Gemara

Mishnah: We do not make children fast on Yom Kipur, but we train them in the Mitzvah a year or two before (adulthood) so they will be used to Mitzvos.

Question: If we train them two years before, there is no need to say one year!

Answer (Rav Chisda): We train a healthy child two years before, and a sickly child one year before.

Rav Huna: We train a child of eight or nine to fast for hours (he eats slightly later than usual). A child of 10 or 11 is Mashlim (fasts the entire day) mid'Rabanan. A 12 year old is Mashlim mid'Oraisa (regarding a girl. There are many different texts in this Sugya; some omit these words.)

Rav Nachman: We train a child of nine or 10 to fast for hours; a child of 11 or 12 is Mashlim mid'Rabanan; a 13 year old is Mashlim mid'Oraisa regarding a boy.

R. Yochanan: No child is Mashlim mid'Rabanan! A child of 10 or 11 fasts for hours; a 12 year old is Mashlim mid'Oraisa.

Question (against Rav Huna and Rav Nachman - Beraisa): We do not make children fast on Yom Kipur, but we Mechanech (train) them in the Mitzvah a year or two before adulthood.

Answer: This refers to Chinuch of Hashlamah.

Question (Beraisa): What is Chinuch? If he usually eats at two hours of the day, we feed him at three hours, if he usually eats at three hours, he eats at four hours.

Answer (Rava bar Ula): There are two kinds of Chinuch.

Kesuvos 50a - Abaye: My (surrogate) mother told me that a boy begins to learn Chumash at six, he begins Mishnah at 10, he fasts the entire day of Yom Kipur at 13, and a girl fasts at 12.

Rishonim

Rif. Rav Huna and Rav Nachman both obligate a boy of 12 to fast mid'Oraisa. The Halachah follows R. Yochanan, who says that the earliest age of adulthood is 12 or 13 for a girl and boy, respectively. Regarding Chinuch the Halachah follows Rav Huna and Rav Nachman, for they are Basra.

(Ran DH Amar): 'A child of eight' means that he finished eight years and started his ninth, and similarly for every age. Rav Huna and Rav Nachman hold that Toch ha'Zeman (pubic hairs brought in the year before they normally come, e.g. 13 years for a boy) is like after the Zeman (it is a sign of adulthood). R. Yochanan holds that Toch ha'Zeman is like before the Zeman.

Rif: Both girls and boys are Mashlim at 11 mid'Rabanan. We train a child of 10 to fast for hours, like Rav Nachman. He eats an hour later than he usually does.

Milchamos Hash-m: Chachamim were not stringent to make girls fast earlier than boys, even though they nature a year earlier, because Chinuch is primarily for boys. Regarding Chinuch and Hashlamah mid'Rabanan the Halachah follows the intermediate opinion, i.e. Rav Nachman.

Rebuttal (of Rif - Rosh 8:12): Since the Rif rules like R. Yochanan regarding Toch ha'Zeman, he requires two years of Hashlamah (even more if the boy will not bring hairs until after 14). He should require only one year of Hashlamah like Rav Nachman, the intermediate opinion! Also, why do Rav Huna and Rav Nachman discuss obligating one who brings hairs early regarding Inuy, and not regarding other Mitzvos? Also, R. Yochanan holds (Nidah 45b) that Toch ha'Zeman is like after the Zeman!

Korban Nesan'el (3): Even though R. Yochanan was refuted in Nidah, if he holds here that Toch ha'Zeman is like before the Zeman, the Gemara should have said that he contradicts himself.

Rebuttal (of Rif - Ba'al ha'Ma'or): All agree that Toch ha'Zeman is like before the Zeman. Rav Huna and Rav Nachman discuss a girl. The Amora'im argue only about Hashlamah mid'Rabanan. R. Yochanan says that there is none, Rav Nachman says that it is one year, and Rav Huna says that it is two years.

Rebuttal (of Ba'al ha'Ma'or - Milchamos Hash-m): If an Amora does not specify, we assume that he discusses males, especially when the conjugation is male (Mashlimim). Also, it is unreasonable to give the law for girls and not for boys!

Question: In Nidah (45b) R. Yochanan says that Toch ha'Zeman is like after the Zeman! If this is his opinion, the Halachah does not follow any opinion here!

Answers (Milchamos Hash-m): There, R. Yochanan obligates one whose 13th year was completed on Yom Kipur to fast mid'Oraisa. Alternatively, R. Yochanan retracted because he was refuted. Rav Nachman and Rav Huna did not retract, because they found that Tana'im argue about this. Alternatively, our Sugya argues with that Sugya about R. Yochanan's opinion.

Rambam (Hilchos Shevisas Asor 2:10): We train a child of nine or 10 to fast for hours, e.g. an hour later than he usually eats. If he is stronger we train him to fast longer. An 11 year old boy or girl completes the fast mid'Rabanan for Chinuch Mitzvos.

Rebuttal (Ra'avad): Rav Huna discusses a boy, and Rav Nachman discusses a girl. They do not argue. The Halachah follows them (there are two years of Hashlamah mid'Rabanan and two years of fasting for hours for each gender).

Kesef Mishneh: The Rambam holds like the Rif. The Rambam calls 'a child of eight' one who finished eight years.

