# Can you be 1/12th Cherokee?

I was watching an old Daily Show clip and someone self-identified as "one twelfth Cherokee". It sounded peculiar, as people usually state they're "1/16th", or generally $1/2^n, n \in \mathbb{N}$.

Obviously you could also be some summation of same to achieve 3/32nds, etc., but will an irreducible fraction with the numerator 1 always need a power-of-two denominator? More generally does it always require a power-of-two denominator?

1. Nobody can trace their lineage infinitely
2. Incest is OK, including transgenerational, but someone can't be their own parent.
• With enough incest you can be any rational you like. – Alec Teal Jan 18 '15 at 18:48
• If your otherwise non-Cherokee grandmother went back in time, found her Cherokee grandfather, and had a baby (her father) with him, then she would be $\frac13$ Cherokee and you would be $\frac1{12}$ Cherokee. That's probably what happened. – Greg Martin Jan 18 '15 at 19:08
• Chuck Norris is 1/12 Cherokee. – Asaf Karagila Jan 18 '15 at 19:21
• I guess someone did the arithmetic $\frac{1}{4} + \frac{1}{8} = \frac{1}{4 + 8} = \frac{1}{12}$. – azimut Jan 18 '15 at 21:07
• Or he could spend 1 month a year as a Cherokee. – fluffy Jan 18 '15 at 21:33

This depends on the model. Instead of arguing that we have only $46$ chromosomes and cross-overs or whatever the mechanism is called are not that common, let us assume a continuous model. That is, a priori, everybody can be $\alpha$ Cherokee for any $\alpha\in[0,1]$ and the rules are as follows

• Everybody has exactly two parents.
• If the parents have Cherokee coefficients $\alpha_m, \alpha_f$, then the child has $\alpha=\frac{\alpha_m+\alpha_f}2$
• In a sufficiently large but finite number of generations ago, people had $\alpha\in\{0,1\}$

It follows by induction, that $\alpha$ can always be expressed as $\alpha=\frac{k}{2^n}$ with $k,n\in\mathbb N_0$ and $0\le k\le 2^n$. Consequently $\alpha=\frac1{12}$ is not possible exactly (though for example $\frac{85}{1024}\approx\frac1{12}$ would be possible). It doesn't matter if there is any type of inbreeding taking place anywhere in the tree (or then not-tree) of ancestors. The only way to obtain $\alpha$ not of this form would involve time-travel and genealogical paradoxes: If you travel back in time and paradoxically become your own grandparent and one of the other three grandparents is $\frac14$-Cherokee and the others are $0$-Cherokee, you end up as a solution to $\alpha=\frac{\frac14+\alpha}4$, i.e. $\alpha=\frac1{12}$.

• Any genealogy solution that requires time travel is a good solution. – fluffy Jan 18 '15 at 20:58
• @DeadMG Inbreeding doesn't do the trick, because there's no difference between having one person appear multiple times in a family tree and having many people with the same coefficient each appear once in the tree. For instance, if your grandfathers are the same person, and he has $\alpha=1/4$ with both grandmothers having $\alpha=0$, that's equivalent to having no inbreeding and having both grandfathers' $\alpha=1/4$, with your $\alpha=1/8$ in both cases. The grandfather appearing twice contributes fully half of your Cherokeeness, not a third. – cpast Jan 18 '15 at 23:16
• Wait, didn't "I'm My Own Grandpa" show you could be your own grandpa without the need for time travel? – Keavon Jan 19 '15 at 6:06
• AFAIK biologists are reasonably in consensus that you can't rigorously define/measure race by DNA. Nevertheless, there's some simple arithmetic behind legal and social definitions of race, and it typically behaves as Hagen modelled it. The fact that it makes no sense in terms of DNA or any other physical process is irrelevant :-) It's biologically possible, of course, that the twelfth-Cherokee person in the question is saying so on the basis of DNA sequencing that reveals they have 1 in 12 of somebody's list of "characteristic" Cherokee genetic markers. If such a list exists or is invented. – Steve Jessop Jan 19 '15 at 11:59
• @Keavon: Technically, the protagonist of that song is his own step-grandfather, not his own biological grandfather. – Dan Jan 26 '15 at 3:16

No, nobody can be $1/12th$ Cherokee.

