The recent issue of Hakirah published my article on the above Mishnah. It will be available online in a a few months after the next issue of Hakirah is published.

Here is my latest version, updated and simplified, at least slightly.

Kinnim (3:2) and the term merubah:

Kinnim (3:2) reads as follows:^[1]

אַחַת לָזוֹ, וּשְׁתַּיִם לָזוֹ, וְשָׁלשׁ לָזוֹ, וְעֶשֶׂר לָזוֹ, וּמֵאָה לָזוֹ, עָשָׂה כֻלָּן לְמַעְלָה, מֶחֱצָה כָשֵׁר וּמֶחֱצָה פָסוּל. כֻּלָּן לְמַטָּן, מֶחֱצָה כָשֵׁר וּמֶחֱצָה פָסוּל. חֶצְיָן לְמַעְלָן וְחֶצְיָן לְמַטָּן, הַמְרֻבֶּה כָשֵׁר. זֶה הַכְּלָל, כָּל מָקוֹם שֶׁאַתָּה יָכוֹל לַחֲלֹק אֶת הַקִּנִּין וְלֹא יְהוּ מִשֶּׁל אִשָּׁה אַחַת, בֵּין מִלְמַעְלָן בֵּין מִלְּמַטָּן, מֶחֱצָה כָשֵׁר וּמֶחֱצָה פָסוּל. כָּל מָקוֹם שֶׁאֵין אַתָּה יָכוֹל לַחֲלֹק אֶת הַקִּנִּין עַד שֶׁיְּהוּ מִשֶּׁל אִשָּׁה אַחַת, בֵּין מִלְמַעְלָן בֵּין מִלְּמַטָּן, הַמְרֻבֶּה כָשֵׁר

If one [pair] belonged to one woman and two [pairs] to another, three [pairs] to another, ten pairs to another and a hundred to another, and he offered all of them above, then half are valid and half are invalid. [Similarly] if he offered all of them below, half are valid, and half are invalid. [If he offered] half of them above and half below, then the [number of birds as there is in the] larger part are valid. This is the general principle: whenever you can divide the pairs [of birds] so that those belonging to one woman need not have part of them [offered] above and part [offered] below, then half of them are valid and half are invalid; but whenever you cannot divide the pairs [of birds] without some of those belonging to one woman being [offered] above and some below, then [the number as there is in] the larger part are valid.

When birds aresacrificed without consultation, we determine the worst possible scenario and then determine how many birds are nonetheless valid even under those conditions. When equal sized kinnim stumot are intermingled, the worst possible case invalidates half of each nest, because the worst possible outcome results when each ken is sacrificed entirely as either olot or ḥatta’ot. When unequal sized nests are intermingled, in all cases, however, more than half of the birds are sacrificed correctly. Mishnah (3:2), conceptually perhaps the hardest Mishnah in Kinnim, deals with a case where (significantly) more than half of the intermingled kinnim stumot are valid; in the case in the Mishnah, 200 of 232 birds are valid.  The difference between what happens with and without consultation can be illustrated in a remarkably simple case when two unspecified nests of unequal size are intermingled.  As demonstrated in the introduction, with consultation, only the number of birds in the smaller nest are permitted to be sacrificed. When sacrificed without consultation, half as hatta’ot and half as olot, the number of birds in the larger nest are valid.[2]

The second half of the Mishnah following זֶה הַכְּלָל distinguishes two situations. In the first situation, it is possible to arrange all the birds in all the kinnim as either olot or ḥatta’ot without any ken having birds sacrificed as both olot and ḥatta’ot.  The ability to arrange the nests exactly that way represents the worst possible outcome.[3] This example is representative of how disqualification is maximized. Kinnim are meant to be sacrificed half as ḥatta’ot and half as olot.  By sacrificing all as one type, disqualification is maximized.

When you cannot divide the nests without one nest[4] having to sacrifice some of its birds as olot and some as ḥatta’ot, more than half of the birds are valid.  The Mishnah uses the phrase הַמְרֻבֶּה כָשֵׁר, whose precise meaning will be defined going forward.[5]

An article by Dr. Phillip Reiss provides an elegant formal proof of the second part of Mishnah (3:2), which specifies that the number of valid sacrifices is the larger amount, הַמְרֻבֶּה כָשֵׁר.  Dr. Reiss defines this expression precisely, corresponding exactly to the halakhic rule with which the Mishnah operates.  That rule, as articulated by many classic commentators, is thatthe way to determine the minimum number of birds correctly sacrificed is to construct a scenario that maximizes the number of birds incorrectly sacrificed. A formula that represents this concept is: (The minimum number of birds correctly sacrificed) = (the total number of birds.) – (the maximum number of birds potentially incorrectly sacrificed).

When we cannot divide the kinnim into two equal sized groups, more than half of the birds will be valid. That larger group, referred to by the phrase הַמְרֻבֶּה כָשֵׁר however, must be the smallest possible larger group.   Dr. Reiss formalized what exactly is meant by “the smallest majority” and I will reframe Dr. Reiss’s approach to make the Mishnah more intuitive and the proof more concise.  It serves as a model for thinking about all the mishnayot of the third chapter, which are examples of this general case as well as also addressing a situation where kinnim mefurashot and kinnim stumot are intermingled.  What follows provides a somewhat less rigorous argument that may capture how the Mishnah may have been conceived by its authors and more traditional commentators.

