Gemara

(Rav Dimi bar Yosef citing Rav): If three ate together and one went out to the market, we call to him, and we can make a Zimun in his absence [if he can hear the Zimun and answer].

(Mar Zutra): This applies only to a Zimun of three. One cannot be counted for a Zimun of 10 (to say "Baruch Elokeinu") unless he comes.

Objection (Rav Ashi): Just the contrary! Nine look like 10, but two do not look like three!

The Halachah follows Mar Zutra.

47b (R. Asi): We may include a baby in a crib for a Zimun.

Question (Mishnah): We do not include women, slaves or children in a Zimun.

Answer: R. Asi holds like R. Yehoshua ben Levi:

(R. Yehoshua ben Levi): Even though we may not include a baby in a crib for a Zimun (of three), he may be included for 10.

(R. Yehoshua ben Levi): Nine Yisre'elim and a [Kena'ani] slave join.

Question (Beraisa): A case occurred in which R. Eliezer found only nine in the Beis ha'Keneses, so he freed his slave to be the 10th. (He would not join if he was not freed!)

Answer: There were eight Yisre'elim and two slaves. He freed one slave, and then the other joined for the Minyan.

(R. Yochanan): A minor Pore'ach (this will be explained) joins for a Zimun.

Support (Beraisa): A boy joins for a Zimun only if he has two hairs. We are not particular about minors.

The latter clause includes a minor Pore'ach.

The Halachah is unlike all these teachings, rather, like the following:

(Rav Nachman): If a minor knows to Whom we bless, he joins for a Zimun.

Rishonim

Tosfos (47b DH Katan): A minor Pore'ach is one who brought two hairs before 13 years. The hairs are less than the required Shi'ur. Rashi says that they are full hairs.

Rif (Berachos 35a): A minor who knows to Whom we bless joins for a Zimun. Rabanan say that this is if he is in his 13th year, even if he was not Pore'ach. Before this, even if he was Pore'ach he does not join. I hold that 'minor' includes a boy of nine or 10.

Rambam (Hilchos Tefilah 8:4): Tefilah b'Tzibur requires 10 adults.

Kesef Mishneh: The Rambam holds that the Gemara allows a minor (who knows to Whom we bless) to join only for a Zimun.

Rosh (Berachos 7:20): R. Tam says that "the Halachah does not follow any of these teachings" includes the teaching about a minor Pore'ach. A Beraisa says "we are not particular about a minor." Rav Nachman establishes it to discuss a minor Pore'ach who does not know to Whom we bless, or he knows, but he is not Pore'ach. "The Halachah does not follow..." does not apply R. Yehoshua ben Levi's teaching, that a baby in a crib counts towards 10, even though he who does not know to Whom we bless or pray. Rav Nachman holds like R. Yehoshua ben Levi regarding Birkas ha'Mazon. R. Yehoshua said that we are not Mezamen on a baby in a crib. This implies that we are Mezamen on a minor who knows to Whom we bless. Also regarding Tefilah, perhaps he does not argue. Perhaps he holds that a minor joins even if he does not know to Whom we bless. He said that we are Mezamen on a minor who knows, but he did not mention including him for 10. This implies that even a minor who does not know joins towards 10. (I.e. if not, he would have said that if a minor knows, he is included for 10 and we are Mezamen on him - PF) It is clear from the entire Sugya that we are more stringent about joining to a Zimun that including for 10. The custom is to put a Sefer Torah or Chumash in a minor's hands (and count him for a Minyan). There is no support for this from our Sugya. However, in Pirkei d'R. Eliezer it says about Ibur Shanah (making a leap year, that a minor holding a Sefer joins). Based on there, people permitted (Ma'adanei Yom Tov 8 fixes the text to say so) also here, even though they are different. There, it is not Devar shebi'Kedushah! Also Rav Hai Gaon includes a minor for 10. It seems that it is because we say that even a baby in a crib joins, for there is Shechinah among any 10. We learn from "v'Nikdashti b'Soch Bnei Yisrael" that we say Devar shebi'Kedushah amidst 10. This is even among minors, as long as nine are adults. We do not allow more than one minor, just like regarding slaves. It does not honor Shamayim to include more than one. There is Shechinah among any 10 obligated in the Mitzvos and to whom Bris Milah applies.

Rosh: All of this is from R. Tam. He permitted including a minor for 10 and for Tefilah even if he is not holding a Sefer, but he did not do so in practice. The only source for this is Pirkei d'R. Eliezer, regarding Ibur Shanah. There, it mentions a Sefer Torah. It says that R. Eliezer requires 10 for Ibur Shanah, and if we lack one, we bring a Sefer Torah. It does not say that the minor holds it. Even if there is such a source regarding Tefilah, this is only for a Sefer Torah or Chumash, i.e. one Sefer of the Torah written on parchment. Chazal did not have [printed] Chumashim like we do. The Yerushalmi says "a minor and a Sefer Torah, we include them for 10." This means a minor or a Sefer Torah, just like nine men and the Aron. The Halachah does not follow the Yerushalmi. Some texts of the Yerushalmi say "a minor and a Sefer Torah, we include Oso (him, or it)." This is like the custom (that the minor holds the Sefer). Or, perhaps it means that a minor may be counted among the seven required Aliyos.

