## The Trollface Math Problem

I’m sure you’ve seen them on social media. Math problems that are trivial but are labeled as if they’re brain-busting conundrums. They often say things like “Only 1% of the people will get this right!” or perhaps “It takes a genius level IQ to solve this!”. These claims are, of course, *lies*. Like the germs on the fixtures of a public restroom, these problems do not exist to challenge your intellect but simply to spread.

As such, the most successful problems are not ones that are difficult, but ones that have ambiguity. Because ambiguity leads to arguments and nothing is more viral than an argument.

I have a low opinion of these sorts of things. Aside from being deliberately designed to start fights, they’re often low-quality images in a terrible font with a poorly expressed problem. I’ve seen a few which were actually wrong. Maybe they showed an incorrect answer. Maybe they were improperly expressed or labeled. Or maybe they suffered from bad grammar and punctuation. So when I glance at one of these and I disagree with the proffered conclusion, I usually assume it’s a garbage attempt at a meme rather than questioning my assumptions.

However, I ran into one over the weekend that was actually kind of clever. As obnoxious trollface problems go, this one was pretty good. It goes thus:

For "Real" "Math Nerds" "Only" |

This is an exemplar of the form. Impact font? Check. Shitty quality and Jpeg artifacting? Check. Entices you to pay attention by questioning your mental prowess? Check. Shitty watermark from an off-brand image hosting site? Check.

`230 - 220 × ½ =`

A lot of these problems exploit the fact that a large segment of the general public doesn’t remember the order of operations from middle school. The naive thing to do is to work left to right. 230 minus 220 is 10, and 10 times ½ is 5. However, this is not correct.

When you’re following the proper order of operations, you calculate things in this order:

- Exponents and roots.
- Multiplication and division.
- Addition and subtraction.

Following these rules would have you multiply 220 by ½ to get 110, then subtract 110 from 230 to get 120.

So your typical art majorI’m sure many art majors are good at math. But sometimes stereotypes are useful. will get 5, your math major will get 120, and the problem itself states that the answer is 5! So now you’re going to have an argument in the comments where the math majors try to explain the order of operations to the art majors while a third, smaller group of provocateur math majors insist the art majors are “right” while snickering.

What you should know is that this problem is, first and foremost, a trollface. Secondly it’s a punctuation problem. Thirdly it’s a math problem.

You can clear up the whole thing by re-wording the conclusion to say something like, “The answer is (5!), but most people won’t believe it!”

See, in mathematics, the exclamation mark is an operator. It means factorial. If you’ve forgotten, *n*! means you multiply all positive integers less than or equal to *n*. So `5!` would be:

`5 × 4 × 3 × 2 × 1 = 120`

Thus `5! = 120`. And now you can see the problem. 120 is the correct answer. Thus *5!* is also correct. But the naive answer of 5 is **not** correct. The conclusion is written so that 5 is at the end of the sentence and the sentence ends in an exclamation mark, so you’re likely to see the mark as emphasis, not factorial.

So if you’re used to seeing people mess up their order of operations problems then that’s what this appears to be.

There’s a bit of strangeness in the other parts of the problem as well. `220 × ½` looks odd to me. It’s totally correct, but typically if you’re trying to divide something in half you’ll either multiply by 0.5 or divide by two. You’re allowed to multiply by ½ if you like (and indeed, that’s what 0.5 is) but it looks unusual to my eyes. When I started to suspect this problem was engaging in a trick, I zeroed in on this bit looking for shenanigans.

The other giveaway that this is a troll problem is the fact that it gives away the “answer”. Most of these don’t, and leave you to work it out for yourself. But here the problem tells you the answer (in an ambiguous way) and then leaves you to argue about it in the comments.

Well played, internet.

EDIT: If you’re curious about the origin, the author stopped by with the goods. Here is the Reddit thread where the trick of punctuation vs. factorials was devised. And here is a blog post where someone else turned it into a troll question by having the problem work out to 5 if you botch the order of operations, thus setting the stage for chaos.

#### Footnotes:

[1] I’m sure many art majors are good at math. But sometimes stereotypes are useful.

## This Game is Too Videogame-y

What's wrong with a game being "too videogameish"?

## The No Politics Rule

Here are 6 reasons why I forbid political discussions on this site. #4 will amaze you. Or not.

## The Plot-Driven Door

You know how videogames sometimes do that thing where it's preposterously hard to go through a simple door? This one is really bad.

## Starcraft: Bot Fight

Let's do some scripting to make the Starcraft AI fight itself, and see how smart it is. Or isn't.

## Another PC Golden Age?

Is it real? Is PC gaming returning to its former glory? Sort of. It's complicated.

## Shamus Plays LOTRO

As someone who loves Tolkein lore and despises silly MMO quests, this game left me deeply conflicted.

Well, I enjoyed the ride. And I’m embarrassed to admit I thought the order of operations was left to right… and I’m a programmer.

Please don’t tell anyone.