To formalize the Mishnah and derive the minimum number of birds sacrificed correctly requires construction of a scenario that provably maximizes the number of birds incorrectly sacrificed. 

Without consultation, the Kohen sacrifices half of the combined nest as ḥatta’ot and half as olot. To develop our model and without loss of generality, assume the Kohen places each of the birds into one of two equal sized storage containers labeled “O” for olot and “Ḥ” for ḥatta’ot. Together the two containers precisely hold the total number of birds originally in all the intermingled nests. The birds in container O are subsequently sacrificed as olot while the birds in container Ḥ are sacrificed as ḥatta’ot. Since the two containers are of equal size, the Kohen assumes, incorrectly of course,[6] that he has fulfilled his obligation to sacrifice half the nest as ḥatta’ot and half as olot.

To fully explain and prove the Mishnah in its most general case requires that we look at the situation prior to nests being combined and examine the original set of nests.  There could have been any number of individual nests, but obviously the number of birds in each of the original nests, (all being ḥovot and therefore containing an equal number of ḥatta’ot and olot,) must be an even number.^[7]

Prior to combining the nests, think of each of the birds in each of the individual kinnim being stored individually in identical cages; the cages of each ken are then stacked vertically one on top of another and not yet mixed with any other ken. That vertical stack of cages from a single ken forms a single package. The height of each package is proportional to the (even) number of bird cages in the package; a package holding a ken of 4 birds is ½ of the height of package holding a ken of 8 birds.

The individual cages are identical, all nests/packages[8] have the same length and width.  The containers are constructed to allow the cages to only be stacked vertically in the O and Ḥ containers. Those containers can hold bird cages stacked vertically, but those cages must not exceed the height of the container, as will become clear later.

Return now to the two storage containers of equal size, (the ḥatta’ot and olot containers respectively,) into which each of the individual cages can be stacked. To match the case in the Mishnah, the two identically sized containers are assumed to have the exact capacity required to store all the bird cages in the N packages (i.e., the kinnim), where N denotes the original number of (now intermingled) kinnim.

Were the birds not intermingled, we can assume that each woman’s ken is divided in half, with each half temporarily stored in one of the respective containers, before being correctly sacrificed.  In the desired scenario, each ken is split equally across the two containers; the total number of bird cages fit precisely into the two equally sized containers.

To construct a worst-case scenario, we attempt to do exactly the opposite.  Instead of dividing each package into an equal number of ḥatta’ot and olot, we try to leave all the packages unsplit.  To the greatest extent possible, something that will be defined formally below, all the birds in each ken are placed in only one container, i.e., all to be sacrificed as either ḥatta’ot or olot.  In terms of the two containers, we will try to fill both containers without splitting any individual ken, effectively sacrificing all birds from each unsplit package identically, thereby disqualifying exactly half of the birds in each ken.  When attempting to do that, we may or may not be completely successful in dividing them evenly without splitting any given ken.  Consider a case of 2, 4 and 6 birds; the worse we can do is put the two smaller kinnim with 2 and 4 birds in one container and the larger ken of 6 birds in the other, a split of kinnim without dividing any single ken.  This case illustrates achieving maximal disqualification, where half (6) of the 12 birds are disqualified, but still allows for half (6) of the sacrificed birds in each nest to be valid. However, if we had 3 kinnim containing 2, 4, and 8 birds in each of the original kinnim, there is no way to divide the (2+4+8=) 14 birds into 2 groups of 7, without splitting a ken.  Note that each ken has an even number of birds and the number of birds in any number of unsplit nests will have an even number of birds as well. In the worst-case scenario, we can put the two smaller kinnim totaling 6 birdsin one container and split the larger ken of 8 birds, putting 7 birds (half the total number of birds) in the other container and 1 bird in the container together with the two smaller unsplit nests, each container thereby holding 7 bird cages. As a result, 2 of the 8 birds in the larger nest are sacrificed correctly, one as an olah and one as a ḥattat, while half of the remaining 12 birds in the two containers are also sacrificed correctly, for a total of 8, correctly sacrificed birds more than half of the 14 intermingled birds.

If we are trying to place unsplit packages into each container, it should be clear that it is always the case, that at most one package needs to be split across the two containers. That only one packet ever needs to be split is fundamental to understanding the proof of the Mishnah and is proven formally in the footnote below.^[9]

There are 2 ways to formalize the Mishnah using the paradigm of storing the maximum number of unsplit packages using either one or both containers. The first method, a 2-container storage solution, maximizes the total number of birds from unsplit packages that are stored entirely in either one of the two containers. However, optimizing the number of unsplit kinnim across both storage containers does not maximize the number of incorrectly sacrificed birds. Rather,^[10] the worst-case arises if we try to maximize the number of birds from unsplit packages that can be stored in only a single container (as opposed to both.)  Therefore, to maximize the number of invalidly sacrificed birds out of the total group of kinnim, we place the largest number of unsplit packages in only one of the two containers.  The other container will hold the remaining unsplit packages whose height is, by definition, less than or equal than the height of unsplit packages in the other container.  If both containers are not full, the one remaining package is split, and it will fill both containers. Of course, even when unlike the case in the Mishnah where the size of one ken is not greater than the sum of all the rest, the package that must be split may not be the smallest package.^[11]

What we will now proceed to prove is that maximizing the number of birds from unsplit packages in only one container creates maximal disqualification.  A case that illustrates why optimizing one container and not two container storage creates maximal disqualification, and a more intuitive discussion of this issue follows the proof.

