#95: Horsing Around

95.1  Imagine a world where the average woman had 10 children over her lifetime…and the average man had over a 1000. What sort of kinship relationships and terminology would develop? For human beings, this would go way beyond plural marriages or even harems. Fortunately, we’re not talking about people, but about horses.

95.2  Among horse breeders, equine kinship consists of a mix of “human” terms and those particular to the field. I’m sure you’ve heard some of the basic ones…a male horse is a stallion, a female is a mare. When young, they are colt and filly. If you’re a horse, your father is your sire…your mother is your dam. Sons and daughters are…well, sons and daughters…altho the overall offspring of a male is his get…of a female, her produce. 

95.3  I thought it would be fun today to compare and contrast human and horsey (their word) systems of kinship, but with this caveat: Altho these terms are ingrained over hundreds of years of use, they are by no means absolutely universal. Fanciers of different breeds look at things in slightly different ways…and indeed what we’ll be talking about today pertains most specifically to thoroughbreds. After all, their stud book dates back to 1751, 46 years before the recording of human births and deaths was required by British law…which should tell you something about how serious they took their horses!

95.4  What’s more, while the nomenclature and the thinking behind it predates scientific concepts of DNA inheritance, there is a small but vocal cadre of nudniks who believe traditional terms should be adjusted to reflect modern genetic principals. Most in the hobby are perfectly happy with the traditional system, but there is occasional agitation…even so, some terms are used by different individuals in different ways…sort of like: Well that’s how I was always told is was supposed to be done. As an outsider looking in, I am left with a somewhat confusing overall picture. Not that much different from human kinship, nez pah?

95.5  Chart 336 shows the pedigree for a horse named Asterisk. Its father’s entire family is called his sire line…its mother’s, his distaff line. Some of its ancestors are marked with an *, and this indicates what’s called the tail male line and the tail female line…that is, the the male line thru the sire and the female line thru the dam. Members of the tail female line are often called 1st, 2nd, 3rd, etc. dams. Less frequently, members of the tail male line are 1st, 2nd, 3rd, etc. sires. The other special term indicated is Asterisk’s maternal grandfather, who is called a damsire or broodmare sire.

95.6  There is no problem with mating horses that are related to each other…in fact, inbreeding as a description generally only applies out to 3 generations…altho some would say as many as 5. Since horses don’t have last names, the most compact way to refer to one is: “by Sire, out of Dam, by Damsire.” A horse is properly BY its sire and OUT OF its dam…the surest way to announce you’re a newbie is to say “out of Sire.” Obviously, this system is tilted towards the males…traditionally, it was thought that the male blood contributed more than the female. Why don’t you also mention the sire’s sire? Because if you’re talking about a particular male horse, you probably already know…seriously, that’s how they think of it. The entire pedigree is important of course, but this shorthand refers to the 3 most important ancestors: father, mother, and paternal grandfather. Its similar to the Spanish custom of double surnames…a combination of your father’s last name and your mother’s maiden name, which of course is her father’s last name, indicating her paternal line.

95.7   Because horses don’t get married, and can be bred to relatives both close and remote, the occurrence of individuals who are less closely related than full siblings, but more closely related than full 1st cousins, is much more prevalent than with humans. Human half-siblings, with CR of 1/4, are what you’ll most likely encounter falling in between full siblings (½) and full 1st cousins (1/8). True, with the breakdown of the traditional family unit, half-siblings are becoming more common…to the point where where the term “quarter siblings” has evolved to describe unrelated half-siblings “on the other side.” But in human kinship, “enhanced half-siblings” are unusual…that is, half-siblings related thru both paternal and maternal lines…not so in the equine sphere, where they are the frequent and of course intentional. Here are some basic definitions I plucked from the internet to get us started…

95.8  Chart 337 diagrams the first 2 definitions. The important thing to notice is that half brothers occur thru the dam only…on the other hand, what in human terms would be half-brothers sharing the same father are in horsey terms called by the same sire. They are not thought of as being “related,” at least to the extent that they cannot be described as “brothers.” This odd dichotomy is the result of the fact that a horse can have a dozen half-siblings thru its mother, but a 1000 or more thru its father. Thus, having a common sire is considered of less significance, at least as far as the terminology is concerned. As the old joke goes: Your horse is a grand nephew of Secretariat? Him and 5000 others…

95.9  Obviously 2 horses with the same sire are related in a genetic sense, and a few cranky reformers insist they too should be called half-brothers. The rest of the fraternity ignores them. As with language in general, it’s just a case of making oneself clearly understood…using the terms in a different, “better” way is going to get you in trouble, pure and simple. But what’s shown in Chart 337 is considered standard.

