Gilgamesh’s lineage and denominators of power two

Question

So I was watching the wonderful show “Um, Actually” and they mentioned that, as bonkers as this sounds from a genealogical perspective, Gilgamesh was said to be 2/3 God. Not half, not 3/4, 2/3. The host jokes that this is mathematically nonsense, but I’ve been musing now about how to approach the problem of finding a series or, I guess, a sequence, that converges to 2/3 using only averages of two numbers that have a denominator that is a power of two. Another way of approaching it is that the initial phase may only be a set of ones and zeroes and they are paired and averaged with themselves and descendants until we can be shown to approach 2/3. I guess if we think about it abstractly we can say that we can just assume that at some level x ancestors away from Gilgamesh, roughly 2 in 3 of these ancestors are gods while the others are mortals. Just a goofy exercise of course, but how might you explain, mathematically, the statement that someone might be not a half-god, or demi-god, but just a bit more. Hercules, step aside, how would you demonstrate that Gilgamesh is 2/3 God?

Answer 1

Suppose at $n$ generations ago, there were only gods and not-gods. If Gilgamesh’s ancestors $n$ generations ago consisted of $k$ gods, then Gilgamesh would be $k/2^n$-god. In order to get this proportion to be close to $2/3$, we set $k$ to be $2^n/3$ rounded to the nearest integer. This proportion is never exactly $2/3$, but gets closer as $n$ gets larger.

But what if we want exactly $2/3$? There is a way to do this if we don’t restrict ourselves to a generation which is entirely god or not-god. The easiest way to do this is to have Gilgamesh’s grandparents consist of two gods, a not-god, and a $2/3$-god. This works because $$ \frac{1+1+0+2/3}{4} = 2/3 .$$ Of course you have an issue because you need to explain the grandparent’s ancestry, but I think we can afford to gloss over this detail of infinite recursion when discussing mythology.