If 2*K is the total number of birds in all the kinnim, then each container will have capacity for K birds.^[12] Let J be the largest number of birds that can be placed in one container without splitting a package. To simplify the presentation, we will assume that the container with J birds (cages) is the O container, as opposed to the Ḥ container.[13]  Of course, the Ḥ container must therefore contain either as many as or fewer than J birds from unsplit containers; it cannot contain more than J birds. Note, if a container can be totally filled without splitting up any package, then J = K.[14]  Whether or not a container can be filled with unsplit packages, the minimal number of correctly sacrificed birds will be proven to equal J + 2*(K-J).^{[15] [16]} Once we have determined J, the formula provides the (minimal) number of correctly sacrificed birds. The entire proof hinges on proving that determining J allows the identification of the minimal number of valid sacrifices, i.e., the “worst-case scenario.”[17]

Before providing an outline of the proof, some examples will help to facilitate understanding of the proof.

1. For 2, 4, 6, 8, and 10 birds (1, 2, 3, 4, and 5 kinnim): In this example, K = 15, J = 14, and a minimum of 16 sacrifices are valid, since J+2*(K-J) = 14 + 2*(15-14) = 14 + 2 = 16. The O container holds either the kinnim with 10 and 4 birds, 6 and 8 birds, or 2, 4 and 8 birds, each group totaling to 14 birds.
2. For 2, 4, 6, and 8 birds J = K =10: One container holds the groups of 4 and 6 birds and the other holds the groups of 2 and 8 birds. Since every group of kinnim can be stored unsplit, the number of correctly sacrificed bids is exactly half (this is the worst result that occurs only when both containers can be filled with unsplit nests.)
3. For the case in the Mishnah of 2, 4, 6, 20 and 200 birds, K = 116 and J = 32 and at least 200 birds were correctly sacrificed. In this example, the O container initially holds (2+4+6+20=) 32 birds in unsplit packages and the Ḥ container holds no birds in unsplit packages.  In this case where one of the kinnim contains more than half of the total number of birds, this large ken must contain K+ (K-J) birds.  K is the complete capacity of the Ḥ container, (116 birds in the Mishnah’s case,). K must be added to the remaining capacity of the O container, i.e., (K-J), (84 birds in the Mishnah’s case,). (K-J) represents O’s original capacity of 116 birds minus the space already occupied by the 32 birds from unsplit packages, already present in the O container. This the total size of this large nest (K + (K-J)) = J + 2*(K-J).
4. In Dr. Reiss’s example of 8, 12, and 14 birds, K= 17 and J = 14, and even in the worst-case scenario, at least (14 + 2*(17-14) =) 20 birds were correctly sacrificed. In a 2-container storage optimization, unsplit nests with a total of 14 birds go in one container, unsplit nests with a total of 12 birds go into the other container, while the remaining nest of 8 birds is split across the two containers – 3 in the container with 14 birds and 5 in the container with 12 birds. In the kinnim case of 1-container optimization however, once 14 birds are placed in the O container, how the remaining three slots in that container are filled (i.e., from the group of 8 or 12 or some combination) is irrelevant in determining the number of birds correctly sacrificed.[18]

An outline of a formal proof follows.

Remember that J is the largest number of birds that can be placed in the O container without splitting any package.^[19] Since those birds are all sacrificed identically as olot, exactly J/2 were correctly sacrificed, as the other J/2 birds in the O container should have been sacrificed as ḥatta’ot.  No remaining ken has fewer than (K-J) birds. Otherwise, it could have been added to the O container that contains J birds from unsplit packages, thereby increasing J by the size of that ken.  After the O container is maximally filled with unsplit nests and the remainder of the unsplit nests are placed into the Ḥ container, then (K-J) is less than or equal to half the number of birds contained in the single remaining nest that now must be split.[20] This is the key point in the proof.  The Ḥ container may also hold unsplit packages, but with less than or equal to J birds.  Hence, after filling the O container with the split nest, there must be room for at least (K-J) additional birds from the remaining split package in the H container.  Thus, for each of the remaining (K-J) birds which are the birds from the split ken placed in the O container, both it and some other member of its ken which were placed in the Ḥ container, were correctly sacrificed, one as an olah and one as a ḥattat. This adds 2*(K-J) correctly sacrificed birds to the J/2 correctly sacrificed birds from the unsplit nests in the O container.  Beyond the K-J birds in the H container that have a mate in the O container and are therefore correctly sacrificed, none of the remaining J birds in the Ḥ container, have a mate in the O container. Those J birds in the Ḥ container cannot be paired with any of the J birds from unsplit nests in the O container. This leaves J/2 additional valid sacrifices from, the Ḥ container, the J/2 ḥatta’ot that are validly sacrificed, (versus the J/2 that should have sacrificed as olot and are invalid.)   The J/2 correctly sacrificed birds in container H are added to the prior 2 correctly sacrificed groups of J/2 birds from container O and the 2*(K-J) correctly sacrificed birds from a single nest half in each of the O and Ḥ containers. The total of correctly sacrificed birds is now J/2 + 2*(K-J) + J/2, which equals J + 2*(K-J) or 2*K – J birds, Dr. Reiss equivalent expression for minimally valid sacrifices.[21]

