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In a given year, a certain number of Yamim Tovim days fall on Shabbos, and a certain number not. What are the minimum and maximum count of Yamim Tovim days that can fall on Shabbos in a single year (Tishrei-Elul), in and out of Eretz Yisrael?

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Why?​​​​​​​​​​​​​​​ – Double AA♦ Mar 14 at 2:56
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@double aa Maybe he has to plan his vacation schedule and his boss asked him. – Seth J Mar 14 at 3:09
Basically what @SethJ said: the question was spawned by my wanting to be able to say "there will be up to n days spread throughout the year like Shabbos" on interviews. My field is web development, and in some positions 24-hour on-call coverage can be an issue. – yitznewton Mar 14 at 3:33
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@yitznewton So you are more interested in the min/max number of days that aren't on Shabbat. (I know you can figure out how many are from how many aren't.) – Double AA♦ Mar 14 at 4:12
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@DoubleAA I guess so; I was treating the question as abstracted from my particular situation, and as such I phrased it in the way I thought clearest, i.e. removing negatives that are not necessary to the question. – yitznewton Mar 14 at 15:01

2 Answers

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Let's denote a year by what day is the first day of Rosh Hashanah and what day is the first day of Pesach. So, for example, "2-3" means Rosh Hashanah on Monday and Pesach on Tuesday. For our purposes, years run from Tishrei through Elul.

We have to consider the following yomim tovim: Rosh Hashanah, for which we're provided the day of the week; Yom Kippur, which is two "week"-days after R"H; Succos, which is on the same day of the week as R"H; Shmini Atzeres, which is also on the same day of the week as R"H; Pesach, which we're provided; Seventh day of Pesach, which is first-day Pesach minus one; Shavuos, which is first-day Pesach plus one.

Now, for each possible year, I'll provide the R"H-Pesach numbers, and we'll calculate the rest:

  • 2-3. Nothing on Shabbos
  • 2-5. In Chutz La'aretz, second day of Shavuos is on Shabbos.
  • 3-5. same as 2-5.
  • 5-7. Yom Kippur is Shabbos, First day Pesach is Shabbos, and in Chutz La'aretz, Eighth day Pesach is Shabbos.
  • 5-1. Yom Kippur is Shabbos, Seventh day Pesach is Shabbos.
  • 7-1. Rosh Hashanah is Shabbos, Succos is Shabbos, Shmini Atzeres is Shabbos, Seventh day Pesach is Shabbos.
  • 7-3. Rosh Hashanah is Shabbos, Succos is Shabbos, Shmini Atzeres is Shabbos.

(The following occur only as leap-years.)

  • 2-7. First day Pesach is Shabbos, and in Chutz La'aretz, Eighth day Pesach is Shabbos.
  • 3-7. Same as 2-7.
  • 5-3. Yom Kippur is on Shabbos.
  • 7-5. Rosh Hashanah is Shabbos, Succos is Shabbos, Shmini Atzeres is Shabbos. In Chutz La'aretz, second day Shavuos is on Shabbos.

So, the most possible yomim tovim that occur on Shabbos in one year in Israel is four, which happens when R"H falls on Shabbos and Pesach on Sunday. In Chutz La'aretz, the most is also four in the same year and also when Rosh Hashana falls on Shabbos and Pesach on Thursday (which only occurs as a leap year).

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If you ran the year Nisan-Adar, there are many fewer permutations. – Charles Koppelman Mar 14 at 23:23
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@CharlesKoppelman, although Nissan is rosh hashana lirgalim question states an Tishrei-Elul year – JNF Mar 17 at 7:26
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In a non-leap year that starts on Monday and has 29 days in Kislev, no yom tov is Shabas in Israel or out of it. So zero days is the least.

In a non-leap year that starts on Shabas and has 29 days in Kislev, the first days of Rosh Hashana, Sukos, and Sh'mini Atzeres are Shabas, as is the seventh day of Pesach. No year has more, so four days is the most.

Source: tables (a perpetual calendar).

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starts? 1 Nisan? 1 Tishrei? 1 Januray? – Charles Koppelman Mar 14 at 23:25
@CharlesKoppelman 1 Tishre of course. – msh210♦ Mar 15 at 0:01

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