I'll prove a stronger statement:

For any natural $p$ and $q$, one can't be a $p/q$ Cherokee if $p$ is not divisible by $3$, and $q$ is divisible by $3$.

First, the statement that a person can be only "rational number of a Cherokee" can be easily proven - I won't waste space for the proof here.

Now, let's suppose a person was such $p/q$ Cherokee, then, supposing that his/hers parents were $p_1/q_1$ and $p_2/q_2$ Cherokee (fractions are reduced to their minimal form), the following would be valid

$$p/q = (p_1/q_1 + p_2/q_2)/2$$

or

$$2\times p\times q_1\times q_2=q\times (p_1\times q_2+p_2\times q_1)$$

Since $q$ is divisible by $3$, and $p$ is not divisible by $3$, this means that one of $q_1$ and $q_2$ must be divisible by $3$. In other words, one of the parents have the same property as the original person.

And than you can go ad infinitum - but since the oldest ancestor can be either 100% or 0% Cherokee, and that person does not satisfy the property of being "$p/q$ Cherokee if $p$ is not divisible by $3$, and q is divisible by $3$", this is a contradiction.

This means nobody can be $1/12th$ Cherokee.

• You're making too many assumptions. Two things interfere with your model. 1) You don't have to go back too many generations before some overlap in relatives will occur. 2nd or 3rd cousin marriages used to be semi-frequent, and don't even cross that "taboo" threshold. 2) Generations aren't static. One side of one's family could have habitually married and reproduced young, to include 5 or 6 generations in a century, while others in the family tree may have waited until late in life, creating only 2 or 3 generations in the same timeframe. – Random Jan 21 '15 at 23:13
• @Random Not at all, I don't make any assumption of that kind... – VividD Jan 22 '15 at 4:54
• In it's simplest form, here's how it could work going back to the great-great grandparents generation. Brothers Don, Dan, David & Duke. They marry four girls, Darla, Destiny, Delilah, & Daisy (unrelated). Don & Darla have a boy, Carl. Dan & Destiny have a boy, Cameron. David & Delilah have a girl, Cathy. Duke & Daisy have a girl, Cindy. Unfortunately, each of the women die in childbirth, leaving their husbands to remarry four other (unrelated) women, and they have more children. – Random Jan 22 '15 at 19:52
• Don & Daphne have a boy, Calvin. Dan & Destiny have a boy, Cuthbert. David & Danielle have a girl, Candy. Duke & full-blood Cherokee Dances-With-Elk have a girl, Cinnamon. Here we have now a group of 8 people, who can constitute the grandparents. While four of the pairings are half-siblings, those can be avoided. So, we start with a pool of 12 individuals. Correct, 4 of them contributed twice, but if we just count the individuals, this is how it COULD work. – Random Jan 22 '15 at 19:58
• Ahahahah that stronger statement rocks! ;) – MattAllegro Jan 24 '15 at 22:03

Base 2 numbers! If someone is $a$ of race A ($a\in[0,1]$) and someone else is $b\in[0,1]$ race A, their offspring is $\frac{1}{2}(a+b)=\frac{1}{2}a+\frac{1}{2}b$.

This immediately screams "base 2" because you'll get this recursive half pattern.

So let's write something in base 2, take 101 this is 1/2+1/8 (there's a 0 in the 1/4 column) which is 5/8ths as binary, we halve it, which in base 2 is just shifting everything right once, to get 0101 and then add it with the other parent (after shifting theirs).

For example: 1x0 (where x means "have baby with") is 01+00=01=0.5.

The operation of x is closed with finite binary strings - which I will call "race strings".

Adding two terminating numbers is a terminating number.

That means this is closed. It's like the integers in the real numbers, using adding you cannot escape the integers, from inside them. Same sort of closure.

To be 1/3rd is not a finite race-string, so cannot have come from two finite race strings. QED.

