Why did the Egyptians not represent $2/3$ as a sum of unit fractions in the Rhind papyrus? The following is taken verbatim from the MathWorld Wolfram page on Egyptian fractions:

An Egyptian fraction is a sum of positive (usually) distinct unit fractions.  The famous Rhind papyrus, dated to around 1650 BC contains a table of representations of $2/n$ as Egyptian fractions for odd $n$ between $5$ and $101$.  ... The unique fraction that the Egyptians did not represent using unit fractions was $2/3$ (The Penguin Dictionary of Curious and Interesting Numbers, Wells 1986, p. 29).

Well, I kind of find this surprising, since it is relatively easy to compute
$$\frac{2}{3}=\frac{1}{2}+\frac{1}{6},$$
which, according to the cited webpage [equation (4)], can be obtained using the greedy algorithm.
I checked the Wells reference, and it only has the following story to tell:

$2/3$ - The uniquely unrepresentative "Egyptian" fraction, since the Egyptians used only unit fractions, with this one exception.  All other fractional quantities were expressed as sums of unit fractions.

I know that the answer to this question may be covered in some History of Mathematics book, but I currently do not have the expertise to gauge which authoritative sources to check.
MOTIVATION
It is currently unknown whether there is a number $m$ such that
$$I(m) = \frac{\sigma(m)}{m} = \frac{5}{3} = 1 + \frac{2}{3},$$
where $I$ is the abundancy index and $\sigma$ is the sum-of-divisors function.  If such a number $m$ exists, then $5m$ is an odd perfect number, where $5 \nmid m$.

For more information on the exact connection between the Egyptian fraction decomposition of $1$ with odd denominators, and odd perfect numbers, I refer the interested reader to the following answer to a closely related question by MSE user Thomas Bloom.

 A: They had a special symbol for 2/3, presumably because of frequent use, so there was no need to work out its representation.  See chapter 7 of Annette Imhausen's Mathematics in Ancient Egypt: A Contextual History, Princeton University Press, 2016. 
A: The first fractions Ancient Egyptians used were $\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{3}$, $\frac{1}{4}$ and $\frac{3}{4}$. They used special words for these natural fractions.
As techniques of calculation developed unit fractions were introduced. Notation was concise except previously established symbols for natural fractions. Since $\frac{2}{3}$ and $\frac{3}{4}$ already had designators, these weren't broken down into unit fractions. Egyptians just continued using them in the old way. Nevertheless, after some time, $\frac{3}{4}$ began to be expressed like other fractions, but somehow $\frac{2}{3}$ remained a exception.
I based my answer Chapter I. The Egyptians from Waerden, B. L. van der. (1988). Science awakening. Dordrecht, The Netherlands: Kluwer Academic Publishers and https://www.bibalex.org.
P.S. It's a curious thing to see what we are talking about. Ancient Egyptians wrote unit fractions by placing ovals (not "1") above numbers. Below regular $\frac{1}{12}$.

And here are the exceptional $\frac{1}{2}$, $\frac{2}{3}$, $\frac{1}{4}$, $\frac{3}{4}$ and normalized $\frac{3}{4}$ respectively.
, , , , 
