IF THE KOHEN DID NOT ASK
(Mishnah 1): These laws (of the first Perek) are if the Kohen asked what to do. If he did not ask, if each of Rachel and Leah gave one Chovah, or each gave two, or three (or any other number):
If he offered all of them above (like Olas ha'Of), half of them are valid (the Olos);
If he offered all below (like Chatas ha'Of), half are valid;
If he offered half of them above and half below, (at least) half of the birds offered above are valid, and (up to) half are Pesulim. (A bird of Leah is Pasul only if all her Olos, half the number of her birds, were already offered properly. Therefore, at least half are valid. If all of her birds are offered above, exactly half will be Pesulim.)
The same applies to those offered below (at least half are valid Chata'os, up to half are Pesulim).
(Mishnah #2): If one woman gave one Ken, another gave two, another gave three, another gave ten, and another gave 100, and all were offered above, half of them are valid;
If all were offered below, half are valid;
If he offered half (116 birds) above and half below --
Of those offered above, (at least) the number of the most Kinim given by one woman (100) are valid Olos. (If at least 100 of the 116 birds were hers, 100 are valid Olos. If at most 100 of the 116 were hers, all of hers are valid. The only Pesulim are from other women. At most half the total number of their birds offered were Pasul, i.e. 16, so at least 100 are valid.)
Likewise, at least this number (100) offered below are valid Chata'os.
The general rule (when half were offered above and half below) is --
If it is possible that no woman had birds offered both above and below, (at least) half (of those offered above) are valid (and similarly for those offered below);
If (not, rather,) we must say that some woman had birds offered both above and below, the number of valid Chata'os is (at least) the greatest number of Chovos of any woman (and similarly regarding Olos. If over half the birds belong to one woman, we learn this from the reasoning above (c:1). There is nothing special about the numbers 100 and 116. The same applies whenever the first number is at least half of the second. Sometimes, even though no woman owns more than half, some woman must have birds offered both above and below, i.e. it is impossible that for some of the women, all their birds were offered above, and for the rest of the women, all their birds were offered below. This is true when there is no group of women who collectively own exactly half the total number of Kinim. (E.g. if the total number of Kinim is odd, or if the numbers of Kinim are 4, 4, and 4, or they are 2,3,4, and 8, no group of women own exactly half.) In such a case, the Mishnah is Machshir the greatest number of any woman. This is true, but it is an understatement. Really, more than half are valid. If we consider the smallest number of Kinim owned by a group of women that is at least half the total number (if they own 4,4,4, this is 8; if they own 2,3,4,8, this is 9), precisely this number of Chata'os (the same applies to Olos) are guaranteed to be valid, just like if all of these birds belonged to one woman.)
(Mishnah 3): If Rachel gave a Chatas to the Kohen and Leah gave an Olah:
If he offered both of them above, half (i.e., one, the Olah) is valid. (Presumably, it is called "half" for parallel structure with the other Mishnayos);
If he offered both of them below, half (the Chatas) is valid;
If he offered one above and one below, both are (Safek) Pesulim. Perhaps the Olah was offered below and the Chatas above.
(Mishnah 4) (according to Rashi Zevachim 67b): If Rachel and Leah (together) bought a Chatas for Rachel, an Olah for Leah, a Ken Stumah and a Ken Mefureshes (each woman also had to bring a Ken Chovah; they specified the Chatas and Olah of the Mefureshes, but not whose they will be).
If all the birds were offered above, or all below, half are valid;
If half were offered above and half below, all are Pesulim except for the Stumah (perhaps every specified bird was offered incorrectly), they "share" the valid birds. (One bird counts for each woman. They jointly bring another Ken and stipulate that the Chatas/Olah is for the woman who fulfilled her Olah/Chatas through the Stumah. Also, Rachel brings her Chatas and Leah brings her Olah.)
(Mishnah 5): If Chata'os became mixed with Chovos, we can offer Chata'os like the number of Chovos (but no more, lest all be from the Chovos).
If there were twice as many Chovos birds as Chata'os, (e.g. two Kinim Stumos (four birds) and two Chata'os, and the Kohen offered half of them above and half below), half are valid. (In the worst case, all the "fixed" Chata'os were offered above and are Pesulim. In our example, one bird offered above is a valid Olah, and two birds offered below are valid Chata'os. the third is Pasul, for there are only two Chata'os in the Chovos. In general, whenever there are twice as many Chovos birds as Chata'os, (in the worst case) all the fixed Chata'os (a third of all the birds) and half the Olos in the Chovos (a sixth of all the birds) are Pesulim. In all, this is half the total number of birds.)