Here's how I got to the answer

It could be a fraction so close to 1/12 it's easier to say 1/12 than say "21/256ths"

Lets take "race A", if someone who is x/y race A and someone who is a/b of race A, their offspring is $0.5\frac{x}{y} + 0.5\frac{a}{b} = \frac{1}{2}(\frac{x}{y}+\frac{a}{b})$

But $\frac{x}{y}$ and $\frac{a}{b}$ must also come from this relation.

Now 1/3 (1/12 = 1/4 * 1/3) is a recurring number expressed in base 2 (in base 10 it goes 0.1s then 0.01s, then 0.001s and so forth, in base 2 it goes 1/2, 1/4, 1/8...)

So say we wanted someone who was 1/2 + 1/8 of race A, that'd be "101" in binary in this form, it terminates, the 3/4 used in the comments, that's "110" it terminates.

Remember the relation above, if someone is "0110" (3/8) and someone else is "1100" (3/4) say, we get the result by shifting one right and adding, in this case

 "00110"
+"01100" which is "01001" or 9/32,


So to be 1/12th would mean someone who was a quarter, and someone who is a third, but as you can see no one can be a third (in finite steps) starting from 1 or 0 of race A

To sum up! To be 1/3rd something (an infinite string of 0s and 1s) you can't have come from the "product" of two people who have finite strings representing their race. We've seen that "finite strings" are closed (2 people of finite-race string produce someone of finite race string) and thus can't have been produced by two people of finite strings.

• On the other hand, if 1/12th is an estimate, he could be 11/128th Cherokee, which is within 4% of 1/12th. or 21/256 is within 3%, 43/512 is within 1%... – Mooing Duck Jan 20 '15 at 23:22
• I used 21/256ths but whatever – Alec Teal Jan 21 '15 at 0:30
• "To be 1/3rd is not a finite race-string, so cannot have come from two finite race strings" - if there are multiple potential fathers with a chance for each of them to be the real father, you can kind of get 1/3rd. :P – Peter Jan 24 '15 at 12:24
• There are other ways of reaching ${ 1 \over 3}$: bbc.com/news/health-31594856. – copper.hat Apr 22 '15 at 23:54

You can get as close as you like to any fraction. For example, you can be between 1 / 12.5 and 1 / 11.5 Cherokee, and then claiming "I'm one 12th Cherokee" in every day language would be correct.

Also, the fractions that we usually use are approximations. If one parent is pure white and one parent is pure black, the child isn't half white and half black. It will be statistically close to 1/2 each, but in reality the genes are not taken exactly half from each parent, so that child will be a bit more in one of the two directions and not exactly in the middle.

• Actually, children are genetically exactly 50% related to their parents, barring genetic problems. For chromosome 1, each parent contributes exactly one copy, and the kid has two, one from each parent. Recombination may occur, but since both chromosomes are from the same parent, genetic contribution is always 50/50. Percentages can be mixed up at the grandparent-grandchild level, though. – fastmultiplication Jan 20 '15 at 15:32
• Makes sense to me. I've always told people I'm 1/3rd French: one grandparent + one great-great-grandparent. It's not literally true, but it's close enough. I guess that'd make my own grandkids 1/12 French, even if the actually number is closer to 1/13 (probably there's some forgotten great-...-great grandparents out there somewhere). – Mark Jan 20 '15 at 21:43
• If only one great grandparent counts as French, then you're exactly 1/8th French. Your own grandkids will be exactly 1/32nd French, assuming no other French descendancy enters the tree. The grandparent doesn't add anything. If you count the grandparent as full French, then those fractions become 1/4th and 1/16th respectively. – Mooing Duck Jan 20 '15 at 23:25
• @MooingDuck If the grandparent is full French and another great-grandparent is full French, that means you have 1/4 from the grandparent and 1/8 from great-grandparent for 3/8 French. However, in his case, it's a great-great-grandparent, so it's $1/16+4/16=5/16$ French, which is close enough to a third. – cpast Jan 21 '15 at 18:58
• @cpast: I had assumed that the grandparent was a descendant of the great-grandparent, but I realize now that this was not necessary a correct assumption, I stand corrected. – Mooing Duck Jan 21 '15 at 19:03

In a "continuous inheritance" model, no, because of the answers above.

In a naive chromosomal model, no, because 46 is not divisible by 12.