Rosh: R. Yosef says that presumably, R. Yehoshua ben Levi holds like Rav Ashi, who is more lenient about 10 than a Zimun with three. We rule like Mar Zutra, who is more stringent about 10. Therefore, the Halachah does not follow R. Yehoshua ben Levi. R. Chananel says that Rav Nachman discusses a boy who entered his 13th year without hairs. Even so, when he is 13 we are Mezamen on him. The episode with Abaye and Rava was not regarding a Zimun. Rather, since we discussed children, it is brought to show that they were sharp. This is unlike the Rif. In the Yerushalmi, R. Yosi says that they were not Mezamen on him until he brought two hairs. It is proper not to join a minor towards 10 for Birkas ha'Mazon or Tefilah, or for a Zimun of three, until he brings two hairs.

Poskim

Shulchan Aruch (OC 55:4): Some permit saying Devar shebi'Kedushah with nine men and joining a minor above six who knows to Whom we pray.

Beis Yosef (DH v'Elu): R. Tam (in Tosfos 48a DH v'Leis) rules like R. Yehoshua ben Levi, that a baby in a crib joins for 10 for Birkas ha'Mazon. The same applies to Tefilah, for we ask from the episode with R. Eliezer, which was for Tefilah. Sefer ha'Manhig says that that the Ba'al ha'Ma'or permitted including two minors for 10, for it says "we make him Snifim (adjuncts, plural) to 10. It did not say "Snif". This is wrong.

Beis Yosef (DH veha'Tosfos): The Rivash (451) says that even those who permit include a minor for (10 for) Tefilah, he must know to Whom we bless and pray. He must be at least nine or 10, or at least six in any case. We do not join a minor less than six even if he is sharp and with knows to Whom we bless and pray, for he did not reach Pa'utos (a certain level of intelligence of children). It is as if he has no understanding. Orchos Chayim says that R. Shimshon excommunicated people of a village for rejecting his ruling. They used to join a minor for Tefilah. R. Shimshon permitted only in pressed circumstances, e.g. there is no Minyan in the city. The Ra'avad and others agree. The Kolbo went overboard, and said that the custom is of Yisrael is to join one or two minors, and in pressed circumstances, even three.

Magen Avraham (4): Some say that we join a minor towards 10 for Keri'as ha'Torah since he may be one of the seven Olim. The end of Siman 690 connotes that this is a Safek.

Levushei Serad (9): The Rema (690:18) was unsure whether women count towards 10 for Keri'as ha'Megilah. The Magen Avraham (24) was unsure about a minor, even though we are more lenient about joining towards 10 for the Megilah than for Keri'as ha'Torah.

Gra (DH Yesh): R. Tam says that "the Halachah does not follow" does not apply to the first two teachings or R. Yehoshua ben Levi, since the Gemara interrupted after them with other matters. Also, he discussed 10, and here we discuss three. The Poskim argue with R. Tam. The Ri says that all agree that a minor Pore'ach who knows to Whom we bless joins to a Zimun. The Rosh and Tur rely on the Yerushalmi to say that a minor does not join until 13. Even though the Bavli argues, the custom is like the Yerushalmi. Alternatively, our Sugya is for joining towards 10, unlike R. Yehoshua ben Levi. The Rema (197:10) rules like the Rosh, that we are more stringent about 10 than three. The Shulchan Aruch rules like the Rif and Rambam, that if a minor reached Pa'utos and knows to Whom we bless, he joins towards three or 10. This is unlike R. Tam, who distinguishes them. Tefilah and Birkas ha'Mazon are the same.

Shulchan Aruch (ibid.): The great Poskim disagree.

Beis Yosef (DH v'Chasav ha'Rosh): The Mordechai citing Avi ha'Ezri, Hagahos Maimoniyos, Semag, R. Yonah and Rashba require Bnei Mitzvah for Kaddish, Kedushah and Borchu, since the Gedolim argue about this. The Ra'avad, cited in Teshuvas ha'Rashba 1:239, proves from Bereishis Rabah (91:3) that Tefilah requires adults. The Tur and Rambam agree.

Gra (DH v'Lo): We are more stringent about Tefilah than Birkas ha'Mazon. Maseches Sofrim (16:12) requires adults for Kedushah, Borchu, Keri'as ha'Torah and Kaddish. The Rambam, Tur and many hold like this.

Rema: We do not join him even if he holds a Chumash. However, some are lenient in pressed circumstances.

Magen Avraham (5): Darchei Moshe says that in pressed circumstances one may join one minor, even without a Sefer. The Levush says that the custom is to join him when he holds a Chumash. This is only for obligations such as Borchu and Kaddish. They should not [rely on a minor to] say Kaddish after Aleinu.

Kaf ha'Chayim (42): The Levush and others say that no one is lenient. The Rema does not say to be lenient, and he did not see people lenient. He merely says that if some place has a tradition to be lenient, we do not protest. Even regarding a Zimun, he says (199:10) not to deviate [and be lenient]! The Magen Avraham misunderstood the Rema.

Mishnah Berurah (24): Many Acharonim do not permit joining even one minor, even in pressed circumstances, until they are 13 years old.