I wrote the following as an entry for an encyclopedia on ancient mathematics.

Tractate Kinnim. Kinnim is one of the tractates of the Mishnah, a late 2nd century compilation of oral rabbinic traditions. The Mishnah in general comments on mathematical and scientific matters very infrequently and only in the context of legal questions. There is no indication of use of symbolic approaches to mathematical problems. In Tractate Kinnim, however, an explication of laws of intermingled bird sacrifices apparently serves as a vehicle for developing a series of combinatorial theorems, albeit in non-symbolic language.

The background is as follows: bird sacrifices come in two varieties, which, following the Mishnah, we may call UP and DOWN. A set of birds might consist of only those that have been designated as UP birds (an “UP set”), as DOWN birds (a “DOWN set”) or of an even number of birds any half of which might be designated as UP birds and the rest as DOWN birds (a standard set). The birds are assumed to all be identical and the Mishnah deals with cases in which a number of sets have intermingled completely (Chapter 1) or partially (Chapter 2). The prior designations of the intermingled birds having been lost, the birds may now be redesignated as UP or DOWN subject to two constraints: 1. No bird originally from an UP (DOWN) set may be redesignated as DOWN (UP); 2. No more than half the birds originally from a standard set may be designated as UP or as DOWN.

The problem of determining the maximal number of birds that can be redesignated without violating either constraint is an entirely combinatorial one. A variety of problems are presented, using numerical examples, and solutions are given in as general terms as non-symbolic language permits. In the first two chapters, the solutions are mostly fairly trivial (with the notable exception of 2:3). For example, in any case of intermingling including an UP set, no bird may be designated as DOWN, and vice versa.

The third, and final, chapter deals with cases in which sets are intermingled but the rules of Chapters 1 and 2 were not followed and half the birds were redesignated as UP and half as DOWN in random fashion. The question is what is the size of the maximal subset of redesignated birds that certainly does not violate either constraint. The most interesting case is where a number of standard sets are fully intermingled. The language of the Mishnah (3:2) is as follows:

One [pair of birds] for this, two for this, three for this, ten for this, and a hundred for this,…the maximum is allowed. This is the rule: wherever you can divide the sets so that no woman has both UP and DOWN, half are allowed and half are not. Wherever you can’t divide the sets so that no woman has both UP and DOWN, the majority is allowed.

(The Mishnah attributes the sets to women because bird sacrifices were commonly brought by women after childbirth [Leviticus 12:8].)

The term “majority” is unclear here, as the Mishnah acknowledges by adding a rather opaque rule to explicate it. Consideration of the correct solution suggests that, though it is non-trivial, it is in fact the solution intended by the Mishnah. The correct solution is as follows: if the number of birds in each standard set are 2x₁,…,2x_n, respectively (in the Mishnah’s example the x_i are 1, 2, 3, 10, 100), then we attempt to partition the numbers {x₁,…, x_n} into two subcollections A and B such that | |A|-|B| | is minimal (i.e., the standard Set Partition problem). If there is a perfect partition (that is, |A|=|B|), then the solution is that the maximal subset consists of |A| (= |B|) UP birds and |A| DOWN birds, which is exactly half the total number of birds. If there is no perfect partition, the solution is that the number of UP birds (and DOWN birds) in the maximal subset is max(|A|,|B|) where |A| and |B| constitute the optimal partition.

The case described by the Mishnah in which “you can divide the sets so that no woman has both UP and DOWN” corresponds exactly to the cases in which there is a perfect partition and the solution offered by the Mishnah ("half") is correct for that case. Moreover, the issue of partitioning having been introduced, the Mishnah's solution for other cases ("the maximum") could plausibly be referring to the correct solution, namely, the maximum of the two partitions.

A number of other cases considered in Chapter 3 of Tractate Kinnim are equally non-trivial and stretch the limits of non-symbolic language for conveying mathematical content. Whether their content suggests that symbolic methods must have been used to arrive at the conclusions is an open question.