Dividing An Estate Among 3 כתובות: A Non-Game Theory Approach

foreign [Music] Dr Shalom kelman from Baltimore he introduced me to the mishnah because when we met at the recent all that Baltimore cm and the mishna has a complicated halukkah a woman or a man who's paying out three separate subists to three different wives and the mishna and igmar they have a complicated halukkah and Professor Robert Allman actually wrote a Theory or a paper on this Dr Kellman will explain this a little bit more but there were still some questions on this and Dr Fred Gross introduced a simpler method of understanding this mathematical equation there's a little bit above my pay grade so I'm going to hand it over to Dr Kellman to explain but before he takes over I just wanna introduce Dr Kellman Dr kelman attended yeshiva's Karen B Avenue in 1973 followed by three years of intensive studies under rehearsal shakter at Yu for 14 years he had a weekly learning session with review hudu cooperman of meshachma Fame and other swarim Dr Kellman has published in the areas of kabulosa aretz and kadushas yushalayim and regularly speaks at Toro institutions including REITs and yadav Herzog and yushalayim he has presented the Shabbos Baltimore since 2000. so it was my pleasure for Dr Kellman to take us through this mathematical equation thank you so much now by schwed it was a delight to meet you and uh for all you've done for the dafiomi and it's my pleasure to share some exciting material with you on this upcoming daf which has been a mystery for almost 2 000 years as no one has really figured out how to truly understand the mishna the mishna we're going to come across like I said is the mission of the begins and it says man is married to three men who mace and he dies and they have different casubas this one has different claims so there's a table here to help us simply uh understand the question of the mishna the case of the mishna you have on the top you have the claims of each woman a become A1 a sub 2 a sub 3 and they each have a different claim on your state the wealthy one has a 300 First value the second 200 and the lady a hundred on this side as you can see you have the value of the estate so in the value of the estate there's a hundred the khaluka it seems very straightforward you take a hundred each one gets a third that makes sense but there's very little there's divide among the three people who are proclaiming it if you look at 300 that's also something of uh because it's proportional each one's claiming let's see 300 to 200 to 100 and there's only and that would add up to 600 right if Adam or if it was 600 everyone get what they wanted but they only have 300 so you have half and what gets half of what they want 150 150. but the middle of 275 75 and 50 doesn't seem to follow any suburb at all and this is a mystery and we're seeing the Gomorrah Memorial has why I'm struggling getting ukintas how can they understood in the end the mirror says at the end of um I don't understand it understand and in fact that's how they'll be shown in and the Mystery has always been but what's the what's the mishna and there are the goes back there's a famous riff who has much longer the bigger Rift than you'd expect on this little piece of Gamora and he quotes about High Ground who had an idea maybe it was related to schneid mocks and the talus but never was able to implement that idea how was actually working in this case and this question I said baffled Shona but the answer only came to light in our time thanks to Professor Robert Allman almond is famous in the world today he won the Nobel Prize in economic Sciences in 2005. there's a famous picture of him taking the prize with his big yamaka of the shashem and the area of his expertise for which he won the prize is called Game Theory Game Theory is a very sophisticated mathematical structure and there are people who understand it I'm not one of them a lot of smart people don't understand it's very very difficult area and the issue is I'll hit pause before I go this is the copy of this paper and you all have by the way hopefully we'll have a PDFs available of the paper and another paper you wrote two papers this is the one he um was written mainly for mathematical audience uh it was an economics journal and as you see it's game theoretic analysis of bankruptcy problem from the talmud he he labeled this a bankruptcy problem because it it's essentially the same kind of thing you have a and the state you have to and and doesn't have enough money so how do you divide if it's available in the state equitably to all the the shareholders and the other set is dedicated to the member you have his son who was a Tamil who has killed an action in Lebanon in 1982 and as he says in other sources it was his son who brought this to his attention said Daddy if you know Game Theory maybe you can apply to this mishnah and Laura behold he was successful he was able to do so and um the other paper I want to make a difference you'll have is a paper that was written more for non-mathematicians and to understand it on an intuitive level and the people in our world of tashiomi so the second paper is also included it was originally written in Hebrew then translated into English published in Bangla and I should be available to you as well as a PDF so what were some of Almond's insights that allowed him to solve this problem so the first thing you recognized was indeed that's what I had suggested this division is based on the principle of schneid mochs in the Taos yeah we'll jump ahead a little bit he also was able to restate rashi's explanation of how that saluka was done in a mathematical form and we'll come to that in a few minutes another thing he recognized that one can expand the rule of schnaim of sin to not from two people to three or four as many as you need using the same Principle as we'll explain in a few minutes the main concept and the main idea of schneid marks on the talus is that each claimant shares the same amount of loss