Kinnim, Chapter 3, Mishnah 5. Our perek is still occupied with the kohen who failed to bring his question to beit din. Confronted with a mixture of birds, he sought no guidance at all: he simply split the lot in two, performing one part up above the chut hasikra (the red stripe painted around the mizbeach) in the manner of olot, and the other part beneath that stripe in the manner of chataot. Beit din is left afterward to sit and tally how many of these offerings happened to be performed in the correct place.
To follow the cases in front of us, it pays to review the second half of Mishnah Bet in the first perek. There a bird that had already been designated as a chatat fell in among pairs that were owed as an obligation but had not yet been assigned (chovah), so that once mixed, no individual bird could be identified any longer. The ruling there was that the kohen may proceed, but subject to a ceiling: he brings as chataot no more than the tally of chataot that those chovah pairs owed in any event. This is the safest route available, since it is beyond doubt that that many chataot are due from the pairs, so bringing precisely that many beneath the line cannot go astray. Where it is an olah that falls into unassigned pairs, the reasoning runs identically, and he may bring no more olot than the pairs themselves owe.
Our mishnah takes the case one step further: the kohen never asked. He divided the whole mixture down the middle, half above and half below, and we are left to salvage whatever we can.
First case: a chatat mixed into an equal number of chovah birds
First case: a chatat mixed into a matching count of chovah birds
"Chatat shenit'arvah b'chovah": the designated chatat birds and the birds of the unassigned obligatory pairs came in matching counts and were mixed together, and the kohen, having asked nothing, performed one half up above the line in the manner of olot and the other half beneath it in the manner of chataot. The mishnah rules "ein kasher ela minyan chatot sheb'chovah": what stands is limited to the tally of chataot that the chovah pairs owed.
Take ten definite chatat birds mixed with ten undesignated birds, which are five pairs. The kohen offers ten above as olot and ten below as chataot. Out of all twenty birds, only five are valid, corresponding to the five chataot that were due from those five pairs. Those five, offered below, are certainly in their proper place. As for the other five that were offered below as chataot, we have no way of knowing whether they were truly chatat birds or birds that were meant to be brought as olot, so they cannot be counted as valid.
Second case: the chovah is double the chatat
"Chovah shnayim b'chatat": the unassigned obligatory pairs supply double the count of the designated chatat birds; everything was mixed together, and the kohen performed one half up above the line in the manner of olot and one half beneath it in the manner of chataot. The mishnah rules "mechetzah kasher umechetzah pasul": one half stands and one half does not.
Picture ten definite chatat birds mixed with ten undesignated pairs, which are twenty birds, for a total of thirty. The kohen offers fifteen above as olot and fifteen below as chataot. Fifteen are valid and fifteen are invalid, and here is how we arrive at that number. We must assume the worst about the ten definite chatat birds: perhaps all ten of them were among those placed above in the spot of an olah. They are therefore all removed from the reckoning as invalid. That leaves the twenty undesignated birds, of which five ended up above and fifteen ended up below. The five above are valid as olot, since five is less than the ten olot that these ten pairs owe. Of the fifteen below, ten are valid as chataot, matching the ten chataot owed by those pairs, while the remaining five below are invalid. Ten chataot plus five olot gives us fifteen valid birds.
Third case: the chatat is double the chovah
"V'chatat shnayim b'chovah": now the designated chatat birds come to double the count of the birds making up the unassigned obligatory pairs; everything was mixed, and the kohen performed one half up and one half down. The ruling is "haminyan sheb'chovah kasher": whatever remains valid equals the tally of the chovah.
Let the mixture consist of eight certain chatat birds together with four unassigned birds, a dozen all told. The kohen sends six up over the line and performs six beneath it. Exactly four come out valid, which corresponds to the tally of the chovah. Reckon the least favorable possibility: suppose all six sent up were drawn from the eight certain chataot, in which case those six are forfeited. The two certain chataot that remain must then be somewhere among the six performed below, and they are in place. The four unassigned birds also went below; two of them stand as the chataot those pairs owe, while the other two were destined to be olot and were performed below, so they fall away. Two and two yield four valid birds, matching the chovah.
The parallel cases with an olah
Having seen these three cases of a chatat mixed into undesignated pairs, it is easy to see that swapping chatat for olah yields the identical rule, and that is exactly what the mishnah proceeds to say.
"V'chein olah shenit'arvah b'chovah": birds set aside as olot were mixed with a matching count of birds belonging to obligatory pairs, and the kohen, asking nothing, sent one half up and performed one half down. The mishnah's ruling repeats the earlier formula with olot in place of chataot: nothing holds beyond the tally of olot that the chovah pairs owed. Ten olah birds together with ten unassigned birds make twenty. Reckoning the worst: all ten designated olot were performed beneath the line and are therefore lost, while the ten unassigned birds all went up above. Only half of those ten, that is five, ought to be olot, so five hold; the remaining five were owed as chataot and are lost. Five valid birds altogether.
"Chovah shnayim ba'olah": the unassigned obligatory pairs furnish double the count of the designated olah birds, and the kohen sent one half up and performed one half down. "Mechetzah kasher umechetzah pasul": half hold and half do not. Ten olah birds mixed with ten unassigned pairs, which are twenty birds, come to thirty. Fifteen go up as olot and fifteen are performed below as chataot. At worst, the ten designated olot all ended up beneath the line, out of place, and are lost. Of the twenty unassigned birds that remain, five landed below and fifteen above. The five below hold, since they can be reckoned as chataot standing where chataot belong. Among the fifteen above, ten hold as the olot those pairs owe, while the last five, which were owed as chataot, are lost. Fifteen valid, fifteen invalid.
"Olah shnayim b'chovah": the designated olah birds amount to double the count of the birds making up the unassigned obligatory pairs, and once again the kohen sent half up and performed half down. "Haminyan sheb'chovah kasher": what holds equals the tally of the chovah. Eight olah birds mixed with four unassigned birds make twelve, six sent up and six performed below. Reckon the worst: all six performed below were taken from the designated olot, so all six are lost. Two designated olah birds are left over, and they went up above, where an olah belongs, so they hold. The other four that went up are the unassigned birds: two of them serve as the olot the pairs owe and hold, while two were owed as chataot and, having been placed above, are lost. Four valid birds in all, precisely the tally of the obligatory pairs.