1) HALACHAH: INCLUDING A CHILD IN A "ZIMUN"
OPINIONS: According to the conclusion of our Gemara, may a minor (a boy under the age of thirteen) be included in a Zimun of three or ten men? There are several opinions among the Rishonim.
(a) The RAMBAM and RIF rule like the simple understanding of the Gemara and conclude that a child may be one of those who comprise a Zimun as long as he knows to Whom the blessings are directed. Such a child may even be the third person of a Zimun of three (and certainly the tenth of ten).
(b) The TUR (OC 199) cites RAV HAI GA'ON and RABEINU PERETZ who say that a child who knows to Whom the blessings are directed may be included in a Zimun of ten but not in a Zimun of three (see also end of TOSFOS DH v'Leis).
(c) RABEINU TAM rules like the Rambam and Rif, that a child may be one of those who comprise a Zimun (of three according to the Rosh's understanding of Rabeinu Tam, but only of ten according to Tosfos' understanding of Rabeinu Tam), as long as he either knows Whom he is blessing, or he is "Pore'ach." (This refers to a boy who has bodily signs of adulthood, but has not reached the age of thirteen and one day (RASHI), or to a boy within his thirteenth year who does not have fully-developed signs of adulthood (TOSFOS, citing Rabeinu Chananel).) Furthermore, Rabeinu Tam adds that even if he does not know Whom he is blessing nor is he "Pore'ach," he may be included to make a Zimun of ten ("Snif la'Asarah").
(d) TOSFOS (DH v'Leis) explains that in order for a child to be included in a Zimun, he must have both qualities -- "Pore'ach" and know Whom he is blessing. (Tosfos interprets "Pore'ach" to mean that his pubic hair has begun to grow but is not yet fully-developed, and he has not passed the age of thirteen.)
(e) The ROSH cites the Yerushalmi that says that a child must have two hairs in order to be included in a Zimun. Therefore, the Rosh disregards the conclusion of our Gemara and concludes that a child may never be a principal member of a Zimun.
HALACHAH: The SHULCHAN ARUCH (OC 199:10) rules like the Rambam. The Acharonim point out that only one child may be a member of a Zimun, and not more.
The REMA rules like the Rosh, that we do not include minors in a Zimun. We are lenient only when the child is thirteen years old but there is a doubt whether or not he has developed signs of maturity (at least two full pubic hairs). In such a case we may rely on the Chazakah that he has developed signs of maturity, and we may include him as a principal element in a Zimun (because a Zimun is mid'Rabanan).
2) THE OBLIGATION OF A MINOR IN MITZVOS ("CHINUCH")
QUESTION: The Gemara says that in order to recite Birkas ha'Mazon on behalf of others, one must have eaten at least a k'Zayis of bread. RASHI (DH Ad she'Yochal) explains that even though his obligation to recite Birkas ha'Mazon is only mid'Rabanan (since he ate only a k'Zayis and did not eat a Kedei Sevi'ah, enough to satiate him), he may recite Birkas ha'Mazon for others because he is considered to be "Mechuyav b'Davar."
Rashi asks that if a person with a Chiyuv d'Rabanan to recite Birkas ha'Mazon may recite it on behalf of others, then a minor should also be able to recite it for others, even though his obligation is only mid'Rabanan. Rashi answers that the Rabanan did not place the obligation of Chinuch in Mitzvos upon a minor, but rather upon his father. This is logical, because the minor is not obligated to accept upon himself the Rabanan's enactments any more than he is obligated to fulfill any other Mitzvah of the Torah. Rashi in Nidah (46b, DH Isura) makes a similar statement, and emphasizes that it is impossible for the Rabanan to obligate a minor in any of their enactments.
Why, then, does Rashi himself (Berachos 20b, DH Shi'ura) rule that a minor may recite Birkas ha'Mazon for a person over the age of Bar Mitzvah if that person ate less than a Kedei Sevi'ah of bread? (This is also the opinion of TOSFOS (15a, DH v'Rebbi Yehudah, and 48a, DH Ad) and the Halachic ruling (SHULCHAN ARUCH OC 186:2) as well.)
ANSWER: Rashi and Tosfos understand that although a minor is not obligated at all in Birkas ha'Mazon in his own right, nevertheless, the Rabanan -- who originally decreed that a person who ate less than a Kedei Sevi'ah must recite Birkas ha'Mazon -- included in their decree permission for a minor to recite Birkas ha'Mazon on behalf of one who ate less than a Kedei Sevi'ah. The reason they enacted their decree in such a manner was to further the cause of Chinuch, by making it look to the minor as though he is indeed obligated in Mitzvos in his own right, so that he should regard his obligation in Mitzvos with austerity. However, the minor himself is indeed exempt from all Mitzvos, even from the Mitzvah of Chinuch. (M. KORNFELD)
3) HALACHAH: "NECHAMAH" IN BIRKAS HA'MAZON ON SHABBOS
OPINIONS: The Gemara says that on Shabbos, the third blessing of Birkas ha'Mazon begins and ends with "Nechamah" ("comfort"). What does this mean?
There are three opinions among the Rishonim:
(a) RASHI explains that one begins the third blessing with either the word "Rachem" ("have mercy") or "Nachem" ("bring comfort"). Both words express the concept of Nechamah. One ends the blessing with the words, "Boneh Yerushalayim," which express Nechamah. The Gemara is not saying that Birkas ha'Mazon has a different text on Shabbos than on the weekdays. Rather, the Gemara means that no change is made in this blessing because of Shabbos.
(b) The RAMBAM also rules that the blessing is the same on Shabbos as during the week. However, according to the Rambam, at both times one may choose how to begin the blessing ("Rachem" or "Nachem") and how to conclude the blessing ("Boneh Yerushalayim" or "Menachem Yisrael"). This opinion differs from Rashi's, because Rashi says that while one may choose how to begin the blessing, he may not choose how to conclude the blessing. He must end the blessing with "Boneh Yerushalayim."
(c) The RIF says that on Shabbos, one starts the blessing differently than on a weekday. On Shabbos, one says "Nachmeinu" at the beginning of the blessing and "Menachem Amo Yisrael b'Vinyan Yerushalayim" at the end.
HALACHAH: The SHULCHAN ARUCH (OC 188:4) cites the opinion of the Rambam, while the REMA cites Rashi's opinion. The VILNA GA'ON rules in accordance with the opinion of the Rif. The widespread practice is to follow the Rema's view and conclude the blessing with the words "Boneh Yerushalayim."