I started Scot McKnight's A Light Among the Gentiles: Jewish Missionary Activity in the Second Temple Period. The topic of this post will be a debate that McKnight discusses on page 81. The sources that McKnight cites are Mishnah Eduyoth 5:2, Mishnah Pesachim 8:8, Tosefta Pisha 7:14, and Babylonian Talmud 92a.
The subject is this: Suppose that a Gentile converted to Judaism on Passover eve. Can he partake of the Passover? The House of Shammai, in a rare display of leniency, says yes, even if that Gentile has not been circumcised. All he has to do is immerse himself and he can eat his Passover offering in the evening. The House of Hillel, however, says that the proselyte must be circumcised to eat of the Passover. And, if he became a proselyte on Passover eve, then he won't be able to partake of it, for a person who is circumcised becomes clean after being sprinkled on the third and the seventh day after the circumcision. That means that the proselyte would become clean (and thus eligible to eat the Passover) a week after the Passover.
There are questions that I have about this. First, how would the House of Shammai address the circumcision requirement for eating the Passover in Exodus 12:44, 48? Second, when does a Gentile become an official proselyte? Apparently, it is not at circumcision, for the Gentile in the discussion appears to be a proselyte before he has even been circumcised, or even immersed. Third, why must the proselyte be sprinkled on the third and the seventh days after circumcision in order to become clean? Numbers 19 prescribes that sort of sprinkling for those who have touched a corpse. Is circumcision a sort of contact with death? And, fourth, would a proselyte who cannot take the Passover be able to eat it the next month, which is what Numbers 9 prescribes for the Israelite and the resident alien who cannot partake of the Passover on Nisan 14 due to corpse impurity or being away on a journey?