To read the RMP, it is unwise to begin at RMP 56 -59, the seked slope problems. Ahmes tended to record shorthand arithmetic proofs, and not clear calculation statements. That is, to fully understand Ahmes' arithmetic statements and proofs in modern rational numbers another translation needs to be made. Gillings and 20th century researchers did not understand Ahmes' red auxiliary calculations. They cited proofs as 'ab initio' calculations. On page 87 "Mathematics in the Time of the Pharaohs" Gillings says:
" 2/7 = 1/4 1/28
The interesting thing about this ostracon is that the decomposition calculated on its is the unexpected 3-term one
2/7 = 1/6 1/14 1/21
and the author's technique in deriving it throws light upon the Egyptian method of using a reference number together with red auxiliaries when adding fractions. The decomposition is
Whoever inscribed the ostracon was doing just what the scribe of the RMP did in problems 28, 32, 32 and several others. The red beneath the 2/7 means "Take 3 as a multiple of 3, to give the reference number 21." He then multiplied the 2 (of line 3) by his multiplier to give 6 the reference number which he then partitioned 3 1/2, 1 1/2 and 1, each of which divides the reference number 21 in integers, and wrote out in red (line 4),"
But, what was actually going on? Was the information describing a calculation, or a proof? Line 3.0 attests the info to be a proof, since the origin of the answer
1/2 1/6 1/14 1/21
had not been previously given. So what was the origin of the answer?
Many of you may agree that Gillings consistently muddled proofs, suggesting them to be "ab initio" calculations.
For those that do not, insert modern rational numbers, and you may also see that the ostracon data proved the following calculation has been previously made:
with red colors of certain lines not specifying the precise details of the optimization method (that used the red auxiliary reference number 6).
That is, we know from the 2/n table and the aliquot parts of the LCM 42: 42, 21, 14, 7, 3, 2, 1 was inspected, with the numerator 12 decomposed by 7 + 3 + 2, facts clearly reported by Fibonacci, though not cited in every detail by Ahmes within shorthand notes.
In the 2/7 ostracon case there was only one possible solution using reference number 6. But in the 2/n table the reference number 4 was used. The alternative reference number created a second solution. But, which alternative was optimal? Neither is my answer. Both solutions had been optimized, the best that scribes could do --showing a few details of their calculations within their proofs.
Again, the RMP calculation for 2/7 can be best reported by:
2/7 (4/4) = 8/28 = (7 + 1)/28 = 1/4 + 1/28
using a type of rule that Fibonacci reported as one of his seven conversion methods 2,850 years later.
In closing, the aliquot parts of LCM 28 was inspected to find additive parts of 8, from 28: 28, 14, 7, 4, 2, 1 (again finding only 7 + 1).
Egyptian scribes were free to chose their own reference numbers, and their own best set of additive solutions to given numerators. Reporting scribal calculations, be it Ahmes work or another, outside of shorthand notational proofs, is a rigorous task that translators have been very slow to focus upon.