(Mishnah): Cases involving lashes require three judges.

(From R. Yishmael): They require 23.

(R. Meir): Extending a month (to become a 30 day month) requires three judges; extending a year (to make a leap year) also requires three.

(Rabban Shimon ben Gamliel): They begin with three, they discuss it with five and they complete it with seven. If they completed with three, it is valid.

(R. Shimon): Giving Sages semichah and breaking the calf's neck (if there was a murdered corspe found between two cities etc.) require three.

(R. Yehudah): They require five judges.

Chalitzah and Miyun require three. (Redemption of) Neta revai (the fruit that grows in the fourth year of the life of a tree) and of maaser sheni (the second tithe) whose values are not known, require three. Redeeming hekdesh requires three and Arachin Hamitaltalin (evaluating movable objects to be given as payment for a person pledging a specific amount to the Beis Hamikdash) requires three.

(R. Yehudah): One of them must be a Kohen.

And regarding fields, there must be nine and a Kohen which is also the case when evaluating the monetary value of a person.

Capital cases require 23; cases of bestiality require 23 as the pasuk states (Vayikra 20:16), "And you shall kill the woman and the animal" and the (previous) pasuk states, "(And a man who will lie with an animal, he shall surely be put to death) and you shall kill the animal."

The ox that is stoned requires 23 as the pasuk says (Shemos 21:29), "the bull shall be stoned, and also its owner shall be put to death (meaning pay atonement money)" - we learn from this that the death of the owner (if he would be liable the death penalty in another situation) is like the death of the ox - both require 23.

A lion, a wolf, a bear a leopard and a bardelus (some translate it as a cheetah) are sentenced to death with 23 judges.

(R. Eliezer): Whoever kills them first has done a good deed (and there is no need to wait for a ruling of the court).

(R. Akiva): Their death requires 23.

(R. Abahu): (Explaining R. Yishmael's view in the Mishnah) Lashes require 23 as he could sometimes die from his wounds, meaning that the original ruling to give him lashes was actually a capital case.

Bar Kafra said that all of the rules (of Rabban Shimon ben Gamliel of requiring 3, 5 and 7 for extending the month) are alluded to in the following pasuk about the Kohens' blessing (Bamidbar 6:24), "May Hash-m bless you and keep you" (which has three words in Hebrew) - from here we learn that they begin with three judges. "May Hash-m spread his countenance upon you and be gracious to you" (which has five words in Hebrew) - from here we learn that five judges discuss it. "May Hash-m turn His face towards you and give you peace" (which has seven words in Hebrew) - from here we learn that they complete it with seven judges.

R. Yehoshua ben Levi derived all of those rules from the following pasuk (Melachim 2:25:18), "And the chief executioner took...the three gatekeepers" - from here we learn that they begin with three judges. (Verse 19) "and five men from those that see the king's face" - from here we learn that they discuss it with five judges. (Yirmiyah 52:25), "seven men from those that see the king's face" - from here we learn that they complete it with seven judges.