First he notes that in every multiplication of fractions we have 6 items (3 numerators and 3 denominators).
Ex. a/b * c/d = e/f
[Ex using hebrew letters: א/ב * ג/ד = ה/ו]
[The using of the hebrew letters instead of a,b,c, etc is how the Gaon writes it and imho is a critical piece of knowledge in regard to other issues]
Now what he points out beautifully is that there are exactly only 72 ways to manipulate the equation above [a/b *c/d = e/f] while still maintaining equational balance according to the laws of mathematics. You can work it out yourself or you can look at the chart he made there (אות שנ"ו) or mine below.
Additionally, he points out, and this was the brainstorm for me, if you work out all 72 possible ways of rearranging the equation you'll find that the possibilities as to how the numbers can be rearranged and swapped among each other is actually fairly simply broken down into just 2 groups:
a,c,f & b,d,e or in the hebrew אג"ו בד"ה
That means to say that it's impossible for an integer of one of the groups to switch places with an integer from the other group alone and for the equation to still be correct. Example:
2/3 * 6/8 = 5/10 2=a, 3=b, 6=c, 8=d, 5=e, 10=f
2,6,10 v 3,8,5 -- you can switch the 6 with the 2 or 10 but you cannot switch it with the 3,8,or 5 (without making other rearrangements) and still have the equation correct and balanced.
It sounds complicated but it's really not, it's just hard to show in a blog.
So I guess in essence what I found so intriguing about this insight is the revelation of this funny new relationship -- אג"ו & בד"ה
[p.s. Note the word of והי"ה only appearing in אג"ו of the עשר מכות - in דם, כנים, ושחין ]
[Made an amateurish list of the 72 expressions]
