Beitzah she'Noldah

(Beis Shamai): An egg laid on Yom Tov may be eaten.

(Beis Hillel): The egg may not be eaten.

Isur Chametz

(Beis Shamai): One violates the Isur Chametz only once one has ka'Koseves.

(Beis Hillel): Like Se'or, Chametz is be'Kezayis.

Yachpor b'Deker

(Beis Shamai): One may dig in order to perform Kisuy ha'Dam.

(Beis Hillel): The dirt must be prepared before Yom Tov.

Both agree that b'Di'eved, one may dig out the dirt.

The ashes of a Kirah are Muchan.

Question: What type of chicken laid the egg in (1:a)?

Answer: A chicken which was designated to be eaten.

Question: Then why would Beis Hillel forbid its egg (Uchla d'Ifras)?

Answer: It is rather a chicken designated to lay eggs.

Question: Then it is Muktzah, and how could Beis Shamai permit it?

Answer: Perhaps Beis Shamai do not hold of Muktzah (like R. Shimon).

Question: Maybe Nolad is worse than Muktzah!?

Answer (R. Nachman): The chicken was designated to lay eggs, and permitting Muktzah implies permitting Nolad as well.

Question: R. Nachman then seems to contradict himself regarding the views of Beis Hillel and Beis Shamai!?

Here he aligns Beis Hillel with R. Yehudah (stringent opinion regarding Muktzah) and Beis Shamai with R. Shimon (lenient view).

In the Mishnah dealing with bones and skins on Shabbos we read that Beis Hillel prohibits and Beis Shamai permits (consistent with our Mishnah).

Yet, R. Nachman there switches the positions to align Beis Hillel with R. Shimon and Beis Shamai with R. Yehudah!

Answer: R. Nachman was only following the cues left by R. Yehudah ha'Nasi when he dictated the anonymous Mishnayos and kept Beis Hillel aligned with the Halachah.

By Shabbos the anonymous Mishnah (cited) follows R. Shimon.

By Yom Tov the anonymous Mishnah (cited) follows R. Yehudah.

Question: Why did R. Yehudah switch the positions?

Answer: There is less concern that people will abuse a leniency by Shabbos than by Yom Tov.

Question: If the Mishnah speaks of a chicken designated for laying eggs, then the Mishnah should teach the Isur regarding the chicken and we would know the Isur regarding the egg!?

Answer: To teach us the extent of the Kula of Beis Shamai (even Nolad is permitted).

Question: But teaching the Isur would inform us of the extent of the Chumra of Beis Hillel!?

Answer: When left with a choice, we teach the strength of a Kula.

Question: Then satisfy both needs by combining the chicken and the egg in the Mishnah!?

Answer (Rabah): Rather, the chicken was designated for eating.

Question: Then why is it Asur?

Answer: Because of Hachanah (Yom Tov after Shabbos).

(Rabah): An egg laid today was prepared yesterday.

(Rabah): From the Pasuk we learn that one may not prepare on Yom Tov for Shabbos, nor the reverse.

Question (Abaye): Then an egg laid on a mid-week Yom Tov should be permitted!?

Answer: It is a Gezeirah lest we permit an egg laid on Yom Tov after Shabbos.

Question: Then an egg laid on a regular Shabbos (not Motzei Yom Tov) should be permitted!?

Answer: It is a Gezeirah lest we permit an egg laid on Shabbos after Yom Tov.

Question: But we learned that eggs found within a slaughtered chicken on Yom Tov may be eaten, and are not subject to a Gezeirah lest we permit eggs born on that day!?

Answer: Rabanan do not apply Gezeiros to improbable cases such as finding a completed egg in a chicken.