95.10   Chart 338 takes the “half sibling” (AC) and “by the same sire” (DF) cases back a generation. In human terms, all these would be called “enhanced half-siblings”…A and D would be “3/4 siblings,” half-siblings on one side, 1st cousins on the other, for a total Coefficient of Relationship of 3/8. The others would be half-siblings and half-1st cousins, a CR of 5/16. But what are they in horse terms…can we match these 6 cases to the definitions we are examining?

95.11  Well, the definition of brothers in blood fits D (same sire, dams full sisters) and A (same dam, sires full brothers.) The definition of three-quarter brothers fits only C (same dam, sires by same sire.) Presumably, B would be nothing beyond half brothers (same dam)…E and F would simply be by the same sire. But does this nomenclature hold across the board? Sadly, no. For example…

95.12  …let’s take Uncle Wiki…thankfully, they agree on the definitions of full siblings, half siblings, and by the same sire. They also concur on brothers in blood. But then all hell breaks loose. Their definition of three-quarter brothers fits B, not C…same dam, sires are half brothers (i.e. sires have the same dam, different sires.) But they go on to say that three-quarter brothers can also mean same dam, sires who are father and son…as shown at left as G. Whoa…didn’t see that coming…but there’s more.  

95.13  Uncle Wiki defines three-quarter brothers in blood
as the flip side of three-quarter brothers…that is, either same sire, dams that are half-sisters (i.e. dams have the same dam, different sires), which would be E…or same sire, dams who are mother and daughter…shown at right as H. Then there’s three-quarter genetic brothers…and that would be same sire, same damsire (i.e. dams have the same sire) and that gives us F. Notice that “dams have the same sire” presumably implies “but different dams,” otherwise we’d also be defining D, which they’ve already called brothers in blood.

95.14  Notice too that C, the only thing that our original definitions labeled as three-quarter brothers, has fallen thru Uncle Wiki’s cracks…it falls under no definition of three-quarters, and so is presumably nothing more than half-brothers. Quite a mess, no? And I’m sorry to say, it gets worse…Chart 339 summarizes our original definitions, Uncle Wiki’s…plus the style book for an organization called American Horse Publications, and 3 more sources off the net. A double dash “–” means the term was not defined.

95.15  Under Source 4, I and J are marked with an asterisk since we haven’t seen them yet. Below, I is what in human terms are called double 1st cousins…the parents’ generation is colored orange and green because they can be either a same-sex pair of siblings or one of opposite sexes. And J in human terms is an irregular double cousin relationship…full 1st cousins thru the dams who are full sisters, and half-1st cousins thru the sires, who are half-brothers (i.e. same dam, different sires)…presumably it wouldn’t count if the sires were by the same sire (i.e. same sire, different dams.)

95.16  Like I said, quite a mess…for three-quarter brothers, not one of our 6 sources agrees. Uncle Wiki and Source 4 label B and E exactly opposite. G and H are considered either the same thing or 2 different things. At least brothers in blood and three-quarter brothers are 2 different things, right? Nope…Source 5 groups them together.

95.17  And the thing to keep in mind is: these terms are used routinely by horsey folk to describe various pedigrees, and they clearly mean something by them, and expect the reader will know what that something is. As sometimes happens, one column has grown into two…so next week I’ll take a stab at sorting it all out…till then, Hi-Yo G4BB, awaaaaay!…

Wicked Ballsy

So anyhow, here are 4  horses of a different color…top right is an especially dark “silver dapple”…bottom left is a “brindle”…you usually see that in dogs…top left and bottom right look to be “chimeras”…individuals with 2 different sets of DNA…like those cats you see whose faces are 2 different colors, split down the middle…

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Copyright © 2012 Mark John Astolfi, All Rights Reserved

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#94: As the Quarter-What-Again? Turns…

