Gemara

(Beraisa): Birkas Zimun (a preface to Birkas ha'Mazon) is with three.

Berachos 42b (Mishnah): If people were sitting, everyone blesses for himself. (Only if they reclined, one blesses for everyone.)

Contradiction (Beraisa): If ten people were travelling, even if all eat from the same loaf, everyone must bless for himself. If they sat to eat, even if everyone eats from his own loaf, one blesses for everyone.

It suffices that they sit. They need not recline!

Answer (Rav Nachman bar Yitzchak): The case is, they said "let us go eat bread in Ploni (a certain place)."

After Rav died, his Talmidim were returning from his burial site, and suggested to eat bread on the Danak River. They were unsure if saying this is as good as reclining? Rav Ada bar Ahavah tore his cloak out of grief - "Rav died, and we do not even know the laws of Birkas ha'Mazon!"

43a - Version #1 (Rav): Only bread requires reclining. Wine does not.

(R. Yochanan): Even wine requires reclining.

Version #2 (Rav): Reclining helps only for bread, but not for wine;

(R. Yochanan): It helps even for wine.

45a (Mishnah): If three people ate (bread) together, they must be Mezamen. A waiter who ate a k'Zayis counts towards the three for a Zimun.

47a: Rav and Shmuel were eating; Rav Simi bar Chiya came, and ate quickly.

Rav: Do you want to join for a Zimun? We already finished!

Shmuel: If they would bring (for dessert) mushrooms for me or young fowl for you, wouldn't we eat?!

Rishonim

The Rif (Berachos 32b) and Rambam (Hilchos Berachos 5:2) bring the Mishnah.

Rosh (Berachos 7:1, citing R. Yonah): We discuss three who fixed themselves together from the beginning for Birkas ha'Motzi. Then, if they (in the days of the Gemara) reclined, or if we (nowadays) sit, they may not divide (to bless Birkas ha'Mazon without a Zimun). If they did not fix themselves from the beginning for ha'Motzi, they may divide. We learn from 42b. Only if they reclined, one blesses for everyone. We asked from a Beraisa which says that if 10 were travelling, and they sat to eat, even if everyone eats from his own loaf, one blesses for everyone. Rav Nachman bar Yitzchak answered that they said 'let us go eat bread in Ploni,' which is like reclining. The Beraisa discusses Birkas ha'Mazon. If we do not need Kevi'us for Zimun, why did Rav Nachman need to say that they fixed a place? Eating together fixes them! Rather, they must be fixed from the beginning. There is another proof from the Mishnah, which teaches that a waiter who ate a k'Zayis joins for a Zimun. Why is a waiter a bigger Chidush than anyone else? Rather, it teaches that even though other people join only if they sat Derech Kevi'us, a waiter joins without sitting Derech Kevi'us, for it is normal for him to eat in this way.

Rebuttal (Rosh): Even if hey did not bless ha'Motzi together, but afterwards they were fixed to eat together, they are fixed for a Zimun. They may not divide. The proof from 42b is not a proof. There, if they did not say 'let us go eat in Ploni,' one could not be Motzi the others. However, if initially some of them reclined, and afterwards a third came and reclined with them, they must be Mezamen, for they were Kevu'im hen they finished eating. The same applies to us even if two initially sat together. There is no proof from a waiter, for someone else who ate like a waiter would not join, since he is not Kavu'a.

Ma'adanei Yom Tov (30): A waiter is not Kavu'a even at the end.

Rosh (8:3): Beis Hillel say that one person should bless on light (on Motza'ei Shabbos) for everyone, due to "b'Rav Am Hadras Melech" (it is most honorable when many do a Mitzvah together). This is unlike bread, for which one may not bless for others unless they recline. There, since they did not recline, they are not Kevu'im. Here it is like wine according to Rav, which does not require reclining. Even R. Yochanan requires reclining only for wine, which is important and it is normally to recline for it. For other Berachos, it is as if they reclined.