It is critical to appreciate the difference between the improper two container optimization versus the one container optimization, in cases with no prior consultation.  Only one container optimization correctly identifies the smallest number of validly sacrificed birds, by disqualifying the maximum number of birds. In examples a) through d) above, the two container and one container optimizations happen to yield equivalent solutions, but this is not always the case. Consider the case of kinnim consisting of 8, 10, 10, 10, 14, and 14 birds, respectively, totaling 66 birds. Note that the O and the Ḥ container are each of size K = 33 and one package (i.e., ken) must be split in the 2-container optimization. It is easy to see that the package of size 8 can be split, resulting in one container holding the three packages of size 10 and the other container holding the other two packages, both of size 14. In total, 58 birds in unsplit packages are stored across both containers. If we set J = 30 corresponding to the largest number of birds from unsplit nests in either container, the formula J + 2(K-J) would incorrectly yield (30 + 2(33-30) = (30 +6 =) 36 valid sacrifices. However, optimizing the number of birds from unsplit packages that can be placed in only one container allows you to put 32 = (8 + 10 + 14) items into one container, and hence J = 32. Note that this would not yield an optimization of unsplit packages across both containers since only 24 items would go into the second container for a total of 56 unsplit birds achieved in the one container optimization, versus the 58 birds that can be stored in unsplit packages with a 2-container optimization. In the 1 container optimization, a nest of 10 birds must be split, 9 birds in the container with 24 birds and only 1 bird in the container with 32 birds.^[22]

Since constructing the above example in 2001, I have tried to find the smallest example that illustrates the difference between the one and two container optimizations.  Five kinnim of sizes 6, 6, 6, 10 and 10 birds, with 38 birds, is apparently the smallest example.  When maximally filling two containers, each with a capacity to store 19 birds, we would put kinnim with 10 and 6 birds into both containers, leaving one of the three smaller kinnim with 6 birds to be split across both containers. Ostensibly, (J+2(K-J) = (16 + 2(19-16) =) 22 birds are correctly sacrificed.  But that result does not comport to universally accepted interpretation of Mishnah that requires that the worst possible scenario be constructed to determine the smallest number of birds validly sacrificed.  Using a one container optimization, all 3 kinnim that contain 6 birds are placed in the O container unsplit, leaving one unsplit nest of 10 birds in the Ḥ container.  One of the 10 bird kinnim are then split with 1 bird placed in the O container and 9 in the Ḥ container.  This results in (J+2(K-J) = (18 + 2(19-18) =) 20 birds, correctly sacrificed, which are valid even in the worst-case scenario, comporting with the universally accepted interpretation of the Mishnah.^[23]

Conclusion:

The only approach that comports with the Mishnah’s conclusion about intermingled nests when a Kohen sacrifices without prior consultation as if was dealing with a single nest was demonstrated to be the one container optimization.  This approach renders more intuitive why the maximization of the number of invalid sacrifices (hence determining the smallest amount valid sacrifices) requires that one maximize the total size of those kinnim that can be placed unsplit in a container that can hold exactly half of the total number of birds in all the intermingled kinnim.  One can imagine that mathematical reasoning like that contained in the proof can have been used in the formulation of the Mishnah by tannaim 2000 years ago.

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[1] The text of the Mishnah and its translation is largely taken from www.Sefaria.com.

[2] This is not the ruling with a ken mefureshet, in which case, even when sacrificed without consultation no birds are valid, because in the worst-case, it is possible that every bird designated as an olah was sacrificed as a ḥattat and every bird designated as a ḥattat was sacrificed as an olah.

[3] For example, if 4 kinnim have 2, 4, 8 and 10 birds respectively.  Combined there are a total of 24 birds, of which 12 should be sacrificed as olot and 12 as ḥatta’ot.  Clearly one can sacrifice the 2 kinnim with 2 and 10 birds entirely as olot and the 2 kinnim with 4 and 8 birds entirely as hatta’ot.

[4] It is shown later in this chapter, that in every case only one nest must be divided.

[5] For example, consider kinnim with 2, 4, 6 and 10 birds respectively.  If the 22 birds had remained in their individual nests, the birds would be sacrificed as 11 olot and 11 ḥatta’ot in total.  When combined they are sacrificed as 11 olot and 11 ḥatta’ot, without the ability to know how any nest was sacrificed.  We could have sacrificed the nest of 10 birds all as olot and the nests of 4 and 6 birds all as hatta’ot. The nest of 2 birds would then have had to be sacrificed correctly, 1 bird as an olah and 1 bird as a ḥattat. Adding the 2 birds in the smallest nest to only half of the birds in the other 3 nests that are validly sacrificed, results in 12 valid sacrifices.  Note that in this specific case, splitting any of the 4 nests would give the same result. Similarly, the case in the Mishnah with 2, 4, 6, 20 and 200 birds, the largest nest of 200 birds must be divided.  The 232 birds must be sacrificed as 116 olot and 116 ḥatta’ot.  The worst-case sacrifices the nests of 2, 4, 6, and 20 birds all as hatta’ot or olot.  The nest of 200 birds is then split, with 116 of the birds sacrificed opposite to how the smaller nests are sacrificed; and 84 the birds sacrificed the same way as the smaller nests. This results in 200 valid sacrifices.

