(Mishnah): If three people ate [bread] together, they must be Mezamen (add a preface to Birkas ha'Mazon, and one of them must say at least the first blessing of Birkas ha'Mazon for all of them to hear);

The following count towards the three for a Zimun:

One who ate Demai (doubtfully tithed produce), Ma'aser Rishon from which Terumah was taken, or Ma'aser Sheni or Hekdesh that was redeemed;

A waiter who ate a k'Zayis;

A Kusi.

The following do not count towards the three:

One who ate Tevel (untithed produce), Ma'aser Rishon from which Terumah was not taken, or Ma'aser Sheni or Hekdesh that was not redeemed;

A waiter who ate less than a k'Zayis;

A Nochri.

We do not include women, slaves or children in a Zimun;

Question: How much must one eat to be included in a Zimun?

Answer #1: He must eat at least a k'Zayis;

Answer #2 (R. Yehudah): He must eat at least a k'Beitzah.

(Gemara) Question: What is the source (that three may bless together)?

Answer #1 (R. Asi): We learn from "Gadlu la'Shem Iti u'Neromemah Shemo Yachdav."

Answer #2 (R. Avahu): We learn from "Ki Shem Hash-m Ekra Havu Godel l'Elokeinu."

Question (R. Chanan bar Aba): What is the source that one who answers "Amen" may not answer louder than the one who blessed?

Answer: "Gadlu la'Shem Iti u'Neromemah Shemo Yachdav."

Question (R. Shimon ben Pazi): What is the source that one who translates [to Aramaic during Keri'as haTorah] may not speak louder than the one who reads the verse?

Answer: "Moshe Yedaber veha'Elokim Ya'anenu v'Kol";

Question: Why did the Torah need to say "v'Kol"?

Answer: This teaches that Hash-m spoke as loud as Moshe. (Tosfos - after the first two Dibros, Hash-m was speaking to Moshe, and Moshe was broadcasting for Yisrael to hear. We must say that Hash-m spoke as loud as Moshe because it was forbidden for Moshe to speak louder than Hash-m.)

Support (Beraisa): One who translates may not speak louder than the one who reads the verse (Rambam - and the reader may not speak louder than the translator);

If the translator cannot speak as loud as the reader, the reader should lower his voice.

(Rav or R. Yochanan): If two people ate together, they may make a Zimun if they want;

(The other of Rav and R. Yochanan): They may not make a Zimun.

Question (Mishnah): If three people ate together, they must make a Zimun.

Inference: Three make a Zimun, not two!

Answer: Three must make a Zimun, two have permission (to make a Zimun if they want).

Question (Beraisa): If three ate together, they must make a Zimun, they may not divide (and bless individually). (If two may make a Zimun, one of the three should be allowed to leave, for the others will still be able to make a Zimun!)

Answer: There making a Zimun is different -- since they were obligated to make a Zimun, it is not enough to make a Zimun in a way that is optional (i.e. with two).

Question (Beraisa): If a waiter was serving two people, he may eat without permission (surely, they want him to eat, to enable them to make a Zimun);

If he was serving three, he must ask permission to eat (since they can make a Zimun without him).

Answer: Even though two have permission to make a Zimun, they want him to eat, to enable them to make an obligatory Zimun.

Question (Beraisa): Women make a Zimun by themselves, slaves make a Zimun by themselves, women, slaves and children may not make a Zimun together.

Many women are like two men (they are not obligated to make a Zimun), and it says that Zimun is optional!

Answer: Granted, many women are not obligated to make a Zimun, but perhaps the permission to make a Zimun depends on number (we need three, in order to fulfill "Gadlu (plural) la'Shem Iti," i.e. one asks at least two others to bless with him)!

Question (Seifa): Women, slaves and children may not make a Zimun together.

Why shouldn't they, if there are three? (Tosfos deletes "I'Hachi" from the text, for this question does not depend on the previous answer.)

Answer: They do not make a Zimun together on account of immorality.

