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 :)
My only complaint is the suggestion that engineers like to be clear. My undergrad classes included far too many things like 2 cos 2 x sin y
I'd say engineers like to be exact, but they like being lazy even more
I think this speaks to why I have a total of 5 years of college and no degree.
Starting at about 7th grade, math class is taught to every single American school child as if they're going to grow up to become mathematicians. Formal definitions, proofs, long sets of rules for how you manipulate squiggles to become other squiggles that you're supposed to obey because that's what the book says.
Early my 7th grade year, my teacher wrote a long string of numbers and operators on the board, something like 6 + 4 - 7 * 8 + 3 / 9. Then told us to work this problem and then say what we came up with. This divided us into two groups: Those who hadn't learned Order of Operations on our own time who did (six plus four is ten, minus seven is three, times eight is 24, plus three is 27, divided by nine is three) Three, and who were then told we were wrong and stupid, and those who somehow had, who did (seven times eight is 56, three divided by nine is some tiny fraction...) got a very different number, and were told they were right. Terrible method of teaching, because it alienates the students who need to do the learning right off the bat. And this basically set the tone until I dropped out of college for the second time.
Meanwhile, I'm over in the corner like 
You guys are doing it all wrong: ask chatgpt for the correct answer and paste it here. Done.
Who needs to learn or know anything really?
I don't remember everything, but I remember the first two operations are exponents then parentheses. Edit: wait is it the other way around?
Hi, I’m stupid, is it 1+2 first, then multiple it by 2, then divide 6 by 6?
Or is it 1+2, then divide 6 by 2, then multiple?
I think it’s the first one but I’ve got no idea.
It's actually "both". There are two conventions. One is a bit more popular in science and engineering and the other one in the general population. It's actually even more complicated than that (thus the long blog post) but the most correct answer would be to point out that the implicit multiplication after the division is ambiguous. So it's not really "solvable" in that form without context.
Forgot the algebra using fruit emoji or whatever the fuck.
Bonus points for the stuff where suddenly one of the symbols has changed and it's "supposedly" 1/2 or 2/3 etc. of a banana now, without that symbol having been defined.
You lost me on the section when you started going into different calculators, but I read the rest of the post. Well written even if I ultimately disagree!
The reason imo there is ambiguity with these math problems is bad/outdated teaching. The way I was taught pemdas, you always do the left-most operations first, while otherwise still following the ordering.
Doing this for 6÷2(1+2), there is no ambiguity that the answer is 9. You do your parentheses first as always, 6÷2(3), and then since division and multiplication are equal in ordering weight, you do the division first because it's the left most operation, leaving us 3(3), which is of course 9.
If someone wrote this equation with the intention that the answer is 1, they wrote the equation wrong, simple as that.
There has apparently been historical disagreement over whether 6÷2(3) is equivalent to 6÷2x3
As a logician instead of a mathemetician, the answer is "they're both wrong because they have proven themselves ambiguous". Of course, my answer would be RPN to be a jerk or just have more parens to be a programmer
When I used to play WoW years ago I'd always put -6 x 6 - 6 = -12 in trade chat and they would all lose their minds.
Adding that incorrect solution usually got them more riled up than having no solution.
While I agree the problem as written is ambiguous and should be written with explicit operators, I have 1 argument to make. In pretty much every other field if we have a question the answer pretty much always ends up being something along the lines of "well the experts do this" or "this professor at this prestigious university says this", or "the scientific community says". The fact that this article even states that academic circles and "scientific" calculators use strong juxtaposition, while basic education and basic calculators use weak juxtaposition is interesting. Why do we treat math differently than pretty much every other field? Shouldn't strong juxtaposition be the precedent and the norm then just how the scientific community sets precedents for literally every other field? We should start saying weak juxtaposition is wrong and just settle on one.
This has been my devil's advocate argument.
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.
My years out of school has made me forget about how division notation is actually supposed to work and how genuinely useless the ÷ and / symbols are outside the most basic two-number problems. And it's entirely me being dumb because I've already written problems as 6÷(2(1+2)) to account for it before. Me brain dun work right ;~;
There's no forms consensus on which one is correct. To avoid misunderstanding mathematicians use a horizontal bar.
I found a few typos. In the 2nd paragraph under the section "strong feelings", you use "than" when it should be "then". More importantly, when talking about distributive properties, you say x(x+z)=xy+xz. I believe you meant x(y+z)=xy+xz.
Otherwise, I enjoyed that read. I'm embarrassed to say that I did think pemdas meant multiplication came before division, however I'm proud to say that I've unconsciously known that it's important to avoid the ambiguity by putting parentheses everywhere for example when I make formulas in spreadsheets. Which by the way, spreadsheets generally allow multiplication by juxtaposition.
I just finished your article and wow! I'm definitely going to save it and share it the next time I come across another one of those viral problems. It was incredibly thorough and well researched, you clearly put a lot of energy and effort into it and it blew me away. It was really refreshing to see someone articulate themselves so passionately with supporting research. I look forward to reading more of your work!! 👏
I guess if you wrote it out with a different annotation it would be
-‐--------‐--------------
=
-‐--------‐--------------
=
--‐--------‐--------------
=1
I hate the stupid things though
I don't have much to say on this, other than that I appreciate how well-written this deep dive is and I appreciate you for writing it. People get so polarized with these viral math problems and it baffles me.
I recall learning in school that it should be left to right when in doubt. Probably a cop-out from the teacher