as the other claimant I'll give you an illustration in just a moment but that was that's the that's the thing which we'll come away with all the participants every claimant when compared to any other claimant will have the same loss and that is that's the fairest way now that's not always the case you know in other types of aluca you have proportional it's not necessary everybody's going to have the same loss it's not going to work out that way is that every week 71 will save the same amount of loss another thing almond was able to do was to suggest that once he was able to see the mishnah and say oh this Mr fits the rules of Muhammad I expect from Game Theory he was able to develop an algorithm to actually accomplish this how do you go about doing it don't have money have this stuff so what's the how do we do it in Practical terms he came up with an algorithm that allowed him to do it all of this though the entire structure of what he created was based on his development of Game Theory and he acknowledges this in his papers he says you know once I had Game Theory I had the key to the garden now I was able to understand what's going wrong but and he's able to work back and figure out algorithms Etc but how did know about it game theory is so sophisticated it's very unlikely to have access to it he struggles with this he gives some theories how possibly intuitively they might have Etc but it still remains a strong question now comes my chance to introduce Dr Fred Gross Dr Gross himself was a noted mathematician he had a PhD in number Theory but in addition he had Samira from tar vidas his family came before the war and he was a major he also he loved learning and he loved mathematics and he published extensively in his field as well known in the field of math but he also made many fedushim in the field of combining his mathematical knowledge with Target insights most of this material unfortunately is still in the hernanduscript form and what I'm sharing with you today is also only a manuscript I don't have a PDF I can share with you at this point the family still wants to keep it until it's really to be published and then we'll be sending it out but in addition to the things he says on this topic he has on so many other topics he always wanted to in fact write a textbook with for high school students showing them how imagine the mathematics of the of Torah can help enhance their understanding of math and vice versa so I was zohra to have a very nice relationship with Fred going back several years and we discussed this topic many times and he sent me a manuscript that he wrote with his from his son Arya that I used as the basis of today's presentation what was Fred's big insight into this topic he also applied the bhava mitzia division rule that that I had previously noted I didn't say that point that Alman called it the contested garment contested garment based on steinmos and uh and if all the khalukas have to be consistent with this contested garment and Fred named it the BMV the brother mitzia revision and what he did was developed using the government's rule schneid mohsen using basic algebra and not least crime in Game Theory thus it was in the reach above no sun and all the other tanoim who lived in the period of time of the Hellenistic time the Roman Greek periods where basic algebra was available solving simultaneous equations which we're going to do today was accessible to them they could do quadratic equations they were very sophisticated in many areas of mathematics so this is something that someone like NASA who we know from the Gomorrah was a very it was she math genius and astronomy has different different quotations of NASA we just had him recently in the daphion they always being bocaoc and so on but nothing was known for his mathematical prowess and more than that now that we have this algebraic Prince approach it's much more accessible to lay audiences like ourselves and that's what I hope to share with you today and by doing this I hope to enhance our learning the dafyami this Friday I should emphasize that both Professor almond and Dr gross's approaches yield the same results there's no difference all the numbers came out exactly the same it's just how about how does one go about getting there that your approaches are different and at the end I will try to show how they actually the two approaches actually intersect beautifully so let's begin with the wheels thrust of today what is the bmd the government see a division rule is where one man says this all belongs to me so I grabbed I found the Hefner I grabbed it's mine and the other gentleman says no we both picked that up at the same time it belongs to both of us so the first claimant says it's all mine the second claimant says no half of it is mine half of it is yours what does arashi explain over there he says a simple explanation which is if I am the owner if I'm began to claim that nay who says it's all mine so I say all of it is mine claim it B is saying but part of it is yours half of it is yours I agree and the other half we're arguing about I say that's mine and you say it's yours but I disagree with you so both agree though that half of it belongs to the first so the time of day therefore he gets it without any problem the contested part the second half is divided between the two of them and that leads to why as he gets half of the half the other one gets a half of the half so the total he gets three quarters one gets one quarter that's how Rashi explains it what Alman was able to show mathematically was that the losses of the claimants are actually equal and in order to explain that let me just explain these few sining equations and we'll get into more understand a little deeper so another way of understanding the government to your rule is that the losses of the claimants must be shared there's a loss to everyone you divide it and each one divide has the same degree of loss to understand how you understand what a loss is a shared loss you look