TITLE: Can you be 1/12th Cherokee?
QUESTION [129 upvotes]: 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?
Assumptions (as per comments):

Nobody can trace their lineage infinitely
Incest is OK, including transgenerational, but someone can't be their own parent.

REPLY [2 votes]: 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.