[6] Presented with one nest, the Kohen was assumed unaware of the nest being the result of combining several individual nests.  However, when he sacrifices half and half in the case of mixed nests, one full nest can easily be sacrificed entirely as either hatta’ot or olot thereby half of such a nest is incorrectly sacrificed.

[7] Any even number can be expressed as 2*X for some value of X.

[8] The terms kinnim, packages and nests are used interchangeably.

[9] Assume that while constructing a worst-case scenario a ken must be split because it is not possible to fit that ken / package into either container.  We now demonstrate that if a package must be split, then one of the containers must be able to be filled completely by that split ken. Were that not the case, i.e., the container was not filled with birds from that split ken, move as many birds as necessary from that split package to completely fill that container. Since that package had to be split, given that the complete package would not fully fit into either container, there must always be enough birds to fill that container from the split ken.  This situation may occur when the containers are still empty of other kinnim as in the case of the Mishnah where the ken of 200 birds must always be split or may occur when some or all packages are already stored in the containers, depending on the size and grouping of the kinnim. Were that not the case and there were not enough birds in that package to fill that container, that entire package would fit into that container.  Thus, at that point, after splitting one package one container must be completely full and unable to hold any more birds. The other container must exactly hold the remainder of the birds from the one split package and any (unsplit) packages not yet stored in a container. Those unsplit packages cannot be split since the other container is already full and no more room for bird cages exists within it. Thus, none of the remaining packages can be split proving that only one package can ever need to be split.

[10] An example demonstrating this is given below.  It is not easy to construct such examples.

[11] In both the case in the Mishnah and in the example above of 3 Kinnin with 2, 4 and 8 birds, one ken was larger than the combination of all the other kinnin. Consider, however, a case of 6 packages of sizes 6, 20, 28, 28, 28 and 30 where to maximize disqualification a package of size 20 must be split. Note that the total of 140 birds, require 70 birds in both the O and Ḥ containers.  The maximum number of unsplit packages contain (6 + 28 +30 =) 64 birds.  The second container will hold the two remaining complete nests (28+28=) 56 birds. This leaves the 20-bird nest to be split, 6 birds going to the nest with 64 birds, and 14 birds going to the nest with 56 birds. The reader can try unsuccessfully to fit more than 64 birds in a container, demonstrating the correctness of the result.

[12] K is the total number of pairs of one ḥattat and olah in the intermingled nest.

[13] This has no impact on the correctness of what follows; we can easily reverse O and Ḥ.

[14] Of course, if one container can be filled by unsplit packages, the other container can be filled by unsplit packages as well.

[15]Note that (J + 2*(K-J)) equals (J+ 2*K – 2*J), which equals (2*K – J), the expression for the smallest number of validly sacrificed birds used in Dr. Reiss’s paper.

[16] The remainder of the O container, that was not filled with unsplit packages, has the capacity to still contain (K-J) birds, that will come from the split nest.  That split nest, which fills the remaining K-J places, must have a mate from its nest in the other container as well. This contributes 2*(K-J) valid sacrifices, half from the O container and half from the Ḥ container.

[17] The worst-case equals the total number of birds minus the maximal number of birds that could have been sacrificed incorrectly.

[18]Once 14 birds are placed in the O container, it does not matter if the 12 or the 8-bird nest is put in the Ḥ container unsplit, with the other nest split to fill both the O and Ḥ containers.  Even if both nests with 12 and 8 birds are both split, and any combination of birds from these nests are used to complete the O and H containers the resulting minimum number of valid birds does not change.

[19] Note that J can always be computed, if necessary, by an exhaustive examination of all options.

[20] Remember the H container cannot hold more than J birds from unsplit packages, leaving K-J or more spaces to fill the rest of the Ḥ container.

[21] The Ḥ container also contains J birds beyond the (K-J) birds with a mate in the J container. K-J birds are matched up in the two containers, one group as olot and one group as ḥatta’ot, the remaining birds in the H container is a mixture of unsplit nests and excess birds from the split nest.  Thus, there are J/2 valid sacrifices coming from the Ḥ container as well. 

[22] The (8+10+14 =) 32 birds in the O container, contribute 16 valid sacrifices.  2 birds from the split nest one in from the O container and one from the Ḥ container are correctly sacrificed and contribute another 2 valid sacrifices.  The remaining 32 birds in the Ḥ container have no mate in the O container, so only half are validly sacrificed as hatta’ot and contribute another 16 valid sacrifices, resulting in a total of 34 valid sacrifices vs. the 36 valid sacrifices achieved with the two-container optimization. This demonstrates the under certain combinations of intermingled kinnim, the two-container optimization will not result in the maximum invalid sacrifices, whereas the single container optimization does, resulting in the worst-case scenario.

