Minyan--why oh why do we need ten? Can we count a child? How about a Torah scroll?

Similarly, Rav Adda bar Ahava stated, in accordance with the second opinion: From where is it derived that an individual may not recite kedusha alone? As it is stated: “And I shall be hallowed among the children of Israel” (Leviticus 22:32), any expression of sanctity may not be recited in a quorum of fewer than ten men. The Gemara asks: How is this inferred from that verse? The Gemara responds: This must be understood in light of a baraita, which was taught by Rabbenai, the brother of Rabbi Ḥiyya bar Abba: It is inferred by means of a verbal analogy [gezera shava] between the words among, among. Here it is written: “And I shall be hallowed among the children of Israel,” and there, regarding Korah’s congregation, it is written “Separate yourselves from among this congregation” (Numbers 16:21). Just as there among connotes ten, so too here, among connotes ten. The connotation of ten associated with the word among written in the portion of Korah is, in turn, derived by means of another verbal analogy between the word congregation written there and the word congregation written in reference to the ten spies who slandered Eretz Yisrael: “How long shall I bear with this evil congregation?” (Numbers 14:27). Consequently, among the congregation there must be at least ten.

(1) Communal prayer is always heard. Even when there are transgressors among [the congregation], the Holy One, blessed be He, does not reject the prayers of the many. Therefore, a person should include himself in the community and should not pray alone whenever he is able to pray with the community...

(4) What is implied by [the term,] communal prayer? One [person] prays aloud and all [the others] listen. This should not be done with fewer than ten adult free males. The leader of the congregation is [counted as] one of them. Even if some of them have already prayed and fulfilled their obligation, they can complete the [quorum of] ten provided the majority of the ten have not prayed. Similarly, we should not recite Kedushah, read the Torah with its blessings before and after it, or read the haftorah from the Prophets except in [a quorum of] ten.

(5) Similarly, one [person] should not recite the blessings associated with the Shema while the others listen and answer "Amen" except [in the presence of a quorum of] ten. This is called poreis al Shema. One only recites Kaddish with ten. The priests do not bless the people except [in the presence of a quorum of] ten. The priests [themselves] may be considered part of the quorum. [Ten are required] because every [group of] ten Jews is called a congregation as [implied by Numbers 14:27]: "How long [must I suffer] this evil congregation." They were ten, for Joshua and Calev were not included [among them].

(7) All [ten members of a congregation] and the leader of the congregation must be in one place. ...

(4) There are those who permit saying a spoken part [of the service] in which [God's] holiness [is proclaimed] (ie. Kaddish, Kedusha, etc.) with 9 men joined by a child that is older than 6 and knows to whom we pray. And their "words were not seen" (ie. not accepted) by the great halachic decisors. And the same law applies: that a slave or a woman is not joined [to the 9]. Rem"a: And even if he has a Chumash [printed Torah] in his hand, we do not join him [to the 9]. However, there are those who are lenient [regarding this] in an extenuating circumstance. [The Rosh and Mordechai and Hagahot Maimoni Chapter 9 of the Laws of Prayer]

Rav Huna said: Nine plus an ark in which the Torah scrolls are stored join to form a quorum of ten. Rav Naḥman said to him: Is an ark a man, that it may be counted in the quorum of ten?!

It has been stated: One may take a minor and a Torah scroll as appendix. Rebbi Judan said: So is the baraita: One may take a minor as appendix for the purpose of Torah reading. When does one start to make him an appendix? Rebbi Avina said: Rab Ḥuna and Rav Yehudah disagreed, both in the name of Samuel. One said, that he has to understand the nature of a benediction; the other one said that he has to understand to Whom one has to recite the benediction. Rebbi Nassa said, many times did I eat with my father Taḥlipha and my uncle Aninia bar Sisai and they did not recite the ‘invitation’ with me until I grew two pubic hairs.

(6) The intercalation takes place in the presence of three; Rabbi Eliezer says that ten (men are required), as it is said, "God stands in the congregation of God" (Ps. 82:1), and if they become less than ten, since they are diminished they place a scroll of the Torah before them, and they are seated in a circle in the court-room, and the greatest (among them) sits first, and the least sits last; and they direct their gaze downwards to the earth and (then) they stand and spread out their hands before their Father who is in heaven, and the chief of the assembly proclaims the name (of God), and they hear a Bat Ḳol (saying) the following words, "And the Lord spake unto Moses and Aaron… saying, This month shall be unto you" (Ex. 12:1, 2).