(Rav Nachman citing Rav): If a ring of idolatry was mixed with 100 rings and one of them fell into the sea, all the rest are permitted. We assume that the forbidden one fell.

Question (Rava - Mishnah): Even if one (Chatas ha'Mesah or Shor ha'Niskal) was mixed with 10,000, all must die.

After the first dies, we do not assume that it was the forbidden animal!

Answer (Rav Nachman): Rav holds like R. Eliezer;

(Mishnah - R. Eliezer): If one of the heads was offered, we offer the rest of the heads.

Question: R. Elazar said that R. Eliezer permits only offering two at a time (since one of them is surely Kosher, we assume that the other is also), but not one at a time!

Answer (Rav Nachman): Indeed, Rav permits the rest of the rings to be sold only two (or more) at a time.

(Rav Nachman citing Rav): If a ring of idolatry became mixed among 100 rings, and 40 rings became separated from the other 60:

If one of the 40 became mixed with other rings, it does not forbid them. If the 60 became mixed with others, it forbids them.

Objection: If one of the 40 became mixed, it does not forbid because we assume that (it is permitted,) the forbidden ring is among the majority (the remaining rings). We should say the same when one of the 60 became mixed with others!

Correction: Rather, if all 40 became mixed with other rings, they do not forbid the new mixture. If all 60 became mixed, they forbid.

(Shmuel): Even if one of the 40 became mixed, it forbids, for we are always stringent about idolatry, no matter how many doubts there are.

Question (Beraisa): A Safek case of idolatry is forbidden. We permit a Sefek Sefeka (there are two doubts, and if at least one of them is (in truth) 'favorable', the matter is permitted).

If a cup of (forbidden due to) idolatry fell into a storehouse of cups, all are forbidden. If one was mixed with 10,000, and one of those became mixed with 10,000, they are permitted.

Answer: Tana'im argue about Shmuel's law (he holds like a different Tana);

(Beraisa - R. Yehudah): (Forbidden) pomegranates of Badan forbid a mixture of any size;

If one was mixed with 10,000 others, and one of this mixture became mixed with 10,000 others, it forbids all of them.

R. Shimon ben Yehudah says in the name of R. Shimon, if one was mixed with 10,000, it forbids them. If one of them became mixed with three, and one of these became mixed with others, they are permitted.

Question: Like which Tana does Shmuel hold?

He is unlike R. Yehudah, for R. Yehudah forbids a Sefek Sefeka regarding all Isurim;

He is unlike R. Shimon, for R. Shimon permits a Sefek Sefeka even regarding idolatry!

Suggestion: (The Beraisa did not explicitly mention idolatry.) Perhaps R. Shimon permits a Sefek Sefeka regarding other Isurim, but forbids regarding idolatry!

Rejection (Beraisa): A Safek regarding idolatry is forbidden, a Sefek Sefeka is permitted.

If so, this Beraisa would not be like R. Yehudah nor like R. Shimon! (What is Shmuel's source to say that there is a three-way argument among the Tana'im?!)

Answer (#1): Shmuel holds like R. Yehudah regarding idolatry. He argues with him (and holds like R. Shimon) regarding other Isurim.

(Beraisa - R. Shimon): If one of the 10,000 became mixed with three, and one of these became mixed with others, they are permitted.

Question: Why must we say that it became mixed with three? Even if it became mixed with two, they are the majority!

Answer (#1): Indeed, it means that it became mixed with two others, a total of three.

Version #1 - Tosfos - Answer #2 (to Question (m)): R. Shimon holds like R. Eliezer according to R. Elazar. (We must sell two at a time, so that surely at least one (and possibly both) of them is permitted. If the entire mixture consist of three, there is no way to benefit from all of them.)

Version #2 - Rashi - Answer #2 (to Question (j)): Shmuel holds like R. Eliezer (who forbids a Sefek Sefeka regarding idolatry. He requires throwing the value of the idolatry into the sea.)

