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. The book specifies this as a simple problem, so I am looking for an answer, because it makes no sense.

Answer 1

I came across your question searching about Pefsu problems. I am interested into it from a mathematics perspective. I like history but I am by far not an expert into ancient Egyptian beer making, so i will make some assumptions. So let me offer you what I believe is the explanation. I believe the issue is the difference on how a modern person would use the formula and how an ancient Egyptian wold.

The ancient Egyptians would have no issue to use Pefsu for the strength of beer. After all, all they cared about is how much raw material (heqats of grain), will be converted into finished product (jugs of beer). Thus you get the Pefsu which for them would be just a number (and most likely approximated to the closest integer or some simple rational) and it is measured in jugs per heqat.

To our modern sensibilities this strikes as somewhat odd. This is for several factors, such as:

a) The strength (alcohol content) of beer will be or less standardised by the type of beer you want to produce. Therefore it will be a constant in the calculation and not a variable as for ancient Egyptians (in fact, that is the quantity they want to determine). Note that nowadays we have a much better theoretical understanding of fermentation and can use simple scientific instruments that can estimate quite accurately the level of alcohol in the mixture. I am not sure of ancient Egyptian methods but I would suspect they would follow a fixed recipe that produced a fixed ratio concentrate, that would then be diluted accordingly. Thus we would accept the strength of the beer and vary the volume, whereas ancient Egyptians would like to keep the volume the same and vary the strength.

In fact, the problem hints to that and that is what is essentially missing from the problem description: note step 3 "One jug of beer requires 2 and 1/6 heqats of grain". Assuming that the process remains stable (number of fermentation etc.), we can assume that the amount of grain is at the very least a good approximation to the alcohol content of the beer.

b) Regardless if the alcohol level is fixed or not, we desire to have meaningful measure units in modernity. Since we wish to compare two volumes, it would make sense to use the same unit of measurement. Thus, we use step 3 to convert heqats of grain to jugs of beer. Then the ratio would be an absolute number, i.e. without units, which is what we desire.

To summarize, a modern person would say that by using 13 heqats of grain and following a recipe that converts 2 and 1/6 heqats of grain to 1 jug of beer, we may obtain 18 jugs of beer of a strength of 3 Pefsu (as an absolute number)

An ancient Egyptian trader might be fine to skip step 3 and give the strength result in jugs per heqat, thus derive that if he is given 13 heqats of grain and must deliver 18 jugs of beer, the resulting beer will have a strength of 18/13 = 1 + 5/13 Pefsu (as jugs per heqat).

In both cases the full amount of 13 heqats of grain is used, what changes is the unit we measure the strength of the beer with.

Note that we may alternate between the two units by multiplying by the strength of the beer recipe, which is 2 + 1/6 = 13/6 heqats per jug.