To be fair, who relies on order of operations when programming? You’ve got parentheses for that.

Every once in a while I get it in my head that I will inflict Reverse Polish form on everyone the next time I want to make a math parser, just because it’s so easy to write a stack to parse it in the code.

RPN is the best.

Bonus point, no one will ever borrow your calculator and never return it.

In undergrad I had a professor who would only let you borrow an RPN calculator if you forgot yours for an exam, to encourage people to not forget calculators XD

You used a calculator in college? When I was an Undergrad, our teachers told us “a calculator won’t help you.” And in high school, they were straight up banned.

Yes, but the high school part is mainly because at earlier levels of education you’re supposed to be developing the more basic skills that these things automate away; once you’ve got an understanding of what the calculator is doing on your behalf then it’s not as much of an issue if you use it. I mean – if you were allowed to use a calculator to do high school maths, how many hours would it take you to burn through the entire curriculum’s worth of exercises? And at the end of that, what exactly would you have learned?

It’s like how you weren’t supposed to rely on dictionaries (or spell-checkers, I assume, if you had computers) in primary school so you’d learn the correct spellings for words, and then as an adult it’s a miracle if you put the time and effort into actually using one at all.

I don’t know when or where you went to school. When I was in school in the states in the 90’s, not only did they let us use calculators, we had programmable calculators and if we programmed something into our calculators to help us with the test we were allowed to use it.

And a mere 15 years later, I decided to start working with Javascript. I’m sure this experience helped.

We all had to buy non-programmable calculators in 6th grade, learn how to use them in math class, and then proceeded not to use them in math exams because the problems were always “well-natured”, i.e. you seldom saw any numbers larger than 10, or weird decimals, and you could just write “1/2 sqrt(3)” rather than calculating the value, which is less accurate. That’s because maths isn’t about calculating numbers. That’s just what everyone starts with to give them something to apply maths to.

…we did, however use the heck out of those calculators in physics and sometimes CS. And at Uni.

These days you get crazy programmable calculators for crazy little money but I still can’t use them properly. But then why would I, I’ve got several real computers around me at almost all times.

To it you have beat me.

What RPN this is for.

Is it bad that I spent two minutes trying to figure out how to write English in RPN in response to this?

Like, would it be (subject, object,verb)? What about indirect objects? Does RPN really work if you can have operators with more than 2 arguments? I’m guessing yes–otherwise things like calculus couldn’t really work, but has anyone really tried to do calculus with RPN?

Well, it’s fundamentally a stack, so you just need to keep defining arguments until you have enough, then apply the operation.

it stack a is fundamentally you arguments defining keep to need you enough have until operation the apply then so well

Without knowing anything more than that RPN exists, this looks hilarious. It also completely screwed me over when I tried to find the “just” in the second sentence.

I figured I messed something up. It should go after “need”.

For a fun time, try lisp, the programming language built on rpn.

Lisp isn’t really… the point of stack-based languages is that they don’t need parentheses because they can just rely on function arity to determine what to consume. Lisp functions can in principle have any arity, and that’s what the parentheses are for.

Basically, math as we’re used to is

infix, because operations go in the middle of what they operate on, RPN ispostfixbecause they go afterwards, and Lisp isprefixbecause the decision of what to do is specified at the beginning.Hey, God wrote in lisp!

‘Stack is a it, fundamentally arguments defining you, need keep enough to have you, apply until the operation, then so well’

I never knew mathematics could be so poetic.

Yes, please! Any non-trivial operation? Apply a generous topping of parenthesis. It ensures I don’t screw up because I goofed on the order-of-operations, and ensures that whoever deals with the code later* clearly knows my intention.

* Statistically, myself. I don’t know about other people, but I’m often baffled at the bizarre decisions Past Alan made. And I’m thankful when Past Alan erred on the side of being explicit, and commenting

whyhe did things. Thank goodness I’m perfect now, and there is no possible way I’ll being saying the exact same thing a decade from now about the code I’m writing today. :-)TFW you need to modify/add an argument in a soup if parentheses and you spend the next ten minutes counting them out to make sure it’s in the right nest…

Having the IDE mark out the matching Parenthesis or curly braces is one of those features I could never go back to not having. Especially for nesting shenanigans

Do programming languages even follow the same order of operations as normal human math? I learned to use parentheses heavily, because you couldn’t guarantee the order otherwise. Maybe modern languages have it right?

Generally, yes. But languages add far more operators and you’re not always going to know where they go. And it changes between languages.

Does & have higher or lower precedence than +? What about &&?

How the hell is (TRUE ? ‘CAR’ : FALSE ? ‘HORSE’ : ‘FEET’) horse? (because php wanted to do something different)

Look, there is logic, there are programming languages, and there’s php. You can’t compare any two of them!

I suspect you know the answer, since example is just lifted straight from the PHP: A Fractal of Bad Design , but it’s because PHP interprets as it like: (TRUE ? ‘CAR’ : FALSE) ? ‘HORSE’ : ‘FEET’, which becomes ‘CAR’ ? ‘HORSE’ : ‘FEET’, which becomes ‘HORSE’.