[23] My nephew, Joshua Blumenkopf, proved that there is no such example with fewer than 5 kinnim.  While I assume there is no example with fewer than 38 birds in total across 5 kinnim, I do not have a simple proof.  One can exhaustively examine 10 through 36 birds to confirm this conclusion.

This is an introduction to depression angles accumulated from various papers I have written and focused on simply explaing depression angles intuitively.

It was published on the Seforim blog recently:

Depression Angles

As a graduate (1965) of Torah Vodaath, I am sadly required to divorce myself from an institution to whom I owe much gratitude.  Read my bio for some details of my Torah Vodaath experience.  Those who read that synopsis will note that in 4 years, one would normally have four Rabbeim; I listed only three.  I omitted one Rebbe where my experiences would have dishonored Torah Vodaath.

Notwithstanding, three major sins (le’Shem Shamayim 😊) have brought about a complete separation.  I thought that the appointment of the Rav ztl’s grandson and RAL ztl’s son would have made a difference, but sadly his energies have been applied in a different direction.  He has and is rebuilding a strong post-HS program leaving TV with a booming elementary school, but a still anemic HS program.

While there is a host of sins, three unrelated occurrences motivated my divorce.

The first occurred about five and one-half years ago in 2015 when I saw the video at the annual dinner where my 50^th anniversary class was recognized. I was uncertain of what I thought I saw; a CD of the dinner allowed me to verify what I in fact saw.  The video hatched the face of Rav Pam ztl’s rebbitzen AH, who was then in her eighties. I remember paying a shivah call to Rochelle (Dershowitz) Zinkin; her late mother was a childhood friend of the rebbitzen.  I saw Mrs. Dershowitz and her friend as young teenagers playing together in camp, dressed as one might expect to see normal young girls.

Other events at my 50^th HS anniversary dinner were disturbing – 1) the choice of dais members, ve’hamaivin yavin, 2) an author of a sefer on a topic that would have by necessity require reading a prior sefer by RAL, of whose existence he was not even aware, 3) reinterpreting “Vodaath” by two separate speakers as referring to daat Torah. That effort to rewrite history pales in significance relative to my next two sins/occurrences. There were other issues not even worth mentioning.

The second occurred about one year or so later.  In the short biography of an unnamed RY who arrived in the US in the early 40’s, the RY was described as bringing the Brisker derech to America.  Forget the obvious insult to both the Rav and his father, Rav Moshe ztl, who arrived more than a decade earlier; perhaps the greatest talmid of Rav Chaim ztl, Rav Shlomo Polachek ztl who Rav Chaim himself named the Meitscheter illui, was the RY at R.I.E.T.S. for about 5 years before his untimely death in 1927, (from a tooth infection.) Rav Polachek’s students included Rav Nissan Wachtfogel ztl, the long-standing mashgiach of Lakewood, and Rav Pinchas Scheinberg ztl among many others.  But Torah Vodaath continually ignores its predecessor in Washington Heights; this shameless lie is just another part of a continual pattern of the blatant rewriting of history.

The third avlah occurred at TV’s 100^th centennial, two+ years ago, something I would not attend.  My brother-in-law (Rabbi Julius Berman) and sister went because of the presence of Rebbitzen Tovah Lichtenstein, who was there given the introduction of her son, Rav Yitzchak Lichtenstein, as the Rosh HaYeshivah.  The yeshivah distributed a brochure honoring its 100 most influential graduates.  I suspect that Rav Dr. Norman Lamm ztl and yibadail le’chaim, my BIL would deserve the honor of being listed even without any mention of Yeshivah University or R.I.E.T.S.; Rav Lamm’s drashot on the parsha or my BIL’s roles at the OU and the Claims Conference alone tower over the accomplishments of anyone mentioned.

Were all this insufficient, I recently thought about another mega-sin mentioned by Rav Rakeffet in a recent shiur. Torah Vodaath likes to call itself Eim haYeshivos, the mother of all the rest of what it considers yeshivot in the Americas. Other older places Eitz Chaim or the Rabbi Isacc Elchonon Talmudic Academy are disregarded as if they did not really matter. The fact that TV’s name was meant to connect it to Rav Reines’s yeshiva, Torah Vodaas, where secular subjects were studied; another inconvenient fact that ought not to be mentioned in heimeshe surroundings. Rav Z. Gold ztl, among the early founders of TV, makes the connection explicit.

Most astounding, the Gadol mentioned at the dedication of a new TV edifice is none other than the former RY of the European TV, Rav Polachek ztl. This was printed in that OO publication, Mishpacha. Infact the ability for TV and others to attract first rate RY to the goldene medinah is often attributed to the presence of the Meitsheter Ilui.

Only an opinion you say, then try listing important graduates of TV before 1925. R.I.E.T.S. can name a fair number of future Haredim among its early students. From that source of misinformation, Wikipedia:

“At age 17 Scheinberg progressed to Yeshiva University‘s Rabbi Isaac Elchanan Theological Seminary (RIETS). There he studied under Rabbis Shlomo Polachek (known as the “Meitcheter Ilui”) and Moshe Soloveichik.^[9] His learning partners included Rabbis Avigdor Miller, Moshe Bick, Mordechai Gifter, and Nosson Meir Wachtfogel, future leaders of American Torah Jewry.^[7]

Significant proof? I think so. The first major RY at TV was likely Rav Dovid Leibowitz, who had Rav Schorr and Rav Pam both ztl as talmidim, but only AFTER he arrived for a roughly 6-year stay in about 1926. So next time you hear the claim, join me in concluding heilege SHEKER.