at what the total claims are you subtract from the estate what's what's available and that is the loss that they have to share between them so the shared loss is divided between them equally so when you end up with the claim the C of the claim minus the loss of the individual that is the Allah that's what that individual gets so these are the concepts you're getting here over and over again claim minus a loss equals an allotment now let's do our case for example A1 Claims One Talus a sub 2 claims the half a tallest So based on our equation the total claim minus the estate is the shared loss the total claim is one plus a half is one and a half what's the estate value there's only one Talus what's the difference a half a half divided between them comes out to a quarter a half times a half is a quarter so each one has a loss of a quarter that is how do we therefore divide it so the allotment what each one receives is their claim the individual claim mind that's their loss so A1 a in the sub K subscript A1 a refers to the allotment that we associate with A1 the big A1 he has a claim of one you subtract a quarter that's his loss he ends up with three quarters A2 the other contestant he claimed only one half and you subtract that which is that's his claim one half you subtract his loss which is the same loss as the other fellow a quarter and you end up with a quarter so this concept I hope because clear as can be we're going to come through it over and over and up to the Hazara you're more comfortable with it so again another way of looking at this equation is let's move things around moving loss to the left and allotment to the right the equivalent is to say a loss is a claim minus the allotment so in this case the loss of the two are equal the loss is a claim minus allotment so loss of A1 is 1 minus three quarters because his claim is one is a lot must be quarter so his loss is a quarter and A2 his law claim was only a half to begin with his allotment was a quarter and therefore his loss is a quarter the emphasis again is that they are the loss here in both it has to be equal now what are we going to do how are we going to outline the approach that we're going to take now as outlined by Dr Groves the general law again the loss of claimers must be equal we assume that we're going to prove a generalization of this rule bmd World exists in other words we can do it not just for two individuals we're going to expand it as three individuals in all of these cases we'll say that the individuals themselves A1 A2 and A3 they have claims C1 C2 and C3 and we always put them in order such that e which represents the entire estate what's available is greater or equal to C1 which is the higher claim which is equal to or greater than the second claim C2 which is going to be greater or equal to C3 the lowest claim so in all these cases we have to assume that they aren't put in a numerical order now what we'll do is assume one person's array one of the claimers is receive something so let's assume A3 the one who's the lower value he receives his allotment a sub three which is uh I hope it's not clear the uppercase a is the individual the lowercase a is his allotment and the subscript identifies who we're talking to so the upper uppercase a is three he gets his allotment of the lowercase A3 that's the value A3 therefore what's left what's left is what A1 and A2 they share the remaining A1 Plus A2 there because A3 has been removed so all we have left is A1 Plus A2 and therefore if the a1a2 have to split it up by the by the salukas MS of the bmd rule their losses must be equal so claim one minus A1 that's the A1 that's what his claim minus his allotment his loss equals the claim of two minus is allotment and that's his loss and the losses have to be equal so I hope this is clear because this is the key of understanding everything that's coming afterwards now take note you might always assume that the claim is whatever the guy asks so so for example in our case in the mishnah the claim of the first of A1 should be 300. but that's not always the case because the claim cannot be greater than what's available in other words even though he says his claim is that she'll claim is C1 which in this case was 300 it can't be 300 if we assume A3 has taken part of a state away he's been he's gotten his haluca so we always have to figure out what is actually the claim and this is the Hop you have to focus on so again the claim cannot be greater than what is available after A3 takes his allotment leaving A1 and A2 so the way we denote this is say the claim of A1 is the lesser of either his actual claim or what's left after A3 gets what he takes which she takes and leaves behind A1 Plus A2 that would be left over this is mathematically this this script is written as denoted as the minimum of A1 Plus A2 comma C1 means what is the lower of these two values either C1 which is his original claim or it can be less than that because A3 has taken his part and therefore leaving behind A1 and A2 which is the lower part The Point again is that the claim can never exceed what's available now that's we having now we have that now we have the way to achieve what will give us the solution how do we solve these equations so let's assume once AC A3 receives her allotment A3 we have one equation of known values because we know C1 and C2 and we have unknowns we don't know what the allotment is A1 and A2 so we can write an equation C1 minus A1 equals C2 minus A2 however now let's assume we don't know that we're going to make another equation based on another assumption let's assume okay and instead of saying A3 receives his or her allotment let's start with A1 let's say one who gets the hives claim received for his allotment first we have a second equation because that's what's left over in whether it's C2 and a and C2 and have to contend they're eight I'm sorry A2 and H we have to contend with each other so what are their what are their haluca we say the losses have to be equal so C2 