Dear Stolf: I’m sort of an off-and-on fan of the soap opera “Days of Our Lives.” Several years ago I tried to do a family tree but gave up. But I did print out something about “quarter” cousins and siblings…can you make any sense of it?  …from Laura, in Luketown 

94.1  Dear Laura: Ah, yes…somebody who never throws anything out…welcome to the club! The trouble with soap genealogies, apart from their bewildering complexity, is that they can change at the whim of the writers. Stuff like: A, who everybody (both viewers and characters on the show) thought was the child of B and C, turns out to be the child of B and D…or X and Y…or who knows who. I can’t find what you printed out to double check it…it’s long gone off the net it seems…but I did find 2 extensive charts at the Brady Family entry at Uncle Wiki…they needed 2 because it was so intertwined.

94.2  What I have done in Chart 334 is pull out the relevant parties, move them around to where they make sense, and change a couple of last names to maiden names. We’ll examine the claims you found based on this information…altho as we shall see, it is apparently not the whole story…big surprise!

94.3  Now the generation in question I have marked with solid blue squares…and their relationships depend on 3 members of the preceding generation, in solid red squares. The reds are pretty straightforward: Roman and Bo are half-brothers, as Caroline is their mother, and they have different fathers. Roman and John Black are 1st cousins, as their parents Colleen and Shawn are siblings. Bo and John Black are not related at all.

94.4  As to the blue generation, the first thing to notice is that left to right among their parents there is a massive half-sibling “tag team” deal going on…from Anna to Roman to Marlena to John Black to Isabella. Thus among the 4 blues…Carrie, Eric/Sami (full sibs), Belle, and Brady…each is a half-sibling to the ones directly to the left or right…and a quarter-sibling skipping one…thus Carrie and Belle are quarters, Eric/Sami and Brady are quarters. And all that checks with the second line of your info.

94.5  Normally, quarter-siblings aren’t related to each other…both are simply related, as half-saiblings, to someone else. But as can happen, these 2 sets of quarter-siblings are also 2nd cousins, since their fathers Roman and John Black are 1st cousins. Likewise, the 2 on the ends, Carrie and Brady, are also 2nd cousins for the same reason, but not fractional siblings of any kind since they’re too widely separated. Belle and Eric/Sami are also 2nd cousins, besides being half-siblings.

94.6   As to the first line of your info, it’s completely wrong, all of it. Belle and Brady are half-siblings, not quarter-cousins. As for Shawn-D, he is a half-1st cousin to Carrie and Eric/Sami, since their fathers are half-brothers…but he is of no relation to Brady (not half-cousin) and no relation to Belle (not quarter-cousin.) In fact, on the show Shawn-D and Belle had a child! That would be kosher for quarter-sibs who are by definition not related, certainly not for quarter-cousins…who are presumed to be related somehow.

94.7  This is all based of course on the Uncle Wiki trees…but there are 2 further complications, one small and one big, not shown on their charts. The small complication is the fact that Bo was raised as Shawn’s son, and thus as Roman’s full brother, altho unbeknownst to all he had the same mother but a different father…so Bo and Roman were not full brothers but half-brothers. But assuming Bo and Roman were full brothers, as everyone thought, that would make Bo and John Black 1st cousins, and their children Shawn-D and Brady 2nd cousins…which, if you’re using terminology that includes quarter-cousins, could mean they were “half-cousins.” Perhaps that’s what your info is saying. But that still doesn’t help with Belle…she is now 2nd cousin or “half-cousin” to Shawn-D (but still not quarter-cousin) and still half-sibling to Brady (not quarter-cousin.) So there’s that.

94.8  The big complication is another website I found here…dating back over 10 years, mind you…which  includes a very extensive DOOL genealogy, in text form as opposed to chart. And they say that Isabella Toscano, Brady’s mother, is in fact the daughter of Victor Kiriakis…who is Bo’s father…altho not thru Bo’s mother Caroline! And that gives us a whole new set of claims to evaluate, below in Chart 334A…

94.9  And sure enough, Shawn-D and Brady now are half-cousins (well, half-1st cousins) since their parents Isabella and Bo are half-siblings…so your original information is now correct. As to Belle is a quarter cousin…I still don’t see it…unless they mean “Your half-cousin’s half-sibling is your quarter-cousin”…and Belle and Brady are half-siblings. But notice also that Belle and Shawn-D are still not blood relatives in any way…now “quarter-cousin” is being used in the same sense as “quarter-sibling”…implying there’s a connection, but not by blood, but only by serial marriages. I’m not saying it’s right, but such an interpretation at least makes some sense….because from Shawn-D’s point of view, Belle is on the “other side” of his half-cousin Brady’s family.

