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[Biblically], one can only be Motzi another in a Mitzvah if he is of equal, or greater, level obligation as the person who is being Yotzei with him. Thus, in order for one who is Biblically obligated in a Mitzvah to be Yotzei with another person’s performance of the Mitzvah, that other person must likewise be Biblically obligated in the general Mitzvah [whether that person is currently obligated or not]. This applies likewise to the Mitzvah of Shofar, (as although the main Mitzvah of hearing Shofar is merely to hear the sound, nevertheless) if the blower is exempt from the Mitzvah then he cannot fulfill the obligation on behalf of others through blowing for them. Based on this, it is clear that a child who is not yet of age of Mitzvos can not be Motzi an adult for a Mitzvah, as the child is only obligated in it due to Chinuch, while the adult is fully obligated in it either Biblically or Rabbinically. Now, when does a child reach the age of Mitzvos and become Biblically obligated in the commands to be able to be Motzi other adults? Only if the child is above Bar/Bas Mitzvah and has reached puberty through the growth of two pubic hairs. Now, what does one do if he knows the child is Bar Mitzvah but does not know if the child has indeed reached this stage of development? Can he be Yotzei with him anyways or not? So if the boy has already grown a nice amount of facial hair, or is above age 18, it is assumed that he is fully developed, and he may therefore be Motzi others in all Mitzvos just as any other adult. If, however he is below 18, and has not yet grown facial hair to this extent, then he maintains a questionable status of which the ruling is that he may be Motzi others for a Rabbinical command, but cannot be Motzi others for a Biblical command. With that said, we will now analyze the law regarding if such a boy may blow Shofar for others on Rosh Hashanah.
B. The law:
The law on the first day of Rosh Hashanah: Men are Biblically obligated to hear Shofar on the first day of Rosh Hashanah. Accordingly, on the first day of Rosh Hashanah a man cannot be Yotzei the blowing of Shofar with the blowing of another man unless that man is verified to have developed the signs of puberty [i.e. above age 18, or has grown a nice amount of facial hair, or one knows that he has grown two pubic hairs]. [If he already did so, he must rehear the Shofar blowing, without a blessing.]
The law on the second day of Rosh Hashanah: Men are Rabbinically obligated to hear Shofar on the second day of Rosh Hashanah. Accordingly, on the second day of Rosh Hashanah a man may be Yotzei the blowing of Shofar with the blowing of a boy above Bar Mitzvah even if the boy is under age 18 and has not begun to grow facial hair. Nonetheless, initially, it is proper to be Yotzei the Shofar of even the second day through a man that is above age 18 or has grown facial hair.
Blowing for women: Although some Poskim rule that a child may not blow Shofar for a woman even if he has reached the age of Chinuch, nevertheless, seemingly, if he is above the age of Bar Mitzvah, he may do so even on the first day of Rosh Hashanah, even if he is under 18 and has not grown facial hair. Practically, this should only be done if a man above 18 or with facial hair is unable to blow for her.
On the first day of Rosh Hashanah, a man cannot be Yotzei the blowing of Shofar with the blowing of another man unless that man is above age 18 or has grown a nice amount of facial hair. On the second day of Rosh Hashanah, a man may be Yotzei the blowing of Shofar with the blowing of a boy above Bar Mitzvah even if the boy is under age 18 and has not begun to grow facial hair. Nonetheless, initially, it is proper to be Yotzei the Shofar of even the second day through a man who is above age 18 or has grown facial hair.
Directives for appointing a Baal Tokeia:
Based on the above, a community or Shul may not appoint as a Baal Tokeia a boy who is below age 18 and has yet to grow a substantial amount of facial hair. Preferably, this applies for even the second day of Rosh Hashanah.
Directives for Mivtzaim:
When performing Mivtza Shofar on Rosh Hashanah, precedence should always be given to a Bochur above age 18 or with facial hair, to perform the blowing. This applies on both days of Rosh Hashanah, but especially on the first day when blowing for men. Thus, it is proper for those below age 18 or without facial hair to partner up with one above 18 or with facial hair, who can blow the Shofar. If this is not possible, then so long as one is over Bar Mitzvah he may blow Shofar on the second day of Rosh Hashanah, and blow Shofar for woman on the first day of Rosh Hashanah. Furthermore, if the boy in truth knows that he has developed the growth of two pubic hairs, he may blow Shofar for men on even the first day of Rosh Hashanah, if there is no other choice available and they would not hear Shofar otherwise. He should not be discouraged from doing so, and meriting other Jews with this great Mitzvah.