minus A2 the loss now the claim of C2 minus is allotment is his loss is equal to the claim of three a three minus his allotment A3 which is his loss or her loss so in other words these two losses have to be equal so we have two independent linear equations with three unknowns we don't know the unknowns are the allotment a1a2a3 but we do know what the claims are so they have two equations this one equation is two equations and we have a third equation we know A1 A2 and A3 when you add up all the allotments it has to be what's the estate was when we have three equations the three unknowns we have the solution we can solve now for A1 A2 and A3 so Fred was able to prove this algebraically and it's not such a hard proof you'll see but the even more astounding was he can use the same method for four or five or six claimants because the number of equations just keeps growing you have four equations and four unknowns so five equations and five unknowns and each that's algebra that is accessible so people understand algebra okay now this is the meat of tonight today's talk and if all you can do is understand the next three slides four slides you'll understand the chaluka of the mishnah and then you'll feel good about what you accomplished because this is something that um cannot understand and now you're going to have the ability to understand it so let's begin with the first case of the mishnah remember the mishta have the first case was a value of a hundred the estate's only worth a hundred and the mishna said we divide everything equally everyone gets 100 divided by three excuse me now let's assume I would say for each one their claim initially was 300 200 and 100. right but the estate is only worth a hundred right so the most anyone can claim is a hundred in other words even though the claim C1 is 300 C2 is 200 200 but the actual claim is only a hundred because that's what's available now A1 Plus A2 but there is the allotment of a1a2 can never exceed 100 why because A3 is taking apart in other words if you look at this this A1 A2 and A3 that's each one's allotment has to equal to 100. if A3 is getting something then there's that's less than 100 so A1 and A2 has to be less than 100. so assume like we said before A3 his allotment gets goes first therefore A1A to each claim A1 and A2 what's ever left now what's the minimum of the claim of A1 is that A1 Plus A2 or is its actual claim what we just said before it has to be less than his claim of 300 because A1 and A2 has to be less than 100 so we're absolutely so the answer to this question has got to be it's a 1 plus A2 it can't be C1 which would have been 300 it's got to be a one and A2 Which is less than 100. the same argument can be made with C2 c2's original claim was 200 and here we have a claim of either it's either the minimum A1 Plus A2 Which is less than 100 or C2 which is 200 so the answer is obviously a1a2 so now let's use our equations the loss of A1 defined by his claim minus his allotment is the claim is A1 Plus A2 minus this allotment which is A1 so simple it's A2 you subtract A1 minus A1 you're left with A2 the loss of A2 the same argument the claim minus the allotment is the loss his claim is A1 Plus A2 as we pointed out his claim is his allotment is it should be a mistake it's a typo it should be A2 and therefore his value should be A1 he should get that's that that was his loss about his loss is A1 so by the rule of the bmd rule that it should be therefore the losses have to be equal so A2 is equal to A1 we can do the same argument when A1 receives a lot in the first and it just compare A2 to A3 because even A3 is claim was 100 but we know it can't be 100 it's got to be less than 100 because a in which we move A1 A2 and A3 have to be less than a hundred so the same argument applies to if a one and a four when A1 receives this allotment so you have basically A1 equals A2 equals A3 which is like three times A1 which is the same as 100 divided by three so that's why everybody divides it equal and it gets 100 divided by three that is the first case of The Michener the second case is a little bit more complicated with an additional hop but you'll be I hopefully you'll be able to follow it relatively easily the mishna's second case was the Estates worth 200. all right so for E equals 200 what's the most A1 and A2 can claim is 200 right that's all there is so the claim of A1 can't be 300 and A2 can claim a 200 and remember A3 is only claiming a hundred now since A1 Plus A2 plus 3 has to be equal to 200 if you took away A3 A1 and A2 cannot exceed 200 because if H we get something it diminishes 200 so A1 and A2 cannot be greater than 200. and similarly a to an A3 cannot kinetics at 200. if we took away A1 from this equation then leaving A2 and A3 well we know a1s has to be value something so it's going to diminish 200. so what's left H1 A3 kinetic c200 so let's begin uh the assessment like we did before we certainly look what happens with A3 as assume AC received his allotment and then we'll do the same thing assume A1 receipts is alignment so if A3 A3 but this is a lot more first as before using the same argument we had before A1 equals A2 that's the same argument behind the previous page but if A1 receives A1 first right if A1 gets A1 first then what's left A2 and A3 so A2 and 8th we have to share an A2 and A3 this A2 plus A3 so let's look at each one of the claims individually what does a2's claim a2's claim is we said the minimum between its actual came C2 200 or A2 and A3 what we just said A2 and A3 have to be less than 200 because A1 is there if it takes this allotment then it has to diminish 200 so we have to be the A2 plus A3 has to be less than 200. so therefore the minimum of this expression is not 200 that's to be less than that A2 plus 2 has to be less so therefore the answer is A2 plus A3 so the claim of A2 is a plus A3 its loss therefore is A2 plus A3 right minus his allotment A2 so the F gives you A3 right A2 minus A2 and A3 so you have A3 left over but with A3 