Kinim, Chapter Two, Mishnah Three
This mishnah continues to illustrate the principle that a bird that leaves a pair disqualifies by flying away and a bird that returns also disqualifies another pair, lest it is not the same bird that flew away. This mishnah uses large numbers, but I think that the principle should be straightforward.
1) If one [woman] had one pair, another two, another three, another four, another five, another six and another seven pairs, and one bird flew from the first to the second pair, [and then a bird flew from there] to the third, [and then a bird flew from there] to the fourth, [and from there a bird flew] to the fifth [and from there a bird flew] to the sixth, [and from there a bird flew] to the seventh, and then a bird returns [in the same order as they flew away] it disqualifies at each flight and at each return.
a) The first and second [women] have none left, the third has one pair, the fourth two, the fifth three, the sixth four, and the seventh six pairs.
2) If again [one from each group] flew away and returned [in the same order as above], it disqualifies at each flight and return.
a) The third and fourth woman have none left, the fifth has one pair, the sixth two pairs, and the seventh woman five pairs.
3) If again one [from each group] flew away and returned [in the same order as above], it disqualifies at each flight and return.
a) The fifth and sixth women have none left, and the seventh has four pairs.
b) But some say that the seventh woman has lost nothing.
4) If [a bird] from those that are left to die escaped to any of all the groups, then all must be left to die.
Section one: This section sets out the scenario, sort of a musical chairs of sacrificial birds. Basically one bird has left each group and then one bird has returned to each group.
The first woman lost her only pair when one left. The second woman lost one pair when one left and one pair when one returned from her pairs to the first woman, so she is left with nothing. The third woman has one left, the fourth woman two and the fifth woman is left with three. The seventh woman loses only one pair because for her the “flying away” and “returning” are the same. She had one bird fly away and return to a set of pairs from which it had come. So she is left with six pairs.
Section two: Remarkably, the same thing repeats itself. One bird leaves the first, flies to the second, and so on up and then down the line. Again, two pairs are disqualified from the valid pairs that each woman has left, except for the last woman, who only loses one pair. Now the women who originally had three or four pairs are left with nothing, the woman with five is left with one, the woman with six is left with two, and the woman who had seven is left with five.
Section three: Incredibly, the same thing happens again (these birds really know what to do!). The only woman left with any valid birds is the last woman. According to one opinion, she again loses one pair. According to the other opinion, since all of the other pairs are already disqualified, the last bird to fly away from her does not disqualify any of her pairs. Therefore, she is left with five, as she was in the beginning.
Section four: If any birds that have to be left to die fly into any of the pairs, all of the pairs into which they may have flown are disqualified, because any of the birds may be this disqualified bird. In other words, here we do not have the problem of determining which bird is a hatat and which is an olah. Rather the problem is that any of the birds may be a bird that had been left to die and such a bird can never be sacrificed.