , so an 'average' year contains approximately one and a half two-Torah weekdays.

Answer 2

Let us try to answer this question in a meaningful, cause and effect way, as opposed to a count of years on the 247 nearly cycle.

As an introduction, one of the basic principles of the current Jewish calendar is that all dates from the Adar adjacent to Nisan until the following Cheshvan can only fall on four days of the week each. This is because the length of all months except for Cheshvan and Kislev are fixed. When we see that the following Rosh Hashanah will fall out on an invalid day of the week (Sunday, Wednesday or Friday), we add a day to the calendar (at the end of the previous Cheshvan, ten or eleven months earlier, to give people the maximum amount of time to find out). The next year will sometimes have a day taken away from Kislev, depending on the moon's exact time of appearance.

The day of the week that any date, say Rosh Chodesh Nisan, would fall out on if not for this postponement is essentially equal. (I'm sure someone who wants to do the statistics will say some are more common that others. But approximately.) But after the postponement, three days of the week are each twice as common as the fourth valid day of the week.

With this in mind, let us work out first what will be for the earlier parshas shekalim and hachosesh and the following Chanukah for each of the days of the week that RC Nisan can fall on, which are Sunday, Tuesday, Thursday and Shabbos.

Sunday: the parshas shekalim leading up to this year is on Rosh Chodesh, but not hachodesh. Rosh Hashanah will be on a Tuesday, so the next year will always be kisidrin, meaning that Cheshvan will have 29 days and Kislev 30, so RC Teves will be on Shabbos and Sunday. So this year has twice a three torah Shabbos and one two Torah weekday. Notice that the two are not the same calendar year.

Tuesday: neither the previous parshas shekalim nor hachosesh were on RC, and the following Rosh Hashanah will be on Thursday, so RC Teves will be during the week, either one day or two. This year has no three torah Shabbos.

Thursday: neither the previous parshas shekalim nor hachosesh were on RC, and the following Rosh Hashanah will be on Shabbos, so RC Teves will be during the week, either one day or two. This year has no three torah Shabbos.

Shabbos: hachosesh is a three torah Shabbos, but not shekalim. RH will be on Monday, which allows RC Teves to be on Friday, or Shabbos and Sunday. Also in this case, there is at least one day of RC on a Chanukah weekday.

(If RC Teves is also Shabbos and Sunday and the year is a leap year, the following RC Nisan will be on Shabbos, so you get a three Torah Shabbos twice in five months.)

We see from this that four of seven years, from the starting point of Adar, don't have any three Torah Shabbosos, and the other three don't always have two of them, but all years have at least one two Torah weekday, and about half (most years for four out of seven days of the week) have two.