94.10  What else do we have…Bo is John’s half-brother-in-law. That’s correct as far as it goes…hard to imagine someone in real life saying “Well, he’s only my half-brother-in-law”  but it’s possible. Bo is Brady’s half-uncle.  OK, that’s fine…since Bo is Brady’s mother’s half-brother. …and Belle’s quarter-uncle Yikes…and what’s next, quarter-father?  I suppose that’s logical in the sense that your quarter-cousin’s father would be your quarter-uncle…I suppose…but please don’t quote me. John is Shawn-D’s half-uncle also. Well, no…here we must put our foot down…John Black is married to Shawn-D’s half-aunt, his father’s half-sister. And the fact that John Black is also Belle’s father probably plays into this line of thinking somehow, but my head is starting to hurt, you know?

94.11  And that final line about Belle’s half- and quarter-sisterhood is correct, quarter-uncles notwithstanding. Anyway, interesting real-world….sorry, fake-world entanglements, nez pah?


Hey Stolf: Back in
91.3…love that indexing!…you were talking about the diagram for double half-1st cousins…and how without lines of descent crossing each other, you couldn’t put individuals of the same generation on the same horizontal line…because then you’d have to go, as you put it, “up and over.” I tried drawing it that way and don’t see what’s so wrong with it.  …from Pythagoria, in Octagon City

94.12  Hey back atcha, Pythy…is that what they call you? Yeah, that’s a pretty clever little diagram, I have to admit. To get everybody else caught up, in Chart 355,  A represents double half-1st cousins with no lines of descent crossing over…B shows everyone of the same generation on the same horizontal line, with the descent from mother A to daughter Z crossing other lines.

94.13  Now what I was thinking of when I wrote “up and over” is what’s shown in C…without those green arrows, it looks like A and Z are siblings, descended from someone above them…since these lines are presumed to proceed top to bottom, ancestor to descendent. In other words, “up and over” would make you think the relationships were what’s shown in D, which they aren’t. Personally, I’m gonna stick with crossed lines, if it’s all the same to you…a few don’t hurt, altho it could start to look like string-art, couldn’t it? But since I went to the bother of drawing it, here’s a pop quiz…in D, how has the relationship between X and Y changed? Next week, a real change of pace…or is that trot?…as we horse around some. À bientôt. 

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Copyright © 2012 Mark John Astolfi, All Rights Reserved

I’ll Tell Ya for $5 !!!!

Dear friends: Here’s the ad I put up on eBay classified today…

How are you and your grandmother’s cousin’s granddaughter related? If you have a PayPal account, I’ll tell you for $5. Yup, 5 bucks to finally know for sure…or confirm what you always thought. What’s $5…a bag of chips or a pack of smokes…almost a gallon of gas! Can you trust my answer? Do I know what I’m talking about? See for yourself…google G4BB (stands for Genealogy for Baby Boomers) and check out over 90 fun-filled posts. Here’s a sample of what you get…a color chart and a complete explanation emailed to you. And I’ll answer follow-up questions for free if there’s something you don’t understand. Send your question to stolfwx@hotmail.com, which is also my PayPal address.


Yup…this means YOU…$5 and I’ll sort it out for you…of course, you might find your answer somewhere here in my 90+  postings, but then sometimes even I can’t find something I’m looking for 😉 😉  So loosen those purse-strings and then at the next family reunion, you can prove them all wrong…ha ha!!

