MISHNAH: TURNING A CEILING INTO SECHACH
(R. Yehudah): Beis Shamai requires the shaking of each plank as well as the removal of every second plank while Beis Hillel permits the Sechach after either of these acts.
(R. Meir): He must remove every second plank (and shaking is irrelevant).
THE RATIONALE FOR THE ARGUMENT IN OUR MISHNAH
Question: We understand that Beis Hillel requires an act to remove the stigma of Ta'aseh v'lo Min he'Asuy, but why does Beis Shamai demand two acts (one should suffice)!?
Answer: To remove the concern of Gezeiras Tikrah.
Question: Then, again, removing every second plank should be sufficient!
Answer: It is owing to Gezeiras Tikrah and Beis Shamai means that even though he shakes the planks, he must remove every second plank in order to make the Sukah usable.
Question: Then R. Meir is not arguing, he is only restating the position of Beis Shamai!?
Answer: R. Meir is telling R. Yehudah that Beis Hillel and Beis Shamai do not argue on the matter.
Question: It seems that R. Meir and R. Yehudah have already argued over Gezeiras Tikrah in the previous Mishnah (14a)!?
Answer (R. Chiya b. Aba citing R. Yochanan): The previous Mishnah speaks of planks which are less than four Tefachim wide (thus are not subject to Gezeiras Tikrah), but which have been smoothed (and may be prohibited lest they be confused with Kelim having a Beis Kibul [Gezeiras Kelim]).
Question: But given the position of R. Yehudah citing Rav (who on 12b permitted male arrows, in spite of the possible decree forbidding them because of female arrows) if they did not issue a Gezeirah there, why should they do so here?
Answer: Rather, both Mishnayos discuss Gezeiras Tikrah with the first Mishnah introducing the Machlokes.
Question: Why would the Mishnah need to discuss it twice?
Answer: In our Mishnah, R. Yehudah points out that Gezeiras Tikrah is the opinion only of Beis Shamai, to which R. Meir responds that it is unanimous.
Question: But, according to Shmuel, why are they arguing in our Mishnah!?
We understand that, according to Rav (who says that R. Meir and R. Yehudah argue regarding four-Tefach planks) that they argue over Gezeiras Tikrah (since most ceilings use four-Tefach planks).
But for Shmuel (who says that all agree to invalidate four-Tefach planks) why should even Beis Hillel (according to R. Yehudah) permit the use of the ceiling (whose planks, presumably, are four Tefachim)!?
Answer: R. Yehudah concedes that planks of four Tefachim invalidate a Sukah, but only if one can do nothing about it.
However, if one does something to be Mevatel their status as planks of a ceiling it is permitted.
This is because the suspicion that one may come to use a ceiling no longer exists.
The Machlokes in our Mishnah is over what is required to be Mevatel the status of the planks as a ceiling.
MISHNAH: METAL RODS AND A SUKAH IN A HAYSTACK
It is permitted to use metal rods or bed-posts provided one leaves space between them equal to the rods themselves.
One who burrowed out a room within a haystack has not made a Sukah.
Question: Our Mishnah seems to refute R. Huna bar R. Yehoshua in his position regarding Parutz Ke'Omed!?
(R. Papa): Parutz ke'Omed is permitted by the walls of Shabbos (if the spaces between the posts of a wall are the same as the posts themselves, the wall is Kosher).
(R. Huna bar R. Yehoshua): It is not permitted to carry there unless the Omed exceeds the Parutz.
Our Mishnah seems to permit with Sechach equal to the rods!?
Answer: The Mishnah, which says 'like spaces,' does not mean identical spaces, but spaces which would allow rods of the same width to be moved in and out easily (so the Sechach covers more than 50% of the surface).
Question: But from the language of the Mishnah it would be permitted even if, indeed, they were exactly equal!?
Answer (R. Ami): Our Mishnah requires him to leave (slightly larger) spaces for the Sechach to allow the Sechach (equivalent in width to the rods) to be inserted easily, thus creating a majority of Sechach (see Tosfos).
Answer (Rava): When he spreads the Sechach over the rods, he lays the Sechach perpendicular (not parallel) to the rods, which will place more Sechach than rods (or the Sechach would fall through the spaces).
(FORMER) BED POSTS ARE NOT VALID SECHACH
Question: Is our Mishnah a support for R. Ami b. Tivyumi (who invalidates a Sukah whose Sechach is vessels/clothes which are no longer fit for use)?
Answer: No, because our Mishnah may be explained (as R. Chanan, citing Rebbi, explained another Mishnah) as speaking when the bed-post (the long side post or the short head post) still has its two legs still attached to it, and such a bedpost is still fit for use (and still subject to Tum'ah).