(Reish Lakish): If a (sealed) barrel of Terumah became mixed with 100 (permitted) barrels, and one of them fell into the sea, the rest are permitted.

We need to hear Reish Lakish's law and Rav Nachman's (1:a, when a ring of idolatry fell into the sea);

Had we heard only Rav Nachman's law, one might have thought that this applies only to idolatry, for Ein Lo Matirim (there is no way to permit it other than Bitul), but we do not permit Terumah in this way, for Yesh Lo Matirim (Kohanim may eat it);

Had we heard only Reish Lakish's law, one might have thought that this applies only when a barrel falls in, for this is recognizable, but people do not notice when a ring falls. If we permit the rest, onlookers will think that a mixture is permitted even if one did not fall.

(Rabah): Reish Lakish permits only regarding a barrel, for this is recognizable, but he would not permit regarding a fig.

(Rav Yosef): He permits even regarding a fig. Just like when it is mixed in, it is significant enough to forbid the mixture, when one falls to the sea, it is significant enough to permit the rest.

(R. Elazar): If a barrel of Terumah became mixed with 100 barrels, we open one of them, remove Kedai Dimu'a (the proportionate amount of Terumah, i.e. one part in 101 (Rashi; Shitah Mekubetzes - one part in 100)) and one may drink the rest of the barrel.

Objection (Rav Nachman): This cannot be! (If so, closed barrels are not Mekadesh.)

Correction: Rather, if one of them was opened, we remove Kedai Dimu'a, and one may drink the rest of it.

(R. Oshaya): If a barrel of Terumah became mixed with 150 barrels, and 100 of them were opened, we remove Kedai Dimu'a from each and the rest of the open barrels is permitted;

The other 50 barrels are forbidden until they are opened. We do not assume that the Terumah was among the majority of barrels that were opened.

(Mishnah): The animal was Nirva or Rove'a...

Question: (With the exception of Treifah,) all of these are not recognizable, so we understand how they are mixed. However, what is the case of Treifah?

If it is recognizably Treifah, we can remove it and offer the others;

If it is not recognizably Treifah, how do we know that a Treifah is among them?

Answer #1 (d'Vei R. Yanai): The case is, a wolf was Dores (scratched with its venomous claws) an animal, and that animal was mixed with animals scratched by a thorn. (The scratches look the same, but only the first animal is a Tereifah.)

Answer #2 (Reish Lakish): An animal that fell (from the roof) became mixed with other animals.

Question: If an animal fell, and later walked, this shows that it is Kosher!

Answer: Reish Lakish holds that even if it stood up, we must wait 24 hours before slaughtering it, and even if it walked, we must check it (internally, after Shechitah. The answer is based on the latter law. Waiting 24 hours is not pertinent to the question.)

Answer #3 (R. Yirmeyah): The child of a Treifah became mixed with other animals. The Mishnah is like R. Eliezer, who disqualifies such an animal for a Korban.

The other (i.e. last two) opinions did not say like D'Vei R. Yanai, for they hold that the scars are recognizable. Drisah is a long scratch, but a thorn makes a circular cut;

The others opinions did not say like Reish Lakish, for they hold that if it stood up, we need not wait 24 hours. If it walked, we need not check it, and it is Kosher. (Therefore, all animals in the mixture that can walk are Kesherim);

The others did not say like R. Yirmeyah, for they do not want to establish a Stam Mishnah like R. Eliezer.

(Mishnah): If Kodshim became mixed with Kodshim, Min b'Mino (we offer each (Tosfos - Stam, which is) l'Shem its owner...)

Question: The owner cannot do Semichah (press down on the neck of the Korban! One may support himself on a Korban only when it is a Mitzvah, i.e. one's own Korban. Here, no one knows which Korban is his!)

Answer (Rav Yosef): The Mishnah discusses Korbanos of women. (They are exempt from Semichah.)

Inference: We would not offer Korbanos of men (because Semichah is required, but it cannot be done)!