37b: The Torah writes "Keranos" two times without a 'Vov', and once with a Vov. The Mikra (the way words are pronounced) connotes six, and the Mesores (the way words are written) connotes four. Beis Hillel expound both, and consider it as if five were written.

Question: Beis Hillel should expound "ba'Sukos" similarly! It is written twice without a 'Vov', and once with a Vov. They should require five walls in a Sukah!

Answer: One is needed for context (and is not expounded). One teaches about the Sechach. The other three teach about walls. A tradition from Moshe from Sinai teaches that one of the three walls can be as small as a Tefach (and a drop, if it is positioned properly).

Sukah 7a (Beraisa): A Sukah wall is like a wall for Shabbos, provided that there are not three Tefachim between poles. There is a stringency of Shabbos over Sukah. A Mechitzah is valid for Shabbos only if Omed Merubah Al ha'Parutz (the width of what stands exceeds the gaps). This does not apply to Sukah.

Eruvin 2a (Mishnah): If (the opening of the Mavoy) is wider than 10 Amos, it must be reduced (through a wall in the middle). If it has Tzuras ha'Pesach (a vertical beam on each side and a beam on top of them), even if it is wider than 10, it need not be reduced.

11a: Rav Yehudah taught Chiya bar Rav (that the Mishnah says) it need not be reduced.

Rav (to Rav Yehudah): (Even if it has Tzuras ha'Pesach, the text says that) it must be reduced!

Rambam (Hilchos Sukah 4:12): If a Sukah has many openings and gaps in the walls it is Kosher. This is even if Parutz Merubah Al ha'Omed (the width of the gaps exceeds the width of what stands). However, if there is a gap of more than 10 Amos, then the Parutz may not exceed the Omed.

Magid Mishnah: The Rambam learns from the Beraisa that requires Omed Merubah for Shabbos, but not for Sukah. The Rambam permits Parutz Merubah only for the third wall of the Sukah. The Rambam forbids this for Shabbos even when there is Tzuras ha'Pesach.

Kesef Mishneh: It is difficult to say that the only leniency of Sukah is the third wall. If so, why did the Rambam need to write this? He already said that a Tefach suffices for the third wall! Rather, the Magid Mishneh means that Parutz Merubah is Kosher only for the third wall. We see that a Tefach suffices, therefore, Chachamim were lenient to allow Parutz Merubah also when the third wall extends across the entire side. However, the two walls must be like for Shabbos.

Chazon Ish (OC 75:5): The Beis Yosef explains that Rashi and the Rosh explain that the Beraisa discusses Omed Merubah in all the walls together. Presumably, the Magid Mishnah disagrees, for otherwise he would have explained the Rambam similarly.

Rosh (Sukah 1:8): The Yerushalmi says that some things are Kosher for a Sukah and Pasul for a Mavuy, and vice-versa, e.g. four Dukranin (sticks), or two walls and a Tefach, and a wall with an opening more than 10 Amos. Shabbos is more stringent than Sukah, for a Tefach suffices for Sukah, but Shabbos requires Omed Merubah. I.e. if the width of the openings without Tzuras ha'Pesach join with what is Parutz to be Mevatel what stands. This does not apply to Sukah, for which two walls and a Tefach suffice, even if the two walls have many openings without Tzuras ha'Pesach. (This is like Rashi.) The Rambam (12, and Sefer ha'Mitzvos) says that if there is a gap of more than 10 Amos, even if it has Tzuras ha'Pesach, then the Parutz cannot exceed the Omed. This is astounding. The Mishnah says that if it has Tzuras ha'Pesach, one need not diminish (the opening)! The Halachah does not follow Rav, who corrected the text to say 'one must diminish.' Regarding Tzuras ha'Pesach, we do not distinguish between more or less than 10 Amos.

Beis Yosef (OC 630 DH u'Moreinu): The Mizrachi answers (for the Rambam) that there is no source that with Tzuras ha'Pesach we do not distinguish between 10 and more than 10. Perhaps the Mishnah permits more than 10 with Tzuras ha'Pesach, just like less than 10, but they are not identical. We equate them to say that the breach does not disqualify the wall, but we can say that less than 10 is permitted even if Parutz Merubah, but more than 10 with Tzuras ha'Pesach is permitted only if Omed Merubah.

Shulchan Aruch (OC 630:5): If a Sukah is Kosher through two walls like a Gam (a Greek Gamma, like an L-shape) and a Tefach with Tzuras ha'Pesach, even if in the two walls there are many openings without Tzuras ha'Pesach, even if Parutz Merubah Al ha'Omed, it is Kosher.

Taz (6): We calculate the Parutz in the two totally Parutz walls with the openings. If it exceeds the Omed in the two walls, this is Parutz Merubah that is Kosher according to Rashi. This implies that the walls themselves are Omed Merubah. However, Rashi wrote above that a Sukah wall is Kosher with just vertical sticks less than three Tefachim apart. The walls themselves are Parutz Merubah! We must distinguish when there is no big breach, just each is less than three Tefachim. Then, we permit even if Parutz Merubah in that wall. Due to Lavud, it is like a solid wall. When there is a breach more than three Tefachim, and we want to permit because it is like an opening, then we require Omed Merubah in (the rest of) that wall. If not, the breach is not considered an opening. This is even if the entire wall (including the breach) is Parutz Merubah. This is explicit in Rashi. Why did the Poskim omit it?