This post was delayed by my desire not to be so critical of what clearly exhibits the consequences of a ḥareidi orientation.  However, after further consideration the Torah-only philosophy reflected is so pernicious to topics in zemanim, I felt I have no choice but to address / confront.

To add more motivation, I noticed that bastion of fairness, Ha(aino)modia, failed to even mention Rav Adin Steinsalt Even-Israel ztl’s passing.  Long after his detractors are at best a footnote to history, his seforim will still be valued.

I received a sefer entitled Z’manim on the Friday following Tisha B’av.  I told my Rav that the yetzer ha’rah would have encouraged me to read it on Tisha B’av had it arrived a day earlier.  After Shabbat, having read / skimmed the book I wrote back to my Rav that reading the sefer on Tisha B’av would have perhaps been permissible since it would only have added to the spirit of the day.

The author is a grandson-in-law of a prominent Rav who wrote an important sefer on zemanim, Munaḥ Yomah. This sefer, however, included something I never saw previously except in an advertisement – a haskamah with (only) a partial quote, omitting words.  It read as follows: “This is a fascinating and scholarly review… and…”  I wonder what was omitted before “and.” Beyond that, the word scholarly coupled with another haskamah from Rabbi Simcha Bunim Cohen seemed somewhat oxymoronic given normal use of the word scholarly.  I would be more inclined to call Rabbi Cohen Open Orthodox before calling this sefer scholarly; yeshivish, (sadly) perhaps, scholarly never.

What causes my negative reaction?  First, and fundamentally, the lack of reference to the science that provided an underpinning to ḥazal’s calculations of a fixed calendar.  To say otherwise is in my mind (close to) kefirah.  One example (of many) suffices. Ḥazal, like astronomers of their time knew that the exact period between “new moons” varied (considerably).  In establishing the calendar, they used the average time between lunations, something they (and scientists of their time) approximated with 6 decimal place accuracy. It is critical to realize that knowledge of the average lunation came from science and not the halakhah / mesorah.  We now know the average lunation not only to 6 but to 8 decimal places. To attribute slight imprecision to ancient science is hardly surprising, but to assign ḥazal’s imprecision to HlMmS, halakhah or mesorah borders on kefirah.  (BTW the organization responsible for the bible codes episode, put out another similar and objectionable video on the time between lunations as well.)  How ḥazal arrived at the molad / average time between lunations scientifically is necessary to better appreciate the source of the very minor error.

Second, to compare (partially and inadequately) Rabbi Adda’s tekufot to those implicit in the Gregorian calendar is either just bad judgement or inability to calculate accurately, (page 71).  Those tekufot implicit in the calendar of Pope Gregory came over a thousand years after Rabi Adda and are much more accurate. Nowhere did I see any statement on the impact of Rabi Adda’s inaccuracy (at this point an approximate 5 – 6-day drift to a later date in the year caused by the calendar) and reasons it should not be of concern until a few thousand years from now.  Without acknowledging the implication of a slightly longer year, its halakhic (non-) import cannot be addressed, a significant omission. Instead, the issue is simply omitted.

Third, the discussion of the Gaon’s critique of Rabbeinu Tam, is at best incomplete. Its implications go well beyond what was discussed.  The Gaon’s ḥush and scientific knowledge in addition to his compelling logic are all not optional but essential to understanding his attacks on the opinion of Rabbeinu Tam.

Fourth, the author’s assertion of the absolute need for a dateline is simply false. RIZM ztl and RTPF ztl disagree; modern logic (and their insightful he’orot based on the positions of rishonim and aḥaronim) supports their halakhic position.

It is of importance to note that well before Rambam, in geonic times, an Arab astronomer details a roughly accurate version of the fixed calendar created by the rabbis, expressing his amazement at its brilliance. Ḥokhmah ba’Goyim timtzah.

Many of the papers included on this site address all the issues raised.

”Only values that arise from within the halakhic system play a role in producing pesak.”  I agree completely.  Traditional poskim are careful to frame a pesak that way and that is what legitimately constrains the range of possible pesakim. However, as is self-evident, more than one halakhic path forward may exist. The question is what points a posek one way versus another? That is a meta-halakhic question and probably a psychological or sociological one as well. Asserting that only Torah influences are legitimate in forming a posek’s orientation is the what is argued. I have not seen any arguments that would support the position restricting legitimate influences only to those that are Torah based.  In any case, many/most poskim may not necessarily be sufficiently self-aware.  Something as simple as one’s empathy for a situation is IMHO the result of various influences; for example, awareness or evaluation of a situation’s consequences, clearly not in the main derivable from Torah sources is clearly a legitimate basis for differing poskim‘s differing orientations.