In a model which includes recombination, yes. Recombination (aka crossing over) between the parent's chromosomes results in a new chromosome type which has parts from one grandparent and parts from the other (usually one section from each). It is frequent and random enough that people by heritage 1/8th Cherokee would actually have a pretty big variation in their amount of genetic relatedness to their one 100% Cherokee g-grandparent. The diagrams here illustrate actual inheritance of one grandparent's genes: http://www.dnainheritance.kahikatea.net/autosomal.html - which shows that the variation in relatedness to a particular grandparent is huge.

The way real recombination works is actually pretty complex, though, and not purely random. Females recombine more than males (it's not just 50% of the time - females are ~45% something percent, males ~35%), and there are recombination hotspots which are locations which "prefer" to be the site of a cross somehow. Plus, there is lots of other selection going on even after fertilization. But these don't make it impossible to be 1/12, since they probably aren't monitoring the whole genotype.

On the actual "active gene" level, though, while you can do it, it may not mean much (since naively in terms of genes only, we are already extremely closely related to all other humans) It would make more sense to look for people who are 1/12 of the way genetically between two groups. i.e. take a Cherokee variant for 1/12 of all the genes which are not fixed in either Cherokee or the target group (being careful to spread them out). If you did the selection in a way which locally (within one chromosome) obeyed a plausible recombination history, this would be indistinguishable from somehow who just got lucky to be exactly 1/12 Cherokee.

Assuming that we define your heritage as 1/2 (father + mother), then it is not possible to come up with exactly 1/12. Assuming we start with people who are full Cherokee or not Cherokee at all, i.e. 0 or 1, then all their descendants are going to be some integer over a power of 2. A power of 2 can never reduce to 12, as 12 is a multiple of 3 and a power of 2 can never be a multiple of 3.

That said, someone could conceivably be a fraction that rounds to something close to 1/12. For example, suppose a Cherokee marries a non-Cherokee. The result is 1/2 Cherokee. This person marries a full Cherokee, result 3/4. Marries non, result 3/8. Marries full, result 11/16. Marries non, result 11/32. Marries non, result 11/64. Marries non, result 11/64. Marries non, result 11/128. That's pretty close to 1/12.

But THAT said, to come up with a number that is even approximately 1/12 you have to have detailed knowledge of your heritage with many inter-marriages going back at least 6 or 7 generations, and few people would have that much detail. So if you were (approximately) 1/12 Cherokee, I'd guess you wouldn't know it.

Years ago I helped develop a computer system for an Indian reservation, and they measured "bloodedness" in 8ths, which requires knowing your ancestry for 3 generations. They didn't cut it any finer than that.

• This should be accepted as the correct answer. Despite the incorrectness of other answers because of certain assumptions, that the Indian tribes themselves don't go any less than a 1/8th means that any self-described 1/12th Cherokee has some likelihood of being incorrect. – Random Jan 21 '15 at 23:02

It depends on how you define being Cherokee.

As others have shown, no possible normal breeding sequence can produce someone who is 1/12th.

However, we now have three-parent children (26 chromosomes from a male, 26 chromosomes from a female, an ovum with only the mitochondrial DNA from another female.) If one of those three parents is 1/4th Cherokee you could call the child 1/12th Cherokee.

Genealogy and ancestry is fun. My sister did one of those DNA tests that places your geographic location genetically. Here are her results:

30% Great Britain, 29% Scandinavia, 20% Ireland, 8% Europe West, 6% Finland/NW Russia, 4% Europe East, 3% Iberian Peninsula.

Obviously, those don't fit nicely into the 1/(2^x) model. Part of this is because they are larger region (Scandinavia covers multiple countries for example). Another part of this is the fact that, genetically, these regions are similar enough to have some crossovers.

• This is anecdotal, not really an answer, and doesn't actually speak to any of the OP's (mathematical) questions. (e.g., '30% Great Britain' could easily mean 5/16ths Great British up to some rounding error). – Steven Stadnicki Jan 21 '15 at 20:36
• I would accept the idea that 30% GB is due to some rounding error if the values didn't total 100% together and the error was less than 1% offf (5/16 is 31.25%). As I stated at the end, a large amount of genealogical errors come from crossover between regions. This question is as much sociological as it is mathematical and that blurs the perfect nature of the science. – tfitzger Feb 2 '15 at 17:35

There is more than one way to count genetics. Suppose that 120 years ago all four pairs of your mother's great grand parents (8 in all) were pregnant with their first child, and both pairs of your father's grandparents (4 in all) were pregnant with their first child.

