1)

(a)Beis Shamai invalidates an old Sukah - Beis Hillel declares it Kasher.

(b)An old Sukah constitutes - one that was built more than thirty days before Sukos (when we assume that the builder did not have the Mitzvah of Sukah in mind).

(c)Even Beis Shamai will concedes that an old Sukah is Kasher - if the owner specifically had the Mitzvah in mind at the time of building.

2)

(a)Beis Shamai learn the Pesul of an old Sukah from the Pasuk "Chag ha'Sukos ... Shivas Yamim la'Hashem" - which implies that a Sukah must be built for the sake of Hash-m - i.e. for the Mitzvah (and not Stam).

(b)Beis Hillel learn from there the Derashah of Rav Sheshes quoting Rebbi Akiva - that the wood of the Sukah is designated for Hash-m (i.e. which practically speaking, means that one may not derive personal benefit from it during Sukos).

(c)Rebbi Yehudah ben Beseira learns this from the word "Chag" - which teaches us that a certain Kedushah descends on the wood of the Sukah, just like it does on a Korban Chagigah (forbidding its flesh to be eaten until after the fat-pieces have been brought on the Mizbe'ach).

(d)We conclude that really, Beis Shamai too, needs the Pasuk for the Derashah of Rav Sheishes, and he learns the Pesul of Sukah Yeshanah from the Pasuk in Re'eh "Chag ha'Sukos Ta'aseh ... " - implying that the Sukah should be made 'Lishmah' (as if the words were inverted to read "Sukos Ta'aseh l'Chag").

3)

(a)Beis Hillel learn from the Pasuk "Chag ha'Sukos Ta'aseh ... " - that one may construct a Sukah on Chol ha'Mo'ed.

(b)Beis Shamai, on the other hand - agree with Rebbi Eliezer, who forbids it.

4)

(a)Rav Yehudah Amar Rav permits a ball of thread to be used as Tzitzis - since they were attached to the garment for the sake of the Mitzvah. He does not however, permit fringes or loose threads from woven or stitched garments - that were not attached to the garment 'Lishmah'.

(b)He learns this from the Pasuk in Ki Setzei "Gedilim Ta'aseh Lach" - implying that they must be specifically made for the purpose of the Mitzvah.

(c)Nevertheless, he says, Beis Hillel will not learn from "Chag ha'Sukos Ta'aseh Lecha" that a Sukah too, must be constructed 'Lish'mah' - because they need that Pasuk to teach us that a stolen Sukah is Pasuk.

(d)"Gedilim Ta'aseh Lach" on the other hand, is not needed to disqualify stolen material from being used as Tzitzis - because for that, we have another Pasuk in Sh'lach Lecha "v'Asu Lahem Tzitzis" (implying 'theirs but not what is stolen).

5)

6)

(a)Rava qualifies the Mishnah by establishing it where the tree casts more shade than sunshine, because otherwise - the Tana ought to have said simply that if someone makes his Sukah underneath a tree, it is Pasul.

(b)Rava resolves this problem by establishing the Mishnah in a case where the tree lets in more sun than shade, in which case the Tana is coming to teach us - that he is speaking where the tree, like the house, casts more shade than sun. Otherwise, the Sukah would be Kasher.

(c)The problem with this is that the Sukah ought to be Pasul even if the tree lets in more sun than shade - since the Sechach Pasul combines with the Sechach Kasher to render it Pasul.

(d)Rav Papa answers 'be'she'Chavtan', by which he means - that the Mishnah is actually speaking about a case where one lowers the branches on to the Sechach, until they become indistinguishable from those of the Sukah. Then they become Batel to the Sechach.

7)

(a)The Tana of our Mishnah nevertheless considers it necessary to teach us the previous Chidush. It is not so obvious that, if there is more shade than sun (as Rava pointed out) and the part that is Pasul is Batel (as Rav Papa explained), the Sukah is Kasher - because the Rabanan might well have decreed even a tree whose sun is more than its shade, on account of a tree whose shade is more than its sun, when Bitul does not apply.

(b)If not for the Tana of our Mishnah, we would have thought that the Mishnah later, which teaches us that if one bent a vine, a pumpkin or a kind of creeper over his Sukah (which let in more sun than shade), his Sukah is Kasher - means b'Di'eved, but that, l'Chatchilah, one should refrain from doing this.

8)

(a)To explain why, despite the plural use of "ba'Sukos", the Pasuk does not imply that, to the contrary, one Sukah underneath the other, is Kasher - Rav Nachman bar Yitzchak reminds us that the ba'Sukos in question is written without a 'Vav'.

(b)Rebbi Yirmeyahu discusses all the possible computations of 'Sukah Tachas Sukah'. He declares both Sukos Kasher if the bottom one allows in more sun than shade, and the top one is within twenty Amos from the ground, and both Sukos Pasul - if they both shed more shade than sunlight, and the top one is higher than twenty Amos (though this specification does not seem necessary).

(c)The bottom Sukah is Kasher and the top one Pasul, if the bottom Sukah casts more shade than sun, and the top one, which allows in more sun than shade, is within twenty Amos of the ground. The bottom Sukah would be Pasul - if the Sechach of the top one was higher than twenty Amos from the ground - because then it would become Sechach Pasul, which invalidates the bottom Sukah whenever it is not Batel to it, as we learnt above.

(d)The top Sukah is Kasher and the bottom one Pasul - if both Sukos cast more shade than sun, and the top Sukah is within twenty Amos of the ground.