I wrote a (very long) blog post about those viral math problems and am looking for feedback, especially from people who are not convinced that the problem is ambiguous.
It’s about a 30min read so thank you in advance if you really take the time to read it, but I think it’s worth it if you joined such discussions in the past, but I’m probably biased because I wrote it :)
Thank you very much 🫶. No it’s not annoying at all. I’m very grateful not only for the fact that you read the post but also that you took the time to point out issues.
The meme refers to the problem of handling implicit multiplication by juxtaposition.
Depending on what field you're in, implicit multiplication takes priority over explicit multiplication/division (known as strong juxtaposition) rather than what you and a lot of people would assume (known as weak juxtaposition).
With weak juxtaposition you end up 9 just as you did, but with strong juxtaposition you end up with 1 instead.
For most people and most scenarios this doesn't matter, as you'd never encounter such ambiguous equations outside of viral puzzles like this, but it is worth knowing that not all fields agree on how implicit multiplication is handled.
You probably missed the part where the article talks about university level math, and that strong juxtaposition is common there.
I also think that many conventions are bad, but once they exist, their badness doesn’t make them stop being used and relied on by a lot of people.
I don’t have any skin in the game as I never ran into ambiguity. My university professors simply always used fractions, therefore completely getting rid of any possible ambiguity.
The ambiguous ones at least have some discussion around it. The ones I’ve seen thenxouple times I had the misfortune of seeing them on Facebook were just straight up basic order of operations questions. They weren’t ambiguous, they were about a 4th grade math level, and all thenpeople from my high-school that complain that school never taught them anything were completely failing to get it.
I’m talking like 4+1x2 and a bunch of people were saying it was 10.
Just saw the image you posted and it’s awesome :-) I’m part of the group that can’t solve it, because I don’t know the 🌭 function from the top of my head. I also found the choice of symbols interesting that 🌭 is analytical continuation of 🍔 and not the other way round 🤣
Kinda. You can’t define a name, but you can get the compiler to interpret literals as a function. If you have a Num instance for (Integer -> Integer) where,
fromInteger i = x -> x * i
the compiler can interpret integer literals as functions like so
The answer realistically is determined by where you place implicit multiplication (or "multiplication by juxtaposition") in the order of operations.
Some place it above explicit multiplication and division, meaning it gets done before the division giving you an answer of 1
But if you place it as equal to it's explicit counterparts, then you'd sweep left to right giving you an answer of 9
Since those are both valid interpretations of the order of operations dependent on what field you're in, you're always going to end up with disagreements on questions like these...
But in reality nobody would write an equation like this, and even if they did, there would usually be some kind of context (I.e. units) to guide you as to what the answer should be.
Edit: Just skimmed that article, and it looks like I did remember the last explanation I heard about these correctly. Yay me!
Exactly. With the blog post I try to reach people who already heared that some people say it’s ambiguous but either down understand how, or don’t believe it. I’m not sure if that will work out because people who “already know the only correct answer” probably won’t read a 30min blog post.
Unfortunately these types of viral problems are designed the attract people who think they "know it all", so convincing them that their chosen answer isn't as right as they think it is will always be an uphill challenge
yeah, our math profs taught if the 2( is to be separated from that bracket for the implied multiplication then you do that math first, because the 2(1+2) is the same as (1+2)+(1+2) and not related to the first 6.
if it was 6÷2x(2+1) they suggested do division and mult from left to right, but 6÷2(2+1) implied a relationship between the number outside the parenthesis and inside them, and as soon as you broke those () you had to do the multiplication immediately that is connected to them. Like some models of calculatora do. wasn’t till a few yeara ago that I heard people were doing it differently.
It’s not ambiguous, it’s just that correctly parsing the expression requires more precise application of the order of operations than is typical. It’s unclear, sure. Implicit multiplication having higher precedence is intuitive, sure, but not part of the standard as-written order of operations.
I’d really like to know if and how your view on that matter would change once you read the full post. I know it’s very long and a lot of people won’t read it because they “already know” the answer but I’m pretty sure it would shift your perception at least a bit if you find the time to read it.
My opinion hasn’t changed. The standard order of operations is as well defined as a notational convention can be. It’s not necessarily followed strictly in practice, but it’s easier to view such examples as normal deviation from the rules instead of an implicit disagreement about the rules themselves. For example, I know how to “properly” capitalize my sentences too, and I intentionally do it “wrong” all the time. To an outsider claiming my capitalization is incorrect, I don’t say “I am using a different standard,” I just say “Yes, I know, I don’t care.” This is simpler because it accepts the common knowledge of the “normal” rules and communicates a specific intent to deviate. The alternative is to try to invent a new set of ad hoc rules that justify my side, and explain why these rules are equally valid to the ones we both know and understand.
They weren't asking you if there are two sets of rules, we're in a thread that's basically all qbout the Weak vs. Strong juxtaposition debate, they asked you which you consider correct.
