The Beraisa (dealing with Chol ha'Moed) offers an expanded list of public needs.
Other Halachos of Chol ha'Moed are included in the Beraisa.
Question: If these (e.g. irrigating or marking graves) were done on the 15th of Adar, why are they repeated on Chol ha'Moed?
Answer: There were torrential rains which washed them out.
Question: Why do the agents of Beis Din need to go out, again, on the first of Nisan, regarding Kilayim?
Answer: The crops were not then recognizable.
Question: Whence the requirement to mark graves?
Answer #1: The Pasuk in Vayikra implies that the Tumah must 'call out.'
Answer #2: The Pasuk in Yechezkel specifically refers to such markers. (The references in the Pasuk are fully expounded:)
Etzem... Even bones without flesh must be marked.
Adam... Skull and spine.
Uvana... Marking fixed stones (since rolling stones can lead to false Tumah in their new place).
Etzlo... Mark the grave in a (proximate) Tahor place.
Tziyun... The obligation to mark graves.
If one hovers directly above a single marked stone he is Tamei.
If there are two marked stones the area between is Tamei.
If the intervening space has been plowed, then the two stones are considered as separate (and the intervening space is Tahor).
We do not mark the place where flesh is found.
This operates with the assumption that the flesh will decompose to less than a KeZayis.
Question: But in the meantime the Taharos will be Tamei!?
Answer: The marking would create indefinite time of Tumah.
MISHNAH: THE ERADICATION OF KILAYIM OVER TIME
(R. Yehudah) Originally, the found Kilayim was tossed in the field.
When unscrupulous people increased, they tossed it on the road.
Eventually, the Beis Din made the whole field Hefker.
HEFKER BEIS DIN
The Beraisa explains the rationale for the increasing stringency of the Mishnah (i.e. the unscrupulous kept on being pleased with the eradication work done by the agents).
Making the field ownerless employs Hefker Beis Din Hefker.
Question: Whence that Hefker Beis Din Hefker?
Answer: We see that Ezra employed Hefker Beis Din.
Question: How do we know that such Hefker is exempt from Ma'aser?
Answer: It can be seen in the Mishnah which restricts making a leap year of a Shemitah (or post-Shemitah) year (however, the produce of such an added month would be exempt from Maaser).
Question: Why not add to a post-Shemitah year?
Answer: Not to increase the prohibition of Chadash.
The restriction of the Mishnah only applies until Rebbi permitted vegetables brought from Chutz la'Aretz.
The Beraisa taught that now that the years not ke'Takanan, adding a month is permitted.
The Beis Din of R. Gamliel made the eighth year a leap year.
Question: The Torah obligates us to make a leap year (in order to keep Nisan in the Spring), so an added month (even in a Shemitah year) is truly part of the year, and this Mishnah does not prove Hefker Beis Din!?
Answer: Rather, it is learned from the Din of one who placed his Gadish on top of Leket.
Even Beis Hillel (who obligate the poor to take Maaser from that which was Hefker only for them) would agree that the bottom layer of such a pile is exempt from Maaseros.
This shows that the penalty imposed on such a person by Beis Din does, indeed, make it ownerless.
MAKING SURE THE HALF-SHEKEL COIN IS AVAILABLE AND IS GIVEN
From the 15th of Adar the money changers set up their shops.
From the 25th of Adar they opened where they were even more accessible to the Mikdash.
From that point they would also begin to extract securities from the homes of the delinquents.
The securities were extracted (even from) the Leviim, Yisraelim, converts, but not from women, Avadim, nor minors.
Once a father begins to give the half-Shekel on behalf of a minor, he must continue to do so each year.
Securities are not taken from Kohanim owing to of Darkei Shalom.
(R. Yehudah): Reported that any Kohen who donates is permitted to do so.
(R. Yochanan b. Zakai): He is obligated (they erroneously inferred from the Pasuk that they must not participate in the Korbanos).
HALF-SHEKEL FROM MINORS
We do not take security from minors, but we do demand the Shekel.
This must refer to a minor who has shown signs of maturity.
We do not demand of a younger minor (but a security is taken only from an older age). [There are two versions of the above steps; the second version reads that the obligation starts at age 20.]
TAKING SECURITY FROM KOHANIM
The correct understanding of Darkei Shalom is Darech ha'Kavod.
THE PARTICIPATION OF KOHANIM IN THE HALF-SHEKEL
The reasoning of R. Yochanan b. Zakai is the word Zeh (12).
The Chachamim would respond to R. Yehudah by pointing out that a Korban Tzibur is not put to death.
Question: But this is not something to which R. Yehudah agrees!?
Answer: R. Yehudah responds that this is not a Minchas Yachid.
They, in turn, respond that once it is turned over to the Tzibur it is reckoned as a Minchas Tzibur.
THE PARTICIPATION OF LEVIIM IN THE HALF-SHEKEL
Kol ha'Over refers to one who passed through the sea (all the tribes) or to one who was counted (and the Leviim were not counted in the usual way).
Each brings a support from the context of the Pasuk.