R. Mano'ach: The Rambam holds that Rav Huna discusses a boy, and Rav Nachman discusses a girl. Regarding Hashlamah; they do not argue. The Gemara said that there are two kinds of Chinuch. Haslamah is Chinuch for the Mitzvah of Inuy, to practice doing it properly. It applies even to girls. Chinuch for hours applies only to boys. It is not a Chiyuv mid'Rabanan. Rather, a man is obligated to ensure that his sons go on a straight path.

Rambam (ibid. 11): If a 13 year old boy or 12 year old girl brought two hairs he or she is a full adult, and must complete the fast mid'Oraisa. If he or she did not bring two hairs he or she is a minor, and must complete the fast mid'Rabanan. We do not Mechanech a child below nine years to fast lest it harm him.

Rosh (ibid.) The Rif's text is difficult, because of questions 2:ii and 2:v. Rashi's text is better. We are Mechanech for hours at ages nine and 10, and for Hashlamah at ages 11 and 12.

Poskim

Shulchan Aruch (OC 616:2): We are Mechanech a (Rema - healthy) child of nine full years or 10 years to fast for hours, e.g. to eat an hour later than he usually eats. If he is stronger we train him to fast longer.

Magen Avraham: The Ri (Tosfos Yeshanim 82a DH Ben) says that the obligation of Chinuch applies only to the father. Nevertheless, a mother should not feed her child (who should be fasting for Chinuch) for it is like giving forbidden food to him. Even though she may feed him outside the Sukah, there the food is fully permitted, and she is exempt from making him enter a Sukah. Also, there is a Lav not to eat on Yom Kipur, whereas Sukah is only an Aseh. However, if he takes food by himself on Yom Kipur she need not stop him.

Mishnah Berurah (5): Some say that also a mother is obligated to be Mechanech her child.

Kaf ha'Chayim (9): One who brings children below the age of Hashlamah to the Beis ha'Keneses should bring food for them, for it is dangerous for them to fast until finishing (Musaf). The children should not eat in front of adults.

Rema: The same applies to a healthy girl.

Shulchan Aruch (ibid.): An 11 year old boy or girl completes the fast mid'Rabanan for Chinuch Mitzvos.

Beis Yosef (DH Kosav ha'Ran): Many say that the Halachah follows R. Yochanan. However, since the Rif and Rambam agree (unlike this opinion), we follow them.

Mishnah Berurah (7): Regarding Hashlamah we do not distinguish between strong and weak children, unless the child is sick and fasting could be dangerous.

Rema: Some say that they need not complete the fast at all. One may rely on this for a weak child.

Source - Gra (DH v'Yesh Omrim): Tosfos (Kesuvos 50a DH u'Vas): Abaye (citing his mother) holds like R. Yochanan that there is no Hashlamah mid'Rabanan (Abaye is Basra with respect to Rav Huna and Rav Nachman).

Rebuttal - Bach (DH Garsinan, citing Rashi Kesuvos 50a DH uv'Tinokes): Surely, Abaye cited her to teach a Chidush! Letter of the law, Hashlamah mid'Rabanan applies to healthy boys and girls two years before adulthood, and one year beforehand if they are weak. However, learning Torah weakens people. Therefore, even if boys (who learn) appear strong they are considered weak. Also the Rosh (Kesuvos 4:16) cites Abaye, even though he rules that there is Hashlamah mid'Rabanan. Nowadays we are not concerned if boys do not fast in their 12th year because they all learn, and we are not proficient to know whether or not it is dangerous for them to fast.

Mishnah Berurah (9): Nowadays no child fasts in the 12th year because everyone is weak. No child is considered healthy for Hashlamah unless we know that he is healthy and can endure the fast. Eliyahu Rabah says that Hashlamah is not done even during the 13th year.

Rema: Whenever we Mechanech to fast, we also Mechanech to refrain from bathing and anointing (this is like the Tur).

Rebuttal (Beis Yosef DH uch'Shem, citing the Ran 1b DH Yom): The Mishnah (78b) permits all of the Inuyim to children; we find Chinuch only regarding fasting. Chachamim did not forbid things that promote children's growth such as bathing and anointing.

Shulchan Aruch (ibid.) If a 13 year old boy or 12 year old girl did not bring two hairs, he or she is a minor, and he or she must complete the fast mid'Rabanan.

Question: Perhaps there were hairs and they fell out (and the fast is mid'Oraisa)!

Answer (Beis Yosef DH Kosav b'Terumas): The case is, he turned 13 on Yom Kipur and they checked him the entire day and did not find hairs.

Magen Avraham (3): A child who did not bring hairs is exempt from the other fast days.

Rema: Even if he is weak he must complete the fast; we are concerned lest he brought hairs and they fell out.

Mishnah Berurah (12): This refers to one who turned 13 before Yom Kipur, or even on Yom Kipur but he was not checked continuously throughout the day.

Source (Beis Yosef ibid.): We do not allow a girl above 12 years to do Mi'un even if she does not have hairs, lest she brought hairs and they fell out. We are similarly concerned regarding eating on Yom Kipur, for it also entails Kares. The same applies to a boy above 13; it is unreasonable to distinguish between boys and girls.

Shulchan Aruch (ibid.): We do not Mechanech a child below nine years to fast lest it harm him.

Rema: If he wants to fast we protest.