Pefsu problem explanation

Question

Problem no. 12 from Moscow Mathematical Papyrus:

1. Example of calculation of $13$ heqats of grain
2. If someone says to you: Take $13$ heqats of grain to make them into $18$ jugs of beer
3. Note that the amount of grain for $1$ jug is $2\frac{1}{6}$.
4. Reckon with $2\frac{1}{6}$ in order to find $13$.
5. The result is $6$ times.
6. Reckon with $6$ to find $18$.
7. The result is $3$ pefsu and this is the solution.

The only ratio that is sufficient for solving these problems is the following:

$$\text{pefsu} = \frac{\text{number of jugs of beer}}{\text{number of heqats of grain}}$$

The problem is essentially looking for the pefsu of those $18$ beers that need to be made. I am stuck at the line $6$ where I don't understand why the author divides $18$ by $6$ to get a pefsu of $3$. With that logic it would be that not all $13$ initial heqats of grain were used to produce $18$ jugs of beer since the pefsu formula would then be $$3 = \frac{18}{6}$$, i.e. only $6$ heqats were used to produce the needed number of beer. Anyone can explain?

Here is a reference from a book.

Answer 1

Some preliminary observations:

Richard Gillings (Mathematics in the Time of the Pharaohs, p. 246) says that this problem on pefsu of beer is unclear (although he contradicts himself on the next page, saying that problems 12, 15 and 16 are "clear and simple"). He also claims that "The scribe of the Moscow Mathematical Papyrus in the view of Egyptologists was a very bad writer".

Annette Imhausen (in the section on Egyptian mathematics in The Mathematics of Egypt, Mesopotamia, China, India, and Islam. A Source Book) says that pefsu of beer varies between 2 and 6. This means that 1 heqat of grain can be made into 2-6 jugs of beer.

Marshall Clagett (Ancient Egyptian Mathematics, the book that is linked to in the original question, p. 63) states: "Some sort of similar diluted beer is found in 8 of the 10 pefsu problems in [the Moscow Mathematical Papyrus]. (...) It usually seems to be made with a strength of pefsu 4. It is ordinarily compared to a stronger beer (and thus one having a lower pefsu number) which is specified as being made with Upper Egyptian Grain and having a pefsu of 2."

The text of Problem 12, linked to in the original question, is more complex than the summary provided above. In particular, it mentions Upper Egyptian grain, Upper Egyptian beer, and malt-date beer.

Suggestion:

One possible interpretation of the problem, which seems to make sense of the different data, is that we are given 13 jugs of beer with a pefsu of $2 {1\over 6}$, made from Upper Egyptian Grain, and we are to dilute in into 18 jugs, which would then be similar to malt-date beer. What is the pefsu of the diluted beer? Answer: 3 pefsu.