However, what we are now witnessing is far worse. Those with a disdain for anything but learning Torah as practiced by say Rav Shach ztl, end up incapable of being a leader and posek. They may be living an ideal existence, but not one that prepares them to opine on the issues of the day. The Sanhedrin did not learn 70 languages from learning Torah. The Metonic relation which hazal adopted is nowhere in the Torah, (thank God, it is faulty) nor is the average lunation (almost but not quite exact and more critical) nor the relationship between stars and darkness. Shmuel did not become a baki be’shivielai de’rakiah by studying Torah. Rabbis without such knowledge did not attempt to create calendars.

There were many great poskim, RMF ztl and especially RSZA ztl for example, who worked diligently to acquire secular knowledge to pasken; today there are defenders for those who do not. That I find problematic.

Stolen from a source I no longer remember.

In the year 2020, the Lord came unto Noah, who was now living in America and said:

“Once again, the earth has become wicked and over-populated, and I see the end of all flesh before me.”

“Build another Ark and save 2 of every living thing along with a few good humans.”

He gave Noah the blueprints, saying:

“You have 6 months to build the Ark before I will start the unending rain for 40 days and 40 nights.”

Six months later, the Lord looked down and saw Noah weeping in his yard – but no Ark.

 “Noah!” He roared, “I’m about to start the rain! Where is the Ark?”

“Forgive me, Lord,” begged Noah, “but things have changed.”

1. “I needed a Building Permit.”
2. “I’ve been arguing with the Boat Inspector about the need for a sprinkler system.”
3. “My homeowners association claim that I’ve violated the construction code.
4. Neighborhood by-laws by building the Ark in my back yard and exceeding the height limitations. We had to go to the local Planning Committee for a decision.”
5. “Then the City Council and the Electricity Company demanded a sh.. load of money for the future costs of moving power lines and other overhead obstructions, to clear the passage for the Ark’s move to the sea. I told them that the sea would be coming to us, but they would hear none of it.”
6. “Getting the wood was another problem. There’s a ban on cutting local trees in order to save the Greater Spotted Barn Owl.”
7. “I tried to convince the environmentalists that I needed the wood to save the owls – but no go!”
8. “When I started gathering the animals, PETA took me to court. They insisted that I was confining wild animals against their will. They argued the accommodations were too restrictive and it was cruel and inhumane to put so many animals in a confined space.”
9. “Then the Environmental Protection Agency ruled that I couldn’t build the Ark until they’d conducted an environmental impact study on Your proposed flood.”
10. “I’m still trying to resolve a complaint with the Human Rights Commission on how many minorities I’m supposed to hire for my building crew.”
11. “The Immigration Dept. Is checking the visa status of most of the people who want to work.”
12. “The labor unions say I can’t use my sons. They insist I have to hire only union workers with ark-building experience.”
13. “To make matters worse, the IRS seized all my assets, claiming I’m trying to leave the country illegally with endangered species.”
14. “So, forgive me, Lord, but it would take at least 10 years for me to finish this ark.”

“Suddenly the skies cleared, the sun began to shine, and a rainbow stretched across the sky.”

Noah looked up in wonder and asked, “You mean you’re not going to destroy the world?”

“No,” said the Lord. ” The Government beat me to it.

1. Do not compare the NY area and Israel. NYC has a similar population to Israel with 70 times the number of deaths. There is much afoot.

2) The NY area MO community suffered greatly. Just in my own immediate circles, my FIL, 2 MO doctors, one who I grew up with, a major RIETS donor, a young man from shul, etc. all succumbed. Others recovered.

3) Haredim do not listen as often to secular news and because of some of their news sources’ abominable behavior, they were insufficiently warned of the danger.

4) Sadly, however, there will be minimal retrospection in many parts of the Haredi world; what went wrong cuts to the heart of hareidi worldviews. Spin-doctors will help to guarantee a minimum level of learning.

5) A benefit that accrues to the MO community is that normal traditional Jews will no longer feel inauthentic given the extremism most hareidi innovators claim as our authentic halakhic heritage; it is not. One grandson’s heard shiurim from RHS on a continuous basis. He was able to learn be’havrutah almost normally. Another heard shiurim from Eretz Yisroel. For all eight grandchildren, their educations may have suffered, but not dramatically.

I have risk factors galore; the worse being the medications I take that suppress my immune system that BH controls my neurological disease. I have ventured out with extreme care (fully only three times in the last 12 weeks, with layers of protection.) OTOH, I have never had so productive a period of reading/learning in decades.

Lesson learned, yes, and no. I have jokingly said that I may be able to return to shul only after the coming of Moshiach, but only if he tests negative.

1. We need a new expression: the road to hell is paved with quotes from the Rav ztl.

2) The OU is dragged right and left; the Agudah seems to have only half of that issue but in spades!! 🙂

3) In the era of instant communications, why do we still have to wait until Hoshanah Rabbah?

One thing is noticeably clear: even the meaning of what is viewed as one of the ikkarim changes over time.  Differing opinions about God’s corporeality, the character of the Messiah, the nature of authoritative texts, etc. are apparent in reading authoritative texts in our mesorah. Even when we seem to pasken on hashkafic issues, the psak is often refined and changes, often dramatically, over time.  This is significantly different from how psak in halakhic matters operates traditionally.

Both our conception of God or understanding of the notion of mesorah are two relatively clear areas, at least to me. This is an area of significant difference between various traditional Jewish streams.