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Copyright © 2012 Mark John Astolfi, All Rights Reserved

#93: Polly-tix-ical Pow Wow

Dear Stolf: Now that Elizabeth “Fauxcahontus” Warren has dislodged Scott Brown from the Teddy Kennedy Senate seat in Taxachusetts…what’s the dang deal? Is she an Indian or isn’t she? …from Princess Fryingpanbottlewasher, in Pioneer Village

93.1  Dear Prinny: Colorful, the way you put it…very colorful. BTW, I remember you from the Howdy Doody show, as the rival of Princess Summerfallwinterspring. Say hello to Dandy Doody and Elephant Bob for me. But for those of you who missed all the hubbub, Bub, here’s a thumbnail sketch…

93.2  Elizabeth “Betsy” Herring was born in Oklahoma City in 1949. Growing up, and all thru grade school and high school, she was white. At college and law school, she was white. As a law professor at various institutions, thru the 70s and 80s, she was white. Then from 1985-1996, we was listed as Native American in various professorial directories. The Spring 1993 Harvard’s Women’s Law Journal went so far as to include her in a list of “women of color.” As late as 1996, a Harvard Law School spokesman was quoted as saying that their faculty of 71 included several blacks and Hispanics, 11 women, and one Native American…her. The Harvard Crimson touted her as the first minority woman to receive tenure there…altho today, all parties concerned say her purported ethnicity played no part in her initial employment and subsequent advancement. At some point thereafter, she went back to being white, and is to this day.

93.3  The obvious question is: how can this happen? Her political opponents branded her a bald-faced liar, who used her supposed heritage to further her academic career. Warren has admitted that there is no documentation as to her Indian roots…it was just “family history” that was told to her as she was growing up. The story was that grandpa Harry Reed and grandma Hannie Crawford had Delaware and Cherokee blood respectively…and this was such a bone of contention in their families that they had to elope to get hitched.

93.4  Chart 332 is her family tree. One Cherokee genealogist says that no-one in this tree was ever listed on any Federal census as anything but white…and none appears on any Tribal roll. That’s pretty much what “no documentation” means, you see. For herself, Warren has provided very few specifics…how much, if any, Indian blood did her grandparents pass on to her? I get the feeling even she doesn’t really know, “high cheek-bones” or no. She defended her listing as a minority as her wanting to “connect with people for whom this heritage is part of their hearts.”

93.5  And in the interest of full disclosure, I am one of those people. But then, a lot of us are. Growing up, there was talk from my French Canadian maternal grandfather that there was “an Indian in the family.” The likely candidate is his paternal grandmother’s father…that would be my 3G grandfather, giving me a 1/32 share, far too little to be granted official tribal membership. But so far no confirmation, no documentation…sadly, not even a name, as my grandfather’s grandmother’s birth is recorded as “illegitimate”…father unknown.

93.6  But to answer your question, I judge this whole deal to be a proverbial tempest in a teapot. What Brown’s camp called a “lie” was really just an exaggeration, pretty much what you’d expect in those first giddy days of the diversity fad. Yes, an exaggeration…I mean, my grandparents were born in Italy, Poland, and Quebec…yet I cannot say I am an Italian, a Pole, a French Canadian…nor certainly an Indian. I am an American, simply that…I was born here, what can I say? I have multi-ethnic heritage…but short of saying “Sure I cry at weddings, all Italian men do,” that’s as far as it can reasonably go.

93.7  The trouble is that the concept of “lying” has been dumbed down to the point where almost anything can be called that. For example, you state that you never took piano lessons. Then you later remember that for several months in the 3rd grade…50 years ago mind you…you did, but hated practicing so much your parents let you stop. So what you said at first was a “lie,” right? No, not right. Like I said, mountain out of a molehill.

93.8  But before we leave this, I’d like to make 2 comments about Chart 332. When the controversy first erupted in April of this year, the New England Historic [sic] Genealogical Society announced at least one instance of possible documentation, altho they later backtracked, since no actual document ever turned up. It was a supposed marriage application on which one William J. Crawford stated that his mother was a Cherokee. On Chart 332, he would be the younger brother of Preston H. Crawford…and it is interesting to note that at that time, the early 1800s, and depending on the prevailing local attitudes, it was not uncommon for someone to claim Indian heritage as a cover for what was actually black ancestry. I’m just sayin’…”of color,” get it?

93.9  And from a purely genealogical standpoint, an interesting article was written by Sally Jacobs in the Boston Globe in September…here. Among professional…and serious amateur…genealogists, the danger of substituting “family lore” for actual documentation is well known and warned against. And this article is from the angle of what other relatives knew or thought they did. But what I especially liked was that Ms. Jacobs got all the kinship connections correct, as far as I could tell. For example, the granddaughter of Everett Reed and Laura Crawford is called Warren’s 2nd cousin…completely correct.