Giving the answer to a question they didn't ask to avoid the one they did is immature.
I can't have stopped because I never started, because I'm not even married... See, even I can answer your bad faith question better than you answered the one @onion asked you.
But I will give it to you that my comment should've stipulated avoiding reasonable questions.
The difference is that there are two sets of rules already in use by large groups of people, so which do you consider correct?
However I still think you need your eyes checked, as the end of this comment by @onion is very clearly a question asking you WHICH ruleset you consider correct.
Unless you're refusing the notion of multiplication by juxtaposition entirely, then you must be on one side of this or the other.
“Which ruleset do you consider correct” presupposes, as the comment said, that there are 2 rulesets. There aren’t. There’s the standard, well known, and simplified model which is taught to kids, and there’s the real world, where adults communicate by using context and shared understanding. Picking a side here makes no sense.
When the @onion said there were two different sets of rules, you know as well as I do that they meant strong vs. weak juxtaposition.
You're right that in reality nobody would write an equation like this, and if they did they would usually provide context to help resolve it without resorting to having to guess...
But the point of this post is exactly to point out this hole that exists in the standard order of operations, the drama that has resulted from it, and to shine some light on it.
Picking a side makes no sense only if you have the context to otherwise resolve it... If you were told to solve this equation, and given no other context to do so, you would either have to pick a side or resolve it both ways and give both answers. In that scenario, crossing your arms and refusing to because "it doesn't make sense" would get you nowhere.
In all honesty, I think you're acting like the people who say things like "I've never used algebra, so it was worthless teaching me it as a kid" as though there aren't people who would learn something out of this.
I'll just say it again, you're the one saying this problem is completely unambiguous, with your only explanation as to why being that real people communicate as though that solves every edge case imaginable.
I'm just saying, if you really believe that to be the case, Good luck.
The expression 6/2(1+2) involves both multiplication and division. According to the order of operations (PEMDAS/BODMAS), you should perform operations inside parentheses first, then any multiplication or division from left to right.
Certainly! The expression 6/2(1+2) is ambiguous due to the implicit multiplication. Let’s solve it in both ways:
Implicit multiplication with higher priority:
[ \frac{6}{2}(1+2) ]
First, solve the parentheses:
[ \frac{6}{2}(3) ]
Now, perform the division:
[ 3 \times 3 = 9 ]
Implicit multiplication with the same priority as division:
[ \frac{6}{2(1+2)} ]
Again, solve the parentheses:
[ \frac{6}{2(3)} ]
Now, perform the multiplication first:
[ \frac{6}{6} = 1 ]
So, depending on the interpretation of implicit multiplication, you can get different results: 9 or 1.
I think it’s funny that ChatGPT figured out 1 and 9 but has the steps completely backwards. First it points out what has high priority and then does the exact opposite, both times 🤣
Having read your article, I contend it should be:
P(arentheses)
E(xponents)
M(ultiplication)D(ivision)
A(ddition)S(ubtraction)
and strong juxtaposition should be thrown out the window.
Why? Well, to be clear, I would prefer one of them die so we can get past this argument that pops up every few years so weak or strong doesn’t matter much to me, and I think weak juxtaposition is more easily taught and more easily supported by PEMDAS. I’m not saying it receives direct support, but rather the lack of instruction has us fall back on what we know as an overarching rule (multiplication and division are equal). Strong juxtaposition has an additional ruling to PEMDAS that specifies this specific case, whereas weak juxtaposition doesn’t need an additional ruling (and I would argue anyone who says otherwise isn’t logically extrapolating from the PEMDAS ruleset). I don’t think the sides are as equal as people pose.
To note, yes, PEMDAS is a teaching tool and yes there are obviously other ways of thinking of math. But do those matter? The mathematical system we currently use will work for any usecase it does currently regardless of the juxtaposition we pick, brackets/parentheses (as well as better ordering of operations when writing them down) can pick up any slack. Weak juxtaposition provides better benefits because it has less rules (and is thusly simpler).
But again, I really don’t care. Just let one die. Kill it, if you have to.
It’s like using literally to add emphasis to something that you are saying figuratively. It’s not objectively “wrong” to do it, but the practice is adding uncertainty where there didn’t need to be any, and thus slightly diminishes our ability to communicate clearly.
I think anything after (whichever grade your country introduces fractions in) should exclusively use fractions or multiplication with fractions to express division in order to disambiguate. A division symbol should never be used after fractions are introduced.
This way, it doesn’t really matter which juxtaposition you prefer, because it will never be ambiguous.
Anything before (whichever grade introduces fractions) should simply overuse brackets.
This comment was written in a couple of seconds, so if I missed something obvious, feel free to obliterate me.
I’m not sure if you read the post yet but I also have a short section about alternative notations which are less ambiguous or never ambiguous. RPN has the same issue as most notations that are never ambiguous namely that it’s hard to read - especially for big expressions.
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