Thus at that moment in time you have 12 ancestors, of mixed generations back, just entering child-bearing years.

If exactly one of those twelve ancestors is full-blooded Cherokee, and the other eleven have no Cherokee blood, then:

Yes, you are clearly 1/12th Cherokee.

Whether you are counting back in even generations or in even years is an arbitrary choice, and it is incorrect to state that only one makes sense. Perhaps there are other valid ways to count ancestry back that would enable other fractions and counting.

• Unfortunately this clever solution does not work. Go back another generation on the father's side. There are eight people, and by assumption, six of them are 0% Cherokee and two of them are 100% Cherokee. Now there are sixteen total ancestors in that generation and two are Cherokee, so you are 1/8th Cherokee, not 1/12th. – Eric Lippert Jan 19 '15 at 21:42
• @EricLippert: It depends on whether you are counting backwards to a point in time, or to a number of generations. For many people it makes more sense to count backwards to a point in time, for which this means of counting makes sense. As others have pointed out, all methods of counting ancestry are approximations at best since 99% of all our genes are identical across the entire human race. – Pieter Geerkens Jan 19 '15 at 23:26
• When calculating ancestry portions, I find it most natural that half of my ancestry comes from my mother and the other half from my father. If you take this starting point with "pure" ancestors a couple of generations back (full Cherokee or fully non-Cherokee), you can only get numbers of the form $a/2^b$. Your model seems to make your ancestry percentages dependent on the time of history you compare yourself to (the moment when you had 12 ancestors alive in your example). – Joonas Ilmavirta Jan 20 '15 at 10:28
• I agree with Joonas -- this characterization doesn't make a lot of sense to me. With this characterization, I could be 1/12 Cherokee one day and some completely different fraction the next, and some completely different fraction the next, and so on. What sense does it make to say "I was 1/12th Cherokee on August 1st 1830, but 2/25ths Cherokee on August 2nd 1830"? – Eric Lippert Jan 20 '15 at 14:13

Not without inbreeding. If we assume not shared ancestry paths, then each generation you go back you double the number of ancestors, thus 2^n where n is the generation. taking the previous 95 generations and dividing each by 12 reveals that there are no members of (this part) of the series that are divisible by 12. Interestingly, all moduli are alternately 4 or 8. While this is not a complete mathematical proof, beyond a few generations it is hard to determine race with much certainty anyway.

• Can you elaborate how this answer brings something new that is not in the other answers already? (It might just be that I'm slow, but I don't see it.) – Joonas Ilmavirta Jan 20 '15 at 10:25
• I was noting that inbreeding might make it possible, not something I spotted in other comments (though maybe I missed it). Also I took a different approach to the answer as my mathematic skills are not as good as others and I thought the recurrent modulus of the series was interesting. Worth a downvote? sorry you thought so! – Blake Jan 21 '15 at 15:16
• @Blake The problem with this answer (I didn't downvote it, but I didn't upvote either despite the passing note about inbreeding) is that everything but the (almost empty) first sentence has already been said by someone else, and by and large said better elsewhere (there are very straightforward mathematical proofs that don't need 95 generations of evidence - you could even have proven your 'alternately 4 or 8' with a quick pair of multiplications!). – Steven Stadnicki Jan 21 '15 at 23:06

Mathematically you can't. Let $a$ and $b$ mark the number of generations since the last time each of your parents had exactly one 100% Cherokee ancestor. Let's assume that

$\frac{\frac{1}{2^a} + \frac{1}{2^b}}{2} = \frac{1}{12}$

$\frac{1}{2^a} + \frac{1}{2^b} = \frac{1}{6}$

$\frac{6}{2^a} + \frac{6}{2^b} = 1$

From now on, just for the sake of simplicity, let's presume $a > b$.