it's not so passionate a3's claim is again the same argument we said before it's either the lesser the minimum of A2 plus A3 which is left after A1 takes this piece with A2 with A3 is left actual claim at this point we don't know every chance we can't we have no way you can't use these arguments earlier to determine what it is so we have to leave it as unknown for now so his loss is therefore the minimum of that expression minus whatever you saved X3 now since we said losses between the two claimants once we've moved day one we have A2 and A3 their losses have to be equal so the loss of A2 is A3 the loss of A3 is this expression and we make them equal losses are equal so A3 equals that expression the lesser of these of either A2 plus 3 versus 100 minus A3 because that's what this expression was you said the loss is this expression minus A3 so these two two this this equation that's what result is results this simple manipulation we add A3 to the side at A3 to this side so you have 2a3 equals what's that minimum of this this expression so here's now the the the the lumbar cup if we assume that the answer to this what is this value is A2 plus A3 let's assume the answer is not 100 but it's A2 plus A3 then we have now the equation resolved so 2a3 this will be here 2a3 equals A2 plus A3 which then comes out because remember we're removing 100 we're saying this is this the Lesser is A2 plus A3 so that being the case we go back to this equation 2 A3 equals A2 plus A3 which then comes out too if you just subtract A3 from this side and from this side you have A3 equals A2 A2 equals A3 all right but we know from the first expression if we assume the A3 receives as a lot in the first we came to the conclusion A1 equals A2 and now we'll come to conclusion A2 equals A3 so if A1 equals A2 so it comes out all of them all the allotments are equal like in the previous case A1 will equal a two equal three which is the same as three times A1 which is equal equals the what's the set the state is 200 or dividing it by three it comes out A1 and all the others all be the same will be 200 over 3 or 66.6 but if that's the case then A2 plus A3 would be 133 and we said over here right which is which is sorry 1833 which is greater than 100 in other words in this expression we said what is the answer what is the lesser of these two expressions so we said let's assume it's this one A2 plus A3 but if we follow that we end up with a contradiction because then it comes out that A3 is greater than 100. so that therefore can't be correct thus our initial assumption that the minimum of these two would be A2 plus 3 is not valid and therefore we have to come to the conclusion it must be the other option which is a hundred is equal to 100. therefore now going back to this original equation 2a3 equals 100. 100 there 100 here what A3 equals 50. so once we now have come to the conclusion that A3 gets 50 so we know 200 equals A2 plus A3 and we know therefore once we A3 gets this piece a 50 what's left is 150 A1 and play two and since we said A1 equals A2 so each one gets 75. 2A equals 150 and you get 75 so this answers now the second case we have the first one getting 50 the the A3 the less we're getting 50 and then the remaining 150 has to be divided equally because we came to the conclusion based on what we said A1 equals A2 and therefore I have now 75 for the A1 A2 and 50 for A3 let's quick a quick check to see if this what I just suggested fits what we're saying all along that the total sums of the claims minus the estate equals the total loss and then has to be shared so let's just let's do a quick a quick check again assume AC A3 received the allotment of 50. so what is the claim of A1 we said it's 300 but it's camping right because what's left only uh it's a 200 right so it was 200 if if A3 gets 50 what's left 150 so the maximum claim of A1 could be 150 same thing A2 it can't be 200 but it's only what's left 150. why because you can't claim more than what is available after A3 is allotment so therefore as to ask the numbers the total claim is 150 150 is 300 right what is the estate's value after A3 takes his allotment 150 not 200 right because we said HB gets his 50. so that's left over is 150. so while do our Quest do our equation the total claim which we said is 300 it's 150 plus 50 minus the estate which is not 200 but 150 because A3 got is a state is equal to the shared loss so 300 minus 150 is 150 that's the shared loss and the shared loss has to be divided between them which is 75 each so therefore individually the claim of one or two a1a2 is 150 minus his loss is 75 minus 150 minus 75 and that explains what each one of them gets an allotment of 75. 75 75 and A3 getting 50. so it all works out from both perspectives that we talked about and now let's look at commissioner of the third missionary A1 and A2 another dimension now is the allotment the the estate available is 300. so again this is the same arguments it's a little easier A1 Plus A2 cannot exceed 300 and A1 Plus n a two plus two you cannot exceed 300. again for the same argument over and again since the total is equal to 300 once we take away A3 then A3 has to diminish what's available he's getting something so A1 Plus A2 cannot be 300 has to be less than that and and the same thing with A3 takes if A1 gets his allotment then A2 and A3 has to be less than 300. so let's start we usually start A3 receives his allotment A3 first so let's look at the claims what are the claims and what's the allotment to determine the loss so a1's claim is the minimum of his claim of 300 or A1 Plus A2 well that's clearly the answer is A1 Plus A2 because we already said that A1 Plus A2 has to be less than 300 because A3 has taken diminished it so this is an easy one to answer so this expression is A1 Plus A2 and therefore the loss is the claim a 1 plus A2 minus his his assignment or allotment A1 is equal to A2 so again if A3 receives first so if a loss of A1 is A2 lowercase A2 if A2 now in the same situation with A3 but