Kaf ha'Chayim (38): When there is a breach more than three Tefachim, the Taz requires Omed Merubah, and he considers gaps less than three to be Parutz. Ma'amar Mordechai disagrees, for less than three is considered Omed, due to Lavud.

Magen Avraham (6): We permit Parutz Merubah overall, but not in the two solid walls.

Gra: Tzuras ha'Pesach does not help for Shi'ur Sukah, even though it helps for Shabbos. If it did, there would be no need for the tradition that the third wall can be a mere Tefach, for through Tzuras ha'Pesach it is like a solid wall! Also, we would not need a Tefach. For Shabbos, sticks (less than three Tefachim apart) suffice! Tzuras ha'Pesach does not help for the corner (between the two solid walls).

Mishnah Berurah (22): The Magen Avraham (Reish ha'Siman) holds that mid'Rabanan, we apply Lavud only if there are four walls, but not if there are only three.

Mishnah Berurah (23): We permit Parutz Merubah Al ha'Omed only if there is a significant Shi'ur in the Omed, i.e. four Tefachim.

Rema: However, it cannot have openings in the corners, for the walls must join like a Gam.

Magen Avraham (7): This refers to all the walls. It suffices if two walls meet in a corner.

Mishnah Berurah (25): The Rema said corners (plural) to teach that when there are four solid walls, he should not make openings in all the corners, for then two walls do not meet in a corner. He may make openings in three corners.

Bi'ur Halachah (DH Ki): There may be a gap less than three Tefachim in the corner, for through Lavud it is considered closed.

Shulchan Aruch (ibid.): However, there cannot be an opening more than 10 Amos. If it has Tzuras ha'Pesach, there can be an opening more than 10 Amos. The Rambam holds that even if it has Tzuras ha'Pesach, an opening more than 10 Amos disqualifies unless Omed Merubah.

Beis Yosef (DH v'Im): Rashi explains that all kinds of walls that Chachamim permitted, e.g. a wall of vertical sticks without horizontal sticks (Eruvin 16b), even if the sticks are only a finger wide and are almost three Tefachim apart and Parutz Merubah, is fine, due to Lavud. It is as if the entire wall stands. Alternatively, a wall can consist of three horizontal ropes (if the sum width exceeds a Tefach, if they are almost three Tefachim apart, they can form a wall of 10 Tefachim) through Lavud. The sticks cannot be more than three Tefachim apart, for then Lavud does not apply. This is if it is more than three Tefachim in many places. If it is only in one or two places, they are like openings. A wall is invalid for Shabbos if there are many openings without Tzuras ha'Pesach, and they Parutz Merubah Al ha'Omed. For Sukah this is Kosher. Even the two walls can be mostly Parutz.

Beis Yosef (DH v'Rabbeinu): R. Yerucham says that according to Rashi, the Beraisa is Machshir for Sukah even if the Omed is less than the seven Tefachim needed for Hechsher Sukah. It seems that the Rosh agrees. Some explain that Parutz Merubah is Kosher only in a big Sukah, when the sum of the Omed is at least Hechsher Sukah. This refers to openings without Tzuras ha'Pesach. If they have Tzuras ha'Pesach, it is Kosher even for Shabbos, even without Hechsher Sukah. The same applies if there are many windows with recognizable walls. My Rebbi, R. Avraham ben Asma'el, said that there are four measures of Sukah walls. 1) An opening is unlike a wall. If there is not Omed as much as Shi'ur Sukah in one place, rather, it is scattered with openings in between, it is Kosher as long as each wall has Omed k'Shi'ur Sukah, even if the opening is more than 10. 2) When there is Parutz more than three Tefachim between the Omed without Tzuras ha'Pesach, if Parutz Merubah, it is Mevatel the Omed if there is no Shi'ur Sukah without a break. 3) If Omed Merubah Al ha'Parutz, if the Omed in each wall is Shi'ur Sukah it is Kosher. 4) If each Parutz is less than three Tefachim, it is Kosher even if Parutz Merubah; the Parutz joins to complete the Shi'ur Sukah.

Sha'ar ha'Tziyon (25): The first opinion in the Shulchan Aruch permits even if Parutz Merubah even when we consider what has Tzuras ha'Pesach to be closed.

Mishnah Berurah (26,27): A gap more than 10 Amos is a breach, unless if it has Tzuras ha'Pesach.Then, it is an opening.

Rema: Nowadays the custom is to make complete walls, because nnot everyone is proficient in the laws of walls. If one does not have enough (for four complete walls), it is better to make three complete walls than four incomplete walls.

Mishnah Berurah (28): 'Complete' walls means without relying on Lavud or Tzuras ha'Pesach. The Ritz Ge'us says that ideally, three walls should be totally closed, and the only gap is the opening.