An expression that has a different value for every interpretation of PEMDAS

PEMDAS (or BODMAS in some other places) is a mnemonic used to teach the order of operations, but there are several possible interpretations, especially when considering implicit multiplication.

The interpretations of PEMDAS we will be considering are:

1. All six types of operation, done one after the other.
2. The same but with BODMAS.
3. The proper groupings with implicit multiplication before exponents.
4. The proper groupings with implicit multiplication between exponents and multiplication.
5. The proper groupings with implicit multiplication grouped with multiplication and division.

For the first two, implicit multiplication will either A) be treated just like ordinary multiplication or B) be placed before exponents. This makes for seven possible interpretations.

The parentheses in the expression a/b(c-d+e)^f get placed in these ways when following each interpretation:

1A: a/(b*((c-(d+e))^f))
1B: a/((b*(c-(d+e)))^f)
2A: (a/b)*((c-(d+e))^f)
2B: a/((b*(c-(d+e)))^f)
3: a/((b*((c-d)+e))^f)
4: a/(b*(((c-d)+e)^f))
5: (a/b)*(((c-d)+e)^f)

All of these are different except 1B and 2B. Let’s just say that’s close enough for our purposes. Next, I want to place actual numbers here. I want all evaluation steps to be only positive integers, and have all possible results be as low as possible.

After some deliberation, I have come to the expression 64/2(4-1+1)^2. The results of each interpretation of this expression are the following:

1A: 8
1B and 2B: 4
2A: 128
3: 1
4: 2
5: 512

I have set up two polls on other social media to see which interpretation people will go with. (Both polls have an option for something else.)

The first of these polls got 27 votes: 11 votes for 512, 10 votes for 2, 2 votes for 4, and 1 vote for all the other options. The second got 26 votes: 11 votes for 2, 5 votes for 512, 5 votes for something else, 2 votes for 1, and 1 vote for all the other options. These votes sum to:

2: 21 votes
512: 16 votes
Something else: 6 votes
1: 3 votes
4: 3 votes
8: 2 votes
128: 2 votes

I am unsure of what conclusion to draw from this, so I will end the article here.