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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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:
(The following occur only as leap-years.)
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).
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).