 Sdei Chemed Mareches Rosh Hashanah in name of Beis Menucha, brought in gloss on Alef Hamagen 589:7
 Meaning, that if one is ideally Biblically obligated, then even if he already fulfilled his obligation, or is not ready to fulfill it yet, he may still be Motzi another. Thus, a man may blow Shofar for another man even if he was already Yotzei, or plans to do so later on. [Admur 591:1; Michaber 591:1; M”B 594:1; Kaf Hachaim 589:1]
 Admur 588:6 regarding Shofar; 37:3; 39:1 regarding Tefillin; 55:6; 128:49; 199:9 regarding Birchas Hamazon; 271:7; M”A 271:2; M”B 271:3; Rav Poalim 1:10; Kaf Hachaim 271:9
The reason: Since Hashem does not desire this person himself to perform the command, how can he fulfill the command on behalf of others. [Levush 589; Kaf Hachaim 589:2]
 The Kuntrus Hashulchan states that in truth this word should read “blowing Shofar”.
 Parentheses is in original.
 Admur 588:6; Michaber 589:1
 See regarding if both are Chad Derabanon and if the child is potentially Chad Derabanon: Admur 186:3; Degul Merivava 271; Rav Akiva Eiger Teshuvah 7; Derech Hachaim ibid; M”B 271:2; Chayeh Adam 5:23, brought in Shaar Hatziyon 271:2; Shaar Hatziyon 271:2 and 4; Kaf Hachaim 271:9
 All Poskim ibid; See Michaber E.H. 155:15 regarding Miun and 169:10 regarding Chalitza
 Facial hair: M”A 271:2; Admur 199:9; Michaber E.H. 169:9 regarding Chalitza; C.M. 35; M”B 271:3; Shaar Hatziyon 271:5
Age 18: Admur 39:1; M”A 39:1; See Miasef Lekol Hamachanos 39:7; Chikrei Halachos 5:62; Yagdil Torah N.Y. 8:44
 See Admur 55:6 regarding Minyan [Rabbinical]; 199:9 regarding Zimun [Rabbinical]; 271:7 regarding Kiddush [Biblcial]; 37:3 and 39:1 regarding Tefillin [Biblical]; Rama 55:5; Maharik 49
The reason: As by a Biblical command, one of questionable obligation cannot be Motzi one of certain Biblical obligation. [Admur 271:7] However, by a Rabbinical obligation, we rely on the Chazaka of Rava [Nida 46a] that a child above Bar/Bas Mitzvah has grown two pubic hairs, even if this has yet to be verified. [Admur 55:6 and Poskim ibid]
 Admur 585:6; 588:6; 592:3-4, 7; 595:1-2
 Mateh Efraim 589:7; Shoel Umeishiv 3:177; M”B 589:2; Sdei Chemed Mareches Rosh Hashanah 2:5; Nitei Gavriel 48:16
 A blessing should not be repeated as perhaps the boy has indeed reached the age of development unbeknownst to him. Vetzaruch Iyun.
 Admur 585:6
 See Poskim ibid in introduction; Shoel Umeishiv ibid regarding Tekios Meumad; Mateh Efraim 589:7 regarding if not possible to get another adult
 Mateh Efraim ibid; Poskim in Nitei Gavriel ibid
 Tiferes Yisrael R”H 3:8
 The reason: As women are not Biblically or even Rabbinically obligated to hear Shofar, and do so simply due to the custom. Vetzaruch Iyun, as perhaps the custom to accept the Mitzvah on the first day is to fulfill it under Biblical standards, and a Neder has the status of a Biblical command.
 The reason: As seemingly, the entire law of verifying the growth of two hairs by a Biblical command is for the person to initially be Yotzei, if however, in truth the boy has two hairs, then even if this was unbeknownst to the person, he is Yotzei Bedieved, in the eyes of Heaven, as in truth the boy is an adult! Now, although it’s possible to learn that until testimony is given regarding his development, he is considered a child even in the eyes of heaven regarding another person, since it is customary for all Jewry to immerse in a Mikveh on Erev Rosh Hashanah, certainly testimony exists, and hence the man is Yotzei with him Bedieved. Accordingly, in Mivtzaim when dealing with an unobservant crowd who would not hear Shofar otherwise, one below age 18 who does not have facial hair should not be discouraged from blowing for other Jews, if they would not hear Shofar otherwise!