To expand on Losethegame's answer
Losethegame answered "I got here by Googling the same question although I'm not sure if (m)any of the commenters have answered it specifically. I think you are right that your rule can't be broken. This can probably be proven algebraically because a*(b/c)=(a/c)*b and a+(b-c)=(a-c)+b...?"
I agree that it can probably be proven algebraically with methods like the one losethegame mentions(though losethegame's example may be flawed since, as user21280 points out, losethegame swaps operands). But elaborating on the idea of proving it algebraically (and without swapping operands!), I can think of some algebra that might prove it. (Aand granted user21280 is of the view that my examples don't account for all possibilities. His answer using logical formulae might).
Given an equation where multiplication comes textually before division, e.g.
3*4/2 it doesn't matter which you do first. So a literal PEMDAS or PEDMAS would be fine. Whereas given an equation where division comes textually before multiplication e.g.
6/2*3 then it does matter which you do first. A traditional reading of PEMDAS or PEDMAS gets it right(because they'd say do the first one first, and that's division), a literal PEDMSA gets it right. A literal reading of PEMDAS would get it wrong.
Subtraction and division share that property. So let's say we make a little equation of the part of the equation that has the operators competing with each other. If substraction is competing with addition, and subtraction occurs first(textually, in the equation), it has to be done first. If division is competing with multiplication, and division comes first, division has to be done first. Whereas if addition were competing with substraction, then regardless of whether or not it occurs first textually, it doesn't matter whether addition is done first or whether subtraction is. Similarly, if multiplication were competing with division, if multiplication comes first textually, then it doesn't matter whether the division is done first, or whether multiplication is done first. So PEDMSA literal always works(i.e. strictly doing division before multiplication, substraction before addition). As does a traditional/proper reading of PEMDAS/PEDMAS i.e. a reading that says multiplication and division are equal priority and do the first one first, similarly with addition and subtraction.
Following PEDMSA literally gives
Following PEDMSA traditionally gives
And we know algebraically that
1*2/3 a* b/c
That's evaluated the same whether following strictly ordered PEDMSA or traditional PEMDAS i.e. whether you do division first as a rule or whether you do the first of the multiplication and division first, that's
(a/b)*c in both cases so clearly the same.
1+2-3 if we do
a+ b-c which is ordered PEDMSA i.e.
a+(b-c) that's the same result as if we do traditional PEMDAS (a+b)-c. We know algebraically
I recall my math teacher pointing out that one thing you want to beware / be aware of is
-(a+b) which we had drilled in was -a-b is so very different to
-a+b. Subtractions always have to be done first and in order.. and if doing subtraction first, or the first of the addition and subtraction first, then we're maintaining that rule.
And for this one it's the same. algebraically
1-2+3 (1-2)+3 (a-b)+c
1-2+3 (1-2)+3 (a-b)+c
And I suppose finally.
1/2/3 And regardless of whether doing strictly ordered PEDMSA or traditional PEMDAS it's
(1/2)/3 so same there.
I'm not sure whether those are all the possibilities.
That may then leave the question of what explains the algebra e.g. the rule that
a*(b/c) = (ab)/c
Also it's a parsing convention that seems to have developed late 20th century, rather than a fundamental rule of mathematics. https://www.quora.com/Is-the-order-of-operations-unclear-for-expressions-like-20-2-5+5