93.10  She even points out, without actually using the word “double,” that the two women are more related than ordinary 2nd cousins…owing to the fact that 2 Reed brothers married 2 Crawford sisters.  Another relative, Robert C. Boraker, is called Warren’s 4th cousin since their great great grandfathers were brothers…again, precisely right. My point is, this stuff isn’t really that difficult…just a little thought, and you can’t go wrong…sez me.

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93.11  The latest from the wiseGeek “Cousins” page…and the Rule of G’s

93.12  Count the G’s to determine cousins. If the numbers are the same, you are simply cousins. Same Grandparents = 1 G = 1st cousins…same Great Grandparents = 2 G’s = 2nd cousins…same Great Great Grandparents = 3 G’s = 3rd cousins, etc. If the numbers are different, the lower one indicates cousins, and the difference between the 2 numbers is the times removed. So in your case, 3 G’s versus 4 G’s…that’s 3rd cousins once removed. Next week, we’ll see if we can soft soap ya…

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Copyright © 2012 Mark John Astolfi, All Rights Reserved

#92: Stump the Band

Dear Stolf: I understand that 2 people can be related only by a fraction which has a power of 2 in the bottom [denominator –ed.]…like 2, 4, 8, 16, 32, etc.  But given that restriction, is ANY fraction less than 1 possible?  …from Matthew Matics Jr. in Space City

92.1  Dear Matt: Off the top of my head, I would say yes. Now granted…”yes” is not a rigorous mathematical proof. I suspect one exists. But maybe not…and if you could show me how it is impossible, say, for 2 people to be 177/512 related, then that would settle it too. But to demonstrate the types of permutations you’d need, I thought I’d sketch out everything from 1/32 to 15/32. 16/32 is of course ½, which is parent/child or brother/sister. Greater than that requires interbreeding…that is, you’d need at least one individual whose parents were related to each other in some way. That would take you from 17/32 up to 31/32…identical twins have the same genetic makeup, so are considered the same person…related by 32/32 or 1.

92.2  I figured this assignment would be tricky but do-able. Then I ran in to a brick wall. Now the cool thing about blogging…as opposed to “dead tree” publishing, like books, magazines, or newspapers…is that if you’re not sure about something, you can still go ahead. No one wants to read a book whose premise is “I have no idea.” But anything goes on the net, and who knows, maybe the answer is out there, and I’d be gratified to hear about it. Or maybe I’ll find the answer myself anyway. But today’s blog is the proverbial “work in progress.”

92.3  So let’s dive right in. 1, 2, 4, and 8 are easily taken care of, in Chart 321. The others will be compounds of these basic relationships…because you and I can be related one way thru our fathers, and another way thru our mothers, and our total CR is the numerical addition of those 2 ways.

Chart 322 outlines what we need, and it looks on the surface to be pretty straightforward.

92.4  The first 3 run pretty much to form, as in Chart 323…father’s side + mother’s side. 7/32 is a little trickier…adding 3 numbers suggests 3 sides? Well, no, 3 lines…and we’ll get to that in a bit.  

92.5  Chart 324 gives you the simplest 8’s…notice on the father’s side, X and Y have the same father, thus they are half-siblings, with a CR of 1/4, which is 8/32.

92.6  And Chart 325 extends that idea, converting the mothers’ side into 2 lines…the mothers of X and Y are half-siblings thru their fathers, 1st cousins thru their mothers.

92.7  Chart 326 sums up what we have so far. Find 13, 14, and 15 and we’re done.

92.8  Only now there’s a problem. Let’s analyze how we got 11…we added 8+2+1…translated, X and Y are related by 1/4 or 8/32 on their fathers’ side…X and Y are half-siblings. So far so good. Now X and Y’s mothers are half-siblings thru their fathers, 1/4 or 8/32…and 1st cousins thru their mothers, 1/8 or 4/32. Since X and Y’s mothers’ are half-siblings, X and Y are half-1st cousins…1/16 or 2/32. And since X and Y’s mothers are 1st cousins, X and Y are 2nd cousins…1/32. So that’s 2/32 + 1/32 = 3/32 from their mothers’ side, 8/32 from their fathers’ side, total 11/32. It checks.