receives it first what is the claim of A2 so we can determine the loss so in this instance we don't know it's a minimum of A1 Plus A2 versus 200. well this could be higher than 200 we don't know so we'll just leave it as we don't know and calculate the loss is the minimum of this a1a2 Plus versus 200 which we don't know minus his actual allotment A2 so this expression is what we know about the loss of A2 this is expression what we know along with the loss of A1 we equate them so A2 from up here equals the expression pump here minimum A1 Plus A2 or 200. right minus A2 because that's what he said the loss is its claim minus is allotment or once we have this expression we just move the two eight the a sub 2 to the bar to the other side so add 2 plus 8 to here and there if you have 2 a sub 2 equals that expression A1 Plus A2 right cannot be less than 200. right why since this means A3 is greater than a hundred if we look at this again if A2 A1 Plus A2 can we say it has to be it cannot be less than 200 because if A3 guesses allotment well A3 can never get more than 100 because he can't he's already claiming a hundred so if A3 is only claiming a hundred it can't get more than a hundred so A1 and A2 has to be it can't be less than a hundred two hundred because otherwise A3 will be getting more than a hundred I hope that's clear I got moment I already lost myself so again A1 and A2 their claim together has to be cannot has to be greater than 200 because if a um the total claim is 300 or I hate when this happens but this is the problem after we do again okay I'm going to start over from this line start from over here so um A1 so now we have claim we have decided that this is the equation of equality between the losses being equal the A2 from over here that was the claim the loss of A1 and the loss of A2 is this expression we trimmed over here it's minimum this minus A2 we equate them and therefore we can now move the A2 on this side to that side and we have 2A 2 equals this now we're trying to determine what is the minimum of this expression so the argument is as follows we can determine this it cannot be less this A1 Plus A2 cannot be less than 200. we're trying to say that this has got to be it can't be less than 200 because if it was less than 200 then A3 is going to get more than 100. if this is less than 200 A1 Plus A2 then A3 has got to be more than a hundred but that can't be because it's only claiming a hundred so therefore means let's proved that A1 and A2 has to be can't be less than 200. so the answer then must be that in this expression the minimum is 200. because this has got this is going to be more than it has to be more than 200. so this has got to be the answer is 200. so if that's the case then we have back to our equation 2a2 equals 200 and therefore A2 is equal to 100. so now we know at least one allotments based on this reason is equal to a number that we know a hundred that is for assuming AC got A3 received his allotment A3 first what okay so now let's now look at the other side uh assuming uh A1 receives his allotment first and what's left is A2 and A3 to receive their share of A2 plus A3 now we know from the previous slide we calculated we've showed that A2 is equal to 50 100. so now we have a two plus three is the same as 100 plus A3 yeah for A2 we're going to try to figure out the same we did before what is the claim and what is the loss for the claim of A2 it's the usual it's the minimum of A2 plus A3 the allotment what's left over after A1 received his allotment versus his actual came of 200 what is the lower value now we can one thing we can plug in is now we know A2 is the same as 100 so this is now equivalent to 100 plus a sub a sub 3 200. now but we have some additional information to help us here since a sub 3 the allotment of three cannot exceed 100 why because he only claims a hundred so the minimum of these thing of 100 plus A3 versus 200 well A3 has to be less than 100 right so what's going to be the lower value this or that is going to be 100 plus A3 so this now becomes the claim 100 plus a sub 3. so a2's claim is 100 plus A3 and now we can clap we can calculate the loss because we know we just said the claim is 100 plus A3 minus his allotment what we calculated earlier A2 is equal to 100 for the previous slide so now 100 plus A3 minus 100 comes out loss is A3 so the loss of A2 is his claim minus his allotment and is equal to A3 now that's for A2 what about A3 he's the other remaining Contender against A2 what's A1 is received as a portion for A3 the same issue the minimum of 100 plus A3 because it's A2 plus A3 so A2 is 100 A2 plus A3 versus 100. what's the lower level what's going to be 100 since he only claims the hundreds it can't be more than 100 so it can't be something 100 plus something has to be a hundred so this now comes down to a hundred so thus a3's loss is his claim which we said is a hundred minus this allotment which is A3 and now we can now configure out the equation of equaling losses we have the loss A3 which we calculate is 100 minus A3 that's here equals the loss of A2 which was A3 so 100 minus A3 equals A3 A3 has to be 50. right 100 minus 50 is 50. and therefore if A3 is 50 and A2 is 100 then we know what's left is a total of 300 so A1 has to be 150 and thus we have figured out all the allotments of each of the individuals on each case of the mishna now what frame this if you come away dressed with this information you'll be able to say I can explain each one of the Kim in the mishna and that was probably something that the tanoim could do like I said what we did today this kind of freaking reasoning through what is the value for each of the situations if you first take what if A1 gets what he gets and make the equations for A2 and A3 and then assume A3 gets what he gets and now A1 versus A2 and knew the equations for them you'll arrive in every case at a very simple sense of equations and it's easy to solve like we just went through and again the principle again is that we