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In certain cases, the Torah prescribes the bringing of a pair of birds as an offering, one as a chatat and the other as an olah. See Lev 12:8, Lev 14:22, Lev 15:14, Lev 15:29, and Num 6:10.

Kinnim 3:1 (3:2 in some versions) describes the case of several women who need to bring various numbers of birds. All the birds were mixed up, and half were offered as chataot and half as olot. The mishna describes how many offerings were valid (translation edited from Sefaria):

אחת לזו, ושתיים לזו, ושלוש לזו, ועשר לזו, ומאה לזו. עשה... חציין למעלן, וחציין למטן: המרובה כשר. זה הכלל: כל מקום שאתה יכול לחלוק את הקינים, ולא יהו משל אישה אחת בין למעלן, בין למטן — מחצה כשר, ומחצה פסול; ובכל מקום שאין אתה יכול לחלוק את הקינים, עד שיהו משל אישה אחת בין למעלן, בין למטן — המרובה כשר.

One [pair] from this woman and two [pairs] from this woman and three [pairs] from this woman and ten [pairs] from this one woman and one hundred [pairs] from this woman: ... If he [the Kohen] did half above and half below, the number of birds in the largest group [one hundred] are valid. This is the rule: Anytime you can divide the pairs so that one woman's birds will not be both above [the mid-line] and below [the mid-line], half will be valid and half will be invalid. Anytime you cannot divide the pairs other than such that one woman's [birds] will both above and below [the mid-line], the number of birds in the largest group are valid.

The rule explains the explicit example well: if five women have 1, 2, 3, 10, and 100 pairs, then it is impossible to arrange the 232 birds such that each women's birds are entirely chataot or entirely olot. Thus, the example agrees with the rule that the number of pairs in the largest group (ie. 100) is valid.

However, this rule only works if the largest number is at least the sum of all the smaller numbers. But if four women bring 1, 1, 1, and 3 pairs, then there are always 4 valid chataot and 4 valid olot (and 4 is neither half of the total number of pairs nor the number of pairs in the largest group). This occurs anytime the largest group has fewer birds than the remaining birds.

What is going on here? How have commentators addressed this issue?

  • Potential answers in Raavad (it's only discussing two women at a time) and R. Zerachia haLevi (yes, OP is correct, and זה הכלל only comes to include the case where it works) (and I might write these up at some point). – magicker72 Apr 19 at 1:55
  • 1. The Q seems to interest only a 0.0% of the audience. 2. Yerushalmi says "אין כלליו של רבי כללים" - we should not call it "rules". – Al Berko Apr 19 at 10:28

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