Suggestion: The following shows that Rav holds that two may not make a Zimun:

(Rav Dimi bar Yosef citing Rav): If three ate together and one went out to the market, we call to him; [if he does not come,] we make a Zimun on [account of] him (in his absence).

Inference: The other two could not make a Zimun without calling him!

Rejection: Since they were obligated to make a Zimun, it is not enough to make a Zimun in a way that is optional.

Suggestion: The following shows that R. Yochanan holds that two may not make a Zimun:

(Rabah bar bar Chanah citing R. Yochanan): If two ate together, one may be Yotzei (fulfill his obligation to bless) by listening to the other.

Question: We already know this from a Beraisa!

(Beraisa): If one heard [a blessing] and did not answer, he was Yotzei.

(R. Zeira): R. Yochanan teaches that they do not make a Zimun.

Affirmation: Indeed, he must hold that two may not make a Zimun.

Question (Rabah bar Rav Huna): But Rabanan that come from Eretz Yisrael (to Bavel) say that two may make a Zimun;

Presumably, they heard this from R. Yochanan, the biggest Chacham of Eretz Yisrael!

Answer (Rav Huna): No, they heard this from Rav before Rav came to Bavel.

(Rav Dimi bar Yosef citing Rav): If three ate together and one went out to the market, we call to him, and (if he does not come) we make a Zimun on behalf of him (in his absence).

(Abaye): This is only if he can hear the Zimun and answer.

(Mar Zutra): This applies only to a Zimun of three, but one cannot be counted for a Zimun of ten (to say "Baruch Elokeinu") unless he comes.

Objection (Rav Ashi): To the contrary -- nine look like ten (we are not concerned if one does not come), but two do not look like three!

The Halachah follows Mar Zutra.

Question: What is the reason?

Answer: It is improper to mention Hash-m's name with less than ten.

(Abaye): The Halachah is if two ate together, each should bless by himself.

Support (Beraisa): If two ate together, each should bless by himself;

This is if both are Chachamim; if one is a Chacham and the other an ignoramus, the ignoramus should be Yotzei by listening to the Chacham.

(Rava and R. Zeira): If three ate together (and two finished), the third must stop [eating to hear Birkas ha'Zimun; if one finished], two need not stop for one (he must wait for them).

Question: But Rav Papa (and another person) stopped for his son Aba Mar!

Answer: Rav Papa went beyond the letter of the law, he was not required to do so.

Yehudah bar Mereimar, Mar bar Rav Ashi and Rav Acha mi'Difti ate together; normally, the most important one is Motzi the others -- none of them was more important than the others.

(Mishnah): If three people ate together, they must make a Zimun.

They thought that perhaps this is only if one of them is more distinguished, but if all are equal, each should bless by himself; they did so, and asked Mereimar.

Mereimar: You fulfilled Birkas ha'Mazon, but you did not fulfill [your obligation to say] Birkas ha'Zimun.

Suggestion: Perhaps they can say it now!

Rejection: A Zimun may not be made retroactively (after Birkas ha'Mazon).

Question: If one finds people saying Birkas ha'Zimun, what should he answer?

Answer #1 (Rav Zevid): He should say, "Baruch u'Mevorach";

Answer #2 (Rav Papa): He should answer, "Amen."

They do not argue -- if one hears them saying, "Let us bless," he answers, "Baruch u'Mevorach"; if one hears them saying Baruch, he answers "Amen."

(Beraisa #1): It is praiseworthy to answer "Amen" after one's own blessing.

(Beraisa #2): It is disdainful.

Resolution: It is praiseworthy regarding Boneh Yerushalayim (in Birkas ha'Mazon -- we will explain why), it is disdainful regarding any other blessing.

Abaye would answer "Amen" (after he said Boneh Yerushalayim) loudly in order that workers would hear it and return to work;

They do not stay to hear ha'Tov veha'Metiv (the fourth blessing of Birkas ha'Mazon), for it is only mid'Rabanan.

Rav Ashi would answer "Amen" quietly, lest people sense that ha'Tov veha'Metiv is only mid'Rabanan and treat it lightly.