assume one person gets what he does and then we have to reassess okay what is the claim because it's a change claim is going to change if someone takes away some from the from the estate so you have to decide what is the actual claim what he says or the Lesser if there's not enough left in the pot his claim can't be his high that's the it's the most important your sowed and based on that you have to have the two left two people are left you calculate their losses you say the the claim minus is allotment of one equals the claim minus allotment of the other and they have to be equal and this will always be the case and in all cases you know whether you're trying A2 versus A3 or A1 versus A2 or A1 versus A3 doesn't make a difference it always comes out equal and this is enough to understand the mishnah and in Practical terms they didn't have to use all the um they would do equations because they had they had numbers to play with they had 300 the estate is 300 he's claiming this he's claiming 100 200 300 it had numbers so it made it easier for them to calculate but the next level from an algebraic standpoint would would be to substitute for numbers the actual symbols of C and Fred did that what I'm going to do next is not to frighten you but just to show you how taking the algebraic level to another level will make it easier to actually program a calculator to do this for us so have you did was here he came up with these nine scenarios because each case is you take the minimum of a102 and versus C1 what is it what has could be it could be all these possibilities and the same thing can be all these possibilities and this and these and then you have you go through all the possible combinations you end up with nine combinations and you can just make What's called the loss equations we know the loss of this versus equal that the loss of this equal that and the loss equation raise points in every scenario we're going to have two equations and then you're going to have a third equation which is the E equals A1 Plus A2 plus A3 and all the way down the same equation and when you solve for these three equations on a um algebraic method you end up with these equations now these equations are easily solved by the way if you use Mathematica or any kind of mathematical program you don't have to use it by hand you can do it you do it quickly on on the on the on the Mathematica once you have this what he did was recognize that some of those cases one two three four cases were really special cases where the claim of two and three were equal so he had resolved with basically five cases five scenarios and in each scenario by studying he was able to determine that the oil based on five different conditions of e depending if e started with e being very low very high and so on without going through the details once you have this then you're able to write a nice program so now that we have the ability to identify the E's the Estates in different values of these inequalities I was able to write a program which takes the opportunity to say take in the value of e and look at what its value is in these different scenarios if it fits this one over here then this is the equations for the allotments if it's these and those and so on and that's how this program was written and once you have it the beauty is you have a little calculator and you can now play with this and this will be available hopefully on the old off website you'll be able to download this it's called a CDF it's a type of PDF that you'll be able to play with and as you see the estate can be January so right now I can move the estate and you'll see below how the values of the very claimants change so if at 100 until we get to value of 150 it states 150 continues and then you see it keeps changing and then the values dynamically changed now to understand this better it may be easier for us to look at a bar chart I found that's a little easier for me to follow this so let me uh do that now and now you can see if we go back to the beginning when the a state is a hundred so just like we expect the division is equal like the mishna says everyone gets a third and that stays that way as we add more the more and more estate increases in value we divide the estate equally among all three claimants but when the value in our reaches a hundred and fifty what happens now is that c has received half of his claim which was our claim of 100 now she gets 50. so that's half the claim so now we just simply divide the rest as the state gets more money we divide that among the two remaining C1 and C2 and then when c 2 gets a hundred which is half of her of her claim she claims 100 200 now she's got a hundred to have her claim then you see the khaluka now just gives money to as the estate and gets more money we now divide it only goes now to C1 to the claim of A1 and here's the interesting thing when it reaches 350 however then we start dividing it among both C1 and C2 the higher values and they share increasing as the value of the estate increases they share that value equally up until about 450 and then C3 starts to increase and shares in the increased value of the estate and as I emphasized earlier in all of these trouble codes the claim minus the allotment is just the loss is always going to remain equal if you cut if you keep C3 alone but if we move that from the estate and just compare C1 and C2 their loss will be equal if you just remove a one like we said earlier and get it and just compare C2 to the C3 it's their losses will be equal and this is what you can dynamically see as we go further and then when the estate has lots of money everybody is increasing until they get their full requirement which was 600 with everyone getting the 300 the 200 and the 100 equals 600 and everybody gets the foja Luca now what's fascinating is when you look at this again this Dynamic I'll just repeat it one more time because it's I want to emphasize these points again we start the lowest value of 100 they all share together then