Poskim

Shulchan Aruch (OC 167:11): If two or more people are eating, one blesses (ha'Motzi) for all of them. This is only if they were reclining, which is Derech Kevi'us (Rema - or a man with his household, which is like reclining). If not, everyone blesses for himself. If they said 'let us eat here (or in Ploni),' since they fixed a place to eat, this is Keva. Nowadays that we do not normally recline, sitting at one table or without a table on one cloth is Kevi'us, even for a group (not a family). For us, even if they fixed a place or are one household, one blesses for everyone only if they sit at one table or one cloth.

Shulchan Aruch (193:2): Even if all three were not initially Kevu'im to eat together, rather, two were Kevu'im and afterwards a third came and was Kove'a with them, or one was Kove'a and afterwards two came and were Kove'a with them, they may not divide, since they were Kevu'im when they finished eating. In any case, if he eats with them not Derech Kevi'us they may divide, unless he is a waiter.

Magen Avraham (8): 'Fixed together' means at one table, like it says in 167:11. If not, one may not exempt another in Birkas ha'Motzi or Birkas ha'Mazon. The Shulchan Aruch (there) says so, and also the Rosh (7:1), Tosfos and R. Yonah. R. Yerucham and the Rashba say that (in such a case) it is a Mitzvah to divide. The Gemara connotes that when one cannot be Motzi another, they may not make a Zimun even if each will bless by himself, for Rav's Talmidim tore their clothes. They should have made a Zimun, and each will bless by himself, like they did for ha'Motzi! Rather, since they were not Kevu'im, they may not make a Zimun. We say so also in 195:1. Also the Bach says so. The Rema says that it is better to be Mezamen, i.e. they should eat a little at one table to be obligated in a Zimun. We must explain the Bach this way, for in Siman 195 he says that they may be Mezamen only if they initially entered for this. This is like the Riva (in Beis Yosef below), who permits relying on the Poskim who do not require reclining. We hold like the Poskim who require it.

Gra (2): The Mechaber rules like the Rosh (8:3), who does not allow a Zimun if they are not obligated.

Mishnah Berurah (18): Kevi'us is the same as in Siman 167. Sitting to eat at one table or on one cloth helps even if each eats from his own loaf. The Gra there holds that a household joins even without one table.

Mishnah Berurah (21): Even if they began and finished together, if it was not Derech Kevi'us at one table, they may divide.

Taz (6): If three fixed to eat together, even if one wants to finished before the others, he cannot uproot his obligation. However, if one began earlier, and a second came and joined him, he may finish earlier and bless by himself. Why did the Beis Yosef omit this distinction?

Bach (DH Al): Three who finish together are obligated to be Mezamen only if they were initially fixed with three, even if they began in separate groups of three.

Mishnah Berurah (19): They are obligated only if they began or finished together. If two finished, but did not bless until the third finished, they must bless together if the ones who finished could still eat if more food was brought. It is as if they finished together.

Kaf ha'Chayim (14): B'Di'eved, even if they were not Kevu'im and one blessed for another, if both intended that he be Motzi him, he was Yotzei.

Rema: In any case, even when they may divide, it is better to be Mezamen, due to "b'Rav Am Hadras Melech".

Beis Yosef (DH Motzasi): The Riva says that if two ate in a corner and one ate in a (another) corner, since they did not intend to eat together, they are not obligated to be Mezamen. It is better if they do, due to "b'Rav Am Hadras Melech".

Mishnah Berurah (23): Likewise, when there are six or 20, they may divide (and each group will be Mezamen by itself), but it is better not to.

Mishnah Berurah (24): It is not clear whether three may be Mezamen when they are exempt. Ten may not do so, for 10 are Mezamen with Hash-m's name.

Kaf ha'Chayim (16): L'Chatchilah, they should eat at one table to be obligated. B'Di'eved they may be Mezamen, especially since nowadays everyone blesses for himself.