Mesorat%20hashas for Yoma 163:4

תנן התינוקות אין מענין אותן ביוה"כ אבל מחנכין אותן לפני שנה ולפני שתים בשלמא לרב הונא ורב נחמן לפני שנה ולפני שתים לפני שנה לדבריהן ולפני שתים לדבריהן

At the age of twelve they must fast to the end of the day by Biblical law, [all this] referring to girls. R'Nahman said: At the age of nine and ten one trains them by hours, at the age of eleven and twelve they must fast to the end of the day by Rabbinic ordinance, at the age of thirteen they must fast to the end of the day by Biblical law, [all this] referring to boys. R'Johanan said: There is no Rabbinic ordinance about the obligation of children to fast to the end of the day.

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