If there were twice as many Chata'os as Chovos birds (e.g. one Ken Stumah and four Chata'os, and the Kohen offered half above and half below), the number of Chovos are valid. (In the worst case, all the birds offered above were fixed Chata'os and are Pesulim. The only valid birds are the remaining fixed Chata'os offered below, and the Chata'os in the Chovos. In our example, this is one fixed Chatas and one Chatas Chovah, two in all. In general, whenever there are twice as many Chata'os as Chovos (in the worst case), the only valid ones are the fixed Chata'os offered below, and the Chata'os in the Chovos. In all, this equals the total number of birds in the Chovos.)
A NEDAVAH AND A CHOVAH
(Mishnah 6) (according to Ra'avad): If a woman said "I will bring a Ken when I have a son," when she has a son (if she is poor) she brings two Kinim, one for her vow and one for the birth;
She gives four birds to the Kohen. He must offer three above (Olos) and one below (Chatas).
If the Kohen did not ask (he assumed that both were Chovos) and offered two above and two below, (one Olah was Pasul, so) she must bring another bird above.
This is if all the birds were the same (all Torim or all Bnei Yonah). If she brought two of each (and we do not know which two were offered below), she must bring (above) another Tor (perhaps both below were Torim. Since the Chatas was a Tor, she must bring a Tor for the Olas Chovah - Mishnah 2:5), and a Ben Yonah (perhaps both below were Bnei Yonah).
If (when she vowed) she picked two birds for her vow (and remembers what species they were) --
If she brought four birds of one species, she must bring three more birds of that species (two above and one below. Perhaps both Chovah birds were offered above, and the Nedavos below, and only the Olas Chovah was valid);
If she brought two Torim and two Bnei Yonah, she brings four more birds, i.e. (if the Torim were Nedavos, perhaps both were offered below, so she must bring another) two Torim for Olos, one Ben Yonah above and one below. (If both were offered above, the Chatas was invalid; if both were offered below, the Olah was invalid. Both birds of the Chovah must be brought from the same species. The Ra'avad is unsure whether or not l'Chatchilah she must bring Olos like she originally selected, or if the Mishnah discusses one who wants to do so.)
If she vowed to bring a particular species (but does not remember which), and told the Kohen which birds are for her vow:
If she brought four birds of one species (let us say Torim), she must bring five new (i.e. additional) birds. (Perhaps both her Nedavos were offered below, so she must bring two more above of each species in order to fulfill her vow. She must bring also a Tor Chatas, lest the Chovah was brought above and the Chatas was Pasul. She need not bring an Olas Chovah. If the Chovah was brought above, she already fulfilled Olas Chovah. If both Nedavos were brought above, the two new Torim Olos are not needed for her vow, for even if she vowed to bring Torim, she already fulfilled it. She stipulates that if she was already Yotzei her vow, one of the new Torim above is for her Chovah.)
If she brought two of each species, she must bring six new birds. (She brings four for her vow, like above, and a Chatas of each species. Perhaps the Olas Chovah was valid, so she must bring a matching Chatas. She need not bring extra Olos for the sake of her Chovah (to match the Chatas). If the Chatas was valid, this means that the Chovah was brought below, so her vow was brought above and was valid, so all four new Olos are Nedavah. She stipulates that that if this is the case, one of them is for her Chovah.) (From here, we explain like Ba'al ha'Me'or (Razah),)
If she forgot also which birds she offered, she must bring four Olos Nedavah (two of each species, to fulfill her vow), two Olos Chovah (one of each species. Perhaps her Chatas was valid, so she must bring a matching Olah), and a Chatas (of either species. This Tana holds that if Olas Chovah was brought before the Chatas; the Chatas need not be from that species, as long as she brought or brings an Olas Chovah matching the Chatas);
Ben Azai says (in addition to the six Olos mentioned above) she must bring two Chata'os. (He holds that whichever Chovah is offered first, even Olah, determines the other. Perhaps the first Chovah was Tor Olah; or perhaps Ben Yonah Olah. Therefore, she brings a Chatas of each species.)