GEMARA

Beraisa: If a Sukah has two walls k'Hilchasan (seven Tefachim), the third wall can be just a Tefach (this is a tradition from Moshe from Sinai).

R. Shimon says, if it has three walls k'Hilchasan, the fourth can be just a Tefach.

Rav: The short wall must be at an exit (perpendicular to the end of one of the other walls).

Question (Rav Kahana and Rav Asi): It would be better to angle the short wall (towards the wall it does not meet, to make the Sukah appear more surrounded by walls)!

Rav did not answer.

Shmuel and Rabanan of the Beis Medrash agreed with Rav.

R. Simon or R. Yehoshua ben Levi: The short wall must be a Tefach Sochek (slightly more than a regular Tefach). One puts it within three Tefachim of the end of one of the other walls; it is considered Lavud (connected, it is like a wall of four Tefachim. This suffices for the third wall, since it is the majority of the Shi'ur for a wall.)

Rav Yehudah: A Sukah like a Mavoy (the two full walls are parallel) is Kosher. The Tefach wall can be at any exit;

R. Simon or R. Yehoshua ben Levi: The short wall must be a bit more than four Tefachim. He puts it within three Tefachim of one of the other walls; it is considered Lavud.

Question: Why does R. Simon require four Tefachim when the full walls are parallel, but only a Tefach Sochek when they are adjacent?

Answer: When there are two walls k'Hilchasan, the third wall can be just a Tefach. When the two full walls are not k'Hilchasan (they do not meet), the third wall must be four Tefachim (so that through Lavud it is like a full wall).

Ran (DH Tanu Rabanan): R. Simon holds that walls 'k'Hilchasan' (in the Beraisa) must be the proper length and must meet. Rav Yehudah holds that k'Hilchasan connotes only the proper length.

Version #1 (Rava): It is permitted only through Tzuras ha'Pesach (a beam on each side and a beam on top of them).

Version #2 (Rava): Alternatively, it is permitted through Tzuras ha'Pesach.

Version #3 (Rava): Additionally, it needs Tzuras ha'Pesach.

Rav Kahana had a wall of a Tefach Sochek and Tzuras ha'Pesach.

Rav Ashi: Why do you need both? Rava taught that alternatively, it is permitted through Tzuras ha'Pesach!

Rav Kahana: I hold that Rava said that additionally it needs Tzuras ha'Pesach.

RISHONIM

Rif and Rosh (1:6): The Halachah follows Chachamim, who allow two full walls and a wall of a Tefach Sochek. The short wall is placed within three Tefachim of one of the other walls.

Ran (DH Tanu Rabanan): Even though just over half a Tefach suffices to obtain the majority Shi'ur for a wall (via Lavud), we never find that less than four Tefachim is significant.

Rif: In version #1, only Tzuras ha'Pesach permits. We do not care whether or not there is a Tefach wall.

Ran (DH l'Hach): This means that we do not care whether or not the Tefach is Sochek. We cannot say that Rava does not require a Tefach wall at all, for the Beraisa requires it.

Rosh (ibid.): The Rif means that we do not care whether the Tefach is in one place or if it split among two beams.

Rif: The Halachah follows the version in which Rava additionally requires Tzuras ha'Pesach, for Rav Kahana followed it.

Ran (DH Amar): We always require two adjacent walls. When the Sukah is like a Mavoy, the third wall is considered adjacent through Lavud. Therefore, a Sukah cannot have openings in all four corners. Even if there is Tzuras ha'Pesach, this is not considered a wall regarding Sukah.

Rosh: In version #1 Rava says that only Tzuras ha'Pesach permits when the two full walls are adjacent. The Tefach wall is split amidst two beams, each a half Tefach. One beam is adjacent to a full wall, the other is opposite it and even with the end of the other wall. The Tzuras ha'Pesach totally closes off the third side. Even though Tzuras ha'Pesach permits regarding Shabbos and Kil'ayim and there is no Shi'ur for the thickness of the beams, a tradition from Sinai requires a Tefach regarding Sukah. Rava had a tradition that one must make Tzuras ha'Pesach from the Tefach.

In version #2 Rava says that alternatively, Tzuras ha'Pesach of any thickness permits. Even though the Beraisa only mentioned a Tefach; all the more so Tzuras ha'Pesach permits, for Tzuras ha'Pesach is considered a wall regarding Shabbos and Kil'ayim. In version #3 Rava requires Tzuras ha'Pesach in addition to the Tefach wall. A beam is erected (even with) the end of the other wall; the top beam is over it and over the Tefach wall. Presumably, since in a Sukah like a Mavoy the third wall must be four Tefachim and through Lavud it is like a full wall, Tzuras ha'Pesach is not required.

Question: If so, Rava should have taught about Tzuras ha'Pesach immediately after the argument about two walls that meet. Why does the Gemara discuss a Sukah like a Mavoy in between?