92.9  Remember from last week: when 2 people who are related have offspring, the offspring are related to each other by 1/4 the relationship of their parents. And that’s just what happened here…as both half-siblings and 1st cousins, X and Y’s mothers have a total CR of 8/32 + 4/32 = 12/32. And 12/32 divided by 4 gives 3/32…and that’s the CR X and Y get from their mothers’ side.

92.10  Now try applying that to 13 = 8+4+1…the CR that X and Y get from their mothers’ side must be 5/32, so their mothers must be related by 4 times that or 20/32. Well, if the mothers were both siblings, 16/32…and 1st cousins, 4/32…that’s 20/32…divided by 4, gives you 5/32. But can you think of a way, without interbreeding, that the mothers of X and Y can be both siblings and 1st cousins? I can’t. I tried…and I just don’t see it. I’m stuck. Then for 14 = 8+4+2, the mothers must be both siblings and half-siblings…and I don’t even want to think about 15 = 8+4+2+1.

92.11  I stared at Chart 326…how do I extend these patterns? I took it to bed with me…I got up in the morning with it. Nothing budged. One thought I had for the mother’s lines was to “eliminate the middle man”…since going down a generation reduces the CR, why not have X and Y’s mothers be of different generations…mother and daughter instead of sisters. That way, instead of “going down” twice, you only go down once, relative to the older one….you divide by 2, not by 4. But as you can see in Chart 327, it was an interesting idea, but it didn’t help.

92.12  At this point, I wondered if 13, 14, and 15 were possible even with interbreeding.Well, I shouldn’t’ve worried, because they are, as you can see in Chart 328where the colors of the sexes have been changed, suggesting we’re now in ancient Egypt or someplace.

And in Chart 329, the inner workings are spelled out. For 13, X and Y have married grandparents who are half-siblings…for 14, their married grandparents are full siblings…and for 15, their married great grandparents are full-siblings, as are their married grandparents. And notice what happened with 14…we said in 92.10 that we would need X and Y’s mothers to be both siblings and half-siblings, which is impossible. What they can be is siblings and double 1st cousins…and double 1st cousins have the same CR as half-siblings, 1/4 or 8/32.

92.13   Now one thing you can do when you’re stuck like this, you can “take a running start at it”…by which I mean, go back to simpler cases…see how they work out…and see if that suggests anything. So I retreated from 32nd’s…first back to 8th’s…and that looked kosher up to ½, nothing missing…

92.14   Next, 16th’s…and sure enough, 7/16 is missing…and we can start to sketch out the problem. For 7, we need a CR contribution of 4 from the fathers and 3 from the mothers. Now the fathers’ side is maxed out…X and Y have the same father…their fathers are related to each other by 1, since they’re the same person. And if you can see how 2 people could be more closely related than by being the same person, I’d sure like to hear about it! Using the divide-by-4 rule, X’s father and Y’s father are related by 1…so X and Y are related by 1/4…they are half-brothers thru their father. (Remember, full brothers are half-brothers thru their mothers and their fathers… 1/4 + 1/4 = ½.)

92.15  That leaves 3 we have to get thru X and Y’s mothers…but from the divide-by-4-rule, their mothers would need to be related to each other by 4 times that much…or 12/16…and full sisters are only 8/16. What it’s starting to look like is, you can’t get the CR’s between “3/4 siblings” and full siblings without interbreeding. So for example, with 64th’s you could get 24 for 3/4 siblings, and 32 for full siblings, but you couldn’t get 25 thru 31 without some kind of interbreeding…and if you’d like to check that  for me, I’d be super-delighted, you betcha.

92.16  So maybe that’s a small insight, one piece of the puzzle…or maybe I’m completely off base. I feel as if I’m missing something obvious, but I can’t put my finger on it. At the beginning, it seemed reasonable to think that if you could get 16/32 without interbreeding, you wouldn’t need interbreeding to get CR’s less than that…but so far, that intuition appears wrong. Next week, the letters will be really EASY, I can just feel it… 😉 😉  buenas nutcase…

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Copyright © 2012 Mark John Astolfi, All Rights Reserved