Kinnim, Chapter 1, Mishnah 3. Masechet Kinnim opens by teaching how bird pairs behave when they become mixed together and can no longer be identified, and this mishnah continues that theme by drawing a sharp line between two very different kinds of mixture: a mixture of obligatory pairs with a voluntary olah, and a mixture of obligatory pairs with each other.
Recall the ruling of the previous mishnah. A designated olah fell into a group of obligatory pairs whose birds had not yet been assigned to chatat or olah. There we ruled that one may offer as olot only the number of olot already contained in the obligatory pairs themselves. In other words, we assume the worst case and offer only what is certainly permitted, so that no korban is ever brought in a way it should not be brought. If a woman had ten obligatory pairs and one designated olah became mixed in with them, ten birds may now be offered as olot, since every one of those ten is either the olah half of one of the obligatory pairs or the stray olah itself. Beyond that number there is no certainty at all about what is on the altar, so we go no further.
Where the earlier rule applies
The mishnah proceeds to fence in that earlier halachah. "Bameh devarim amurim," concerning which situation was this taught? "Bechovah unedavah," it was taught exclusively about obligatory pairs that merged with a bird donated as a voluntary olah. Only in that pairing does the quota of olot permitted on the altar match the olot already built into the obligatory pairs themselves.
Obligatory pairs mixed with obligatory pairs
"Aval bechovah shenit'arvah zo bezo," when obligatory pairs are what got confused with one another, an entirely different halachah takes over. The mishnah spells out the cases one by one: "achat lazo ve'achat lazo," this woman set aside a single pair and so did the other; "shtayim lazo ushtayim lazo," each of them set aside two pairs; "shalosh lazo," three pairs from the one, "veshalosh lazo," three pairs from the other. Notice what these cases share: the two quantities match. Both women brought an identical count of pairs, none of the birds having yet been designated, and then everything got mixed together.
Here the mishnah rules "mechetzah kasher umechetzah pasul," one half of the mixture may be brought and the other half may not. The permitted half is then divided evenly between the two categories, part serving as chataot and part as olot. Working it out in fractions: a quarter of the total count ends up as chatat, a quarter as olah, and the remaining fifty percent is simply lost to the altar.
Work through the final case on the list. Three pairs came from each woman, which gives us a pool of twelve birds, none of them designated, six traceable in principle to each woman. Absent any confusion, each woman's six would have yielded three chataot and three olot. But since no bird can now be traced to its owner, only three chataot may be brought from the twelve. Why stop at three rather than four? Because it is possible that all four of those birds originated with one and the same woman, whose obligation extends to three chataot only. Her fourth chatat would be a korban for which she carries no obligation whatsoever, and it would be disqualified. The same logic caps the olot at three, since a fourth olah might likewise emerge from one woman's six, and she owes no more than three. Between them, then, the two women bring three chataot and three olot, and the other six birds are unusable. Here too the mishnah routes us along the path of least risk, permitting only what is beyond doubt.
When the groups are unequal
Everything until now presumed two sources contributing matching quantities. The mishnah now takes up mismatched quantities: "achat lazo," a single pair from this woman, "ushtayim lazo," two pairs from a second, "veshalosh lazo," three pairs from a third; "ve'eser lazo," ten pairs from one woman, "ume'ah lazo," a hundred from another. Throughout all such combinations the ruling is "hamu'at kasher," we go by the lesser quantity, which may be brought and is split evenly, one portion as chatat and one as olah.
Picture one woman who set aside a single pair, two birds, alongside a woman who set aside two pairs, four birds, for a total pool of six. The lesser quantity governs, namely the lone pair belonging to the first woman. Accordingly two birds may go up, serving as one chatat and one olah. Even on the most pessimistic assumption, that both birds came out of the stock of the woman who contributed just one pair, each offering stands, because a single pair furnishes precisely one of each category. What may not be added is a further chatat or a further olah, for either might be drawn from that same woman's pair, and her pair cannot supply two of either kind. Once more the mishnah directs us to the arrangement that carries no halachic risk.
The scope of the rule
What the mishnah has taught us is how to handle sets of pairs that resist being treated as one combined pool, and it illustrated this with two women whose pairs got mixed together. That illustration, however, is not the whole of it. The same halachah operates in every situation where something prevents the confused birds from being counted as a single unified group.
"Bein mishem echad," whether the pairs were designated for a single category of obligation, "bein mishnei shemot," or for two distinct categories, the principle stands that each set must be assessed independently. Pairs earmarked for two distinct obligations obviously can never be folded into one pool. And even pairs earmarked for one identical obligation stay apart as separate sets if the women who designated them were not the same, or if the designations were made at separate times. "Bein me'ishah achat," whether a single woman set them aside, "bein mishnei nashim," or two women did. The mishnah that follows will unpack these particulars.