Answer (Ran ibid.): After explaining what is required for the third wall when two walls meet, the Gemara discusses whether or not this suffices for a Sukah like a Mavoy. Then Rava discusses the need for Tzuras ha'Pesach.

Rosh (ibid.): Others say that the superior Tikun (four Tefachim for the third wall) needed when the full walls are parallel makes the Sukah equivalent to one with full adjacent walls and a third wall of a Tefach. The first opinion is better, but we must follow the Rif who holds like the latter opinion; the Rambam agrees with him.

Hagahos Ashri: When there are three or four walls Tzuras ha'Pesach is not necessary. The custom is to make Tzuras ha'Pesach for beauty. If the side beams of Tzuras ha'Pesach reach the Sechach there is no need for a beam on top.

Rosh (7): The Rif omits Rava's teaching that a wall of a Sukah is considered a wall for Shabbos. This is because he requires Tzuras ha'Pesach even when the Sukah is like a Mavoy, and Tzuras ha'Pesach is a proper wall for Shabbos.

Rambam (Hilchos Sukah 4:2): If a Sukah lacks three walls it is Pasul. If it has two full adjacent walls like an 'L' shape, one makes the third wall just over a Tefach (wide) and puts it within three Tefachim of one of the other walls. Because there are not three full walls Tzuras ha'Pesach is needed, i.e. a beam on each side and a beam over them. The top beam need not touch the side beams.

Rambam (ibid. 3): If the two walls are parallel, one makes the third wall just over four Tefachim and puts it within three Tefachim of one of the walls. Tzuras ha'Pesach is required.

POSKIM

Shulchan Aruch (OC 630:2): If a Sukah has two walls like an 'L' shape, one makes a third wall over a Tefach wide and puts it within three Tefachim of one of the two walls. He stands up a beam even with (the end of the other wall and) the Tefach wall, and makes Tzuras ha'Pesach, i.e. a beam on top of that beam and the Tefach wall. The top beam need not touch the beam and wall underneath.

Source (Gra DH Af): Eruvin 11b says that the top beam need not touch the side beams.

Eshel Avraham (7): Mid'Oraisa, a wall a Tefach wide suffices. Mid'Rabanan, we require a Tefach Sochek that is like four Tefachim through Lavud, and Tzuras ha'Pesach (Bach DH v'Da, Tosfos 7b DH Sichech (1)).

Mishnah Berurah (9): Some say that the Tzuras ha'Pesach must be at least four Tefachim wide (even if the third wall is less than four Tefachim from the other end of the Sukah).

Kaf ha'Chayim (21): If one forgot to make Tzuras ha'Pesach before Yom Tov, he may use the Sukah if he does not have another Sukah.

Rema: If the side beam and the Tefach wall (this should be the text) reach the Sechach one does not need a beam on top of them.

Question (Magen Avraham 2): The Rema does not stipulate that the Sechach was placed with intent for Tzuras ha'Pesach. This is difficult, for the Yerushalmi requires intent!

Answer #1 (Eshel Avraham 2): The Yerushalmi discusses a Sukah for shade. When Sechach was placed to permit a Sukah, it also helps for Tzuras ha'Pesach. Alternatively, the Yerushalmi asks why the beams must reach the Sechach, for this is not normally required for Tzuras ha'Pesach.

Answer #2 (Gra DH v'Im): The Rema holds that the Bavli does not require intent. Tzuras ha'Pesach is a Mechitzah, and all agree that a Mechitzah made without intent is valid (Eruvin 15a).

Rema: The custom to make an arched Tzuras ha'Pesach in this case is only for beauty.

Kaf ha'Chayim (24,25): It is proper to make Tzuras ha'Pesach for beauty, because of "Zeh Keli v'Anvehu".

Shulchan Aruch (ibid. 3): If the two walls are parallel, one makes the third wall just over four Tefachim and puts it within three Tefachim of one of the walls and it is Kosher.

Source (Gra DH Hayu): The Halachah always follows R. Yehoshua ben Levi.

Mishnah Berurah (15): The Magen Avraham (3) is unsure whether or not the Sukah must be small enough so that the third wall is within three Tefachim of both walls. Many Acharonim say that it need not be.

Shulchan Aruch (ibid.): Also this needs Tzuras ha'Pesach, i.e. a beam on top of the opposite wall and the four Tefachim wall. Some do not require Tzuras ha'Pesach in this case.

Mishnah Berurah (16): One should be stringent like the first opinion.

Rema: If the wall is seven Tefachim without Lavud, one does not need Tzuras ha'Pesach until the end of the wall, because seven Tefachim is the Shi'ur for the width of a Sukah.

Mishnah Berurah (17,18): In this case no one requires Tzuras ha'Pesach. This is only if the third wall is not more than 10 Amos from the other wall.

Rema: The custom to make Tzuras ha'Pesach when there are complete walls is only for beauty.