$6 + 6*2^{a - b} = 2^a$

$3 + 3*2^{a - b} = 2^{a - 1}$

$2^{a - 1} - 3*2^{a - b} = 3$

This last equation is true if you manage to find two different powers of 2 (both even numbers), either one a multiple of 3 (it's still even), whose difference is an odd number. You won't find one, we have a contradiction.

But as stated, this is a mathematical statement. Genetics work quite different.

I believe that strictly from a statistical-genetic perspective, the most we can do is to compare genes and give a probabilistic value of somebody belonging to a certain genetic pool. Just consider mutations and any inter-race breeding that has happened during milleniums to highlight the fuzziness of the notion of defining somebody's race in any exact quantifiable terms. In addition, the notion of being exactly 1/2 Cherokee from statistical-genetic perspective does not make sense since the distribution of genes is not 50%:50% as very insightfully pointed out in the comments by @SteveJessop. There seems to be no mathematically sensical way to say somebody is 100% or 0% Cherokee, even less 1/n Cherokees for any choice of integer value n > 1, by just looking at their genetic markup.

Thus I would say the definition of being a proud member of Cherokee is a matter of social and personal identification. This definition is gray, but at least it gives us sample members of Human Race that we can meaningfully call fully Cherokee or totally not being a Cherokee (while ignoring the grey area cases), which validates the use of fractions in defining somebody's degree of "Cherokeeness".

After thus laying foundational definitions, we refer to the curious cases of children with three parents http://en.wikipedia.org/wiki/Three-parent_baby:

Three-parent babies are human offspring with three genetic parents, created through a specialized form of In vitro fertilisation in which the future baby's mitochondrial DNA comes from a third party.

Such a child would have DNA from three people, and in a sense have three genetic parents. Let's assume one of the parents of such a child would be a Cherokee and two others "totally not Cherokees", and furthermore to make the case stronger, let's assume all three genetic parents would also participate in the upbringing and support of the child so that there really would be three parents in the strongest imaginable and possible sense. We could argue (though not with any mathematical rigor) that the resulting offspring would be 1/3 Cherokee. Now if she or he would produce an offspring, it would make some sense (even if not mathematically very rigorous) to say that offspring would be 1/6 Cherokee. Paring 1/6 Cherokee with non-Cherokee parent would give us an offspring that might want to call himself/herself 1/12 Cherokee.

Logically speaking, though, the mentioned case cannot be 1/12 Cherokee by this avenue because this controversial treatment option has not been in existence long enough for any such three parent child to have a grandchild.

• If the DNA comes from three parents, does it come in equal parts? And how much of all DNA is mitochondrial DNA, if we want to include it in our model? – Joonas Ilmavirta Jan 19 '15 at 6:51
• We should probably ask this in biology SE. As I mentioned, this all is rather loose-minded mathematical reasoning and reduces the complexity of genetics (I was asking the same question as you did in my mind). – FooF Jan 19 '15 at 7:17
• @JoonasIlmavirta: it's a small proportion, and it's not generally included in calculations of what proportion Cherokee someone is. If it were, then someone with a full-Cherokee mother and a not-at-all-Cherokee father would be slightly more than half Cherokee (and 2/3 Cherokee by the counter-factual assumption in this question of even distribution of DNA), since in the usual case we get all mitochondrial DNA and half nuclear DNA from our mother and half nuclear DNA from our father. – Steve Jessop Jan 19 '15 at 9:55
• It seems to me also that the question is non-sensical when attempted to be viewed with any mathematical rigour (unless we want to resort to obscurely ridiculous definitions). The simplistic talk of fractional Cherokeeness kind of implies the notion of somebody being 100% Cherokee and somebody 0% Cherokee in genetical sense which does not make any factual sense. The suggestion per comment on the question of somebody spending 1 month per year as Cherokee is the only definition of "1/12 Cherokee" that makes sense to me and that has nothing to do with genetics. – FooF Jan 19 '15 at 10:27
• @SteveJessop: Thanks for the insightful comment. I rewrote the answer to appear more rigorous, using social identity as the definition of one's "Cherokeeness" as there is obviously no genetical/statistical way to say somebody is 100% Cherokee. – FooF Jan 19 '15 at 11:18

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