when it reaches the half value of the of the claim of C3 which is 100 which is 50 then we just add to the values that we divide the estate only between C1 and C2 until C2 reaches half and then we just get add add if we want to the C1 until which is the heat is 200 when they get to it and then they still have to share again the fascinating thing for me is this is almost ident this is the exact approach of the algorithm that almond described okay now we can look at the Almond algorithm we'll see how it's identical it's what we just saw with the bar charts but you suggest again you order the credit just from lowest to highest like we did a 1828 8382 A1 it divided State equally among all the parties that's what we're doing from the start when we have the low values when the whole that stays rather hydrated by each one equally and the and you do that until the lowest creditor receives half of the claim and that's what we saw in the bars it went up to 100 of the of the C3 got up to A3 got up to 50 and then it stopped moving and now divide the estate equally among the remaining parties right except the lowest creditor you don't give him anything just give the other A1 and A2 and so the next lowest creditor was A2 in our case we see it's half the claim and that's exactly what happened when he received his half his claim but his claim was 200 when he saved 100 then his bar didn't move up anymore and then we proceed until the next credit to reach us half of the original claim now you work in Reverse start giving the highest claim more and more money until his loss actually to be defined before the difference between the claim and the award equals the loss of the next creditor and that's what we'll see in a minute how that works out it's always going to be the same loss and continuing this until you're finished so this is why it's exciting to see how using almonds approach is identical to using Dr gross's approach from a um algebraic standpoint next slide shows another way looking at this is simply a diagram a graph diagram as you look again you'll see as we State increases on the bottom here at the x-axis it increases until it gets to 50 because we we add money into the system and these all three are receiving a1a2h3 receiving the same amounts until we get to 50 when it hits us 50 A3 the lowest one that's no longer gets anything but a 1 and A2 continue to divide equally until A2 divides it gets his 100 and doesn't get any more as the stake increases in value because as the state cruises in value it will be here because this increases we see that he doesn't get any extra but rather we give all the money up to A1 and but astute Observer will say wait if anyone's getting all this money how can the losses be the same it seems that distance to be intuitive they're not right how can the loss be the same if this one's keeping getting more and more money the answer is how we defined a loss and that'll be a final slide to play all together from the beginning and once we get up to the 300 of the when it gets up to this value of a 350 then we see A1 and A2 continue to share until it gets to 450 and then all of them share at the same rate to explain how it's possible that this can keep gaining money by himself and yet maintain the same loss as A2 let's go one last slide remember clay minus allotment equals loss when the value of the estate was 300 remember we had sine the claim based on what's available assume that we A3 got his portion was 50. so the estate even though it's unlisted on the fars and so it's 300 but in reality it's 250 is available so A1 his claim was 300 but what's available is 250 as we did earlier so it's 250 and his allotment is 150 100 that's his loss A2 is he his maximum is 200 he can't ask him whether he asked so it's 200 minus his allotment of 150 so the loss is remain equal when we go to the next level when the estate is 325 remember we always have to take away the allotment of A3 that's because we're comparing A1 to A2 so A3 is received his allotment of 50. so what's the value of the estate is really 275. a1's claim was two seven was three hundred if you can't ask for more than there is so he's a kind is 275. he receives 175 based on the analysis and the algebra we did earlier and therefore his loss is a hundred A2 continues to have the same doesn't change as 100 so the law stays the same so therefore you see an opportunity here as A1 is getting more and more uh his allotment of 175 he wants 150 went from 150 to 175 yet because his claim continues to increase the cane increases this increases and leaves the loss at 100 which is equal to the loss of A2 so this is how the A1 can get more money and yet stay made the same loss as A2 when we get to three evaluate with 350 which is beyond emissions was then the initiative was at 300 and now we're at 350. again the same process we take away A3 left out the value of the estate is 300. A3 a1's claim was only 300 so that's a 300 is allotment is 200 value and loss is a hundred and that's the same as A2 it hasn't changed what if we go beyond that and it's the last one if we go beyond that let's say just to make the numbers easier um the estate now is up to 372. so again comparing A1 to A2 A3 is removed that leaves the 15 removed that's the estate is 4322 so A1 his name is always 300 and the allotment based on the system is 211 his loss is 18 high and that's the same a A2 is 200 his his claim is 200 but he's now getting more money now he's starting to share as we saw the par c2's graph is starting to move up and he gets 111 and therefore the loss Remains the Same and this could be you can look at this or with your calculator and play with it and the insight into what's happening okay fine so thank you very much Dr Kellman for math Fanatics this is certainly going to be something which they'll find very very intriguing for the regular person this is something that if they want to spend the time and go through it they'll certainly be able to have a much better understanding of the mishnah