Rather than every other language (that supports non-boolean ternaries) that I’ve ever seen, which is going to interpret it like TRUE ? ‘CAR’ (FALSE ? ‘HORSE’ : ‘FEET’), which becomes TRUE ? ‘CAR’ : ‘FEET’, which becomes ‘CAR’.

Pretty much always yes, but they tend to involve stacking so many operators I use parenthesis even when they aren’t necessary.

I have a good memory of the order of operations… but I still heavily (overly) use parenthesis “just to be sure”

Don’t worry, your secret is safe with the Internet!

I remember my dad telling me that he was in his second year of college before he was introduced to order of operations. It was kinda cool that we ended up basically learning lots of math together. (He had a masters in psychology, so he probably understood statistics way better than I ever did, but lots of other areas were kinda new to him)

Interesting, order of operations was one of the first examples when I was learning formal grammars, since they’re a great foray into how to deal with ambiguity when parsing. I recommend looking into those, having a bit of theory can help when writing parsers, if only to recognize the situations where regular expressions are incapable of doing what you want on a fundamental level (e.g. matching parentheses).

That reminds me, it’s going to be 40320 Day in a couple of weeks.

And I will be

shockedif anybody gets that joke.Well a quick calculation shows that the unix epoch was roughly 17000 days ago, so it can’t be the numerical value of a date type calculated in whole days (these do exist, typically using a double as the data type, so 12 hours would have the value 0.5).

It’s going to be 8! Day?

I’m trying to remember how many times Deadpool said ‘Bang!’ during his gunfire hyper combo in MvC3.

Yes, but the question is why.

This statement isnt necessarily a lie,because only people who are very into math will actually do the problem in its correct fashion,not due to them being that smart,but rather because only they are that interested to follow the proper rigor of math to find the solution in the proper way.

Its a nifty way to write things to never confuse the order of operations.You can completely eliminate subtraction,division and roots if you start considering them as inverse operations.So a-b is actually a+(-b).And instead of a/b you write a*(1/b).It seems like its more laborious,but actually its much easier when you deal with long equations,and also lessens your margin for error.

I don’t really see how that helps. If I’m doing

`a - b * c`

, either I remember that`b * c`

comes first, or I don’t. Writing it as`a + (-b) * c`

isn’t going to make it any more obvious that I need to do the multiplication first. If I forget about order of operations, I’m still going to get the wrong answer.It seems like you need a very specific level of forgetful for that to be helpful.

I think Lucifer’s method allows the brain to immediately realise that math is a lot more modular and malleable than most people realise? For example -4+3=3-4=3+-4. It’s a conceptual reminder that’ll make everything more consistent in your head potentially.

There are actually cases where it does help.For example,Redingold mentions 6Ã·2(1+2).People are often confused as to whether to first multiply or divide in that case.But,if you write it as 6*1/2(1+2),the confusion is gone.Here,no matter which order you pick,youll get the same answer.

Yeah, that doesn’t actually make it any less ambiguous at all. You still get a different answer depending on whether you do the 2(1+2) first or the 6*1/2 first, that is, if you take multiplication by juxtaposition to have a higher precedence than division. Some mathematicians do and some do not. There is no consensus and thus no unambiguously correct answer.

In fairness to his point, I think he’s means writing it like

`6 * Â½(1 + 2)`

, where Â½ is clearly a single number, not an operation that can be split.I still don’t think it’s a great example, as you said, the original:

`6 / 2(1 + 2)`

is pretty ambiguous. (Depends on whether you consider`xy`

to be implicitly`(xy)`

). Damien’s method is one way to disambiguate.But if the method is only useful for very specific examples that are ambiguous in the first place, I’m still not convinced of the general utility of the method.

In small equations like this,you are right,there is no (practical) usefulness.But in big equations,say describing a rotation of a 3d object,its very useful to differentiate numbers like this.

If you’re using handwriting or something like Latex, then yes, it would be better to use a horizontal divider, but if you’re just typing in regular old text like we are, then you can’t really use horizontal dividers. In such a scenario, I claim it makes more sense to use my method of taking every implied product after the division sign to be part of the denominator, so you don’t have to include any brackets to denote such fractions, as opposed to just going left to right, where you do need brackets. I can’t think of any situations where this makes writing out expressions more complicated, and there’s plenty of occasions where it makes them less complicated – there’s no end of formulas in physics that are the ratio of products of various constants and variables. Being able to write those out in a single line without the use of brackets is very useful, as a physicist.

Yet you do not apply the same thing to the power sign (^).You already caused me to do a double take when you wrote down there e^2somethingsomething,instead of sticking that e^2 in the end or in parentheses.

Well, that’s because exponentiation has a higher precedence than division. If I write e^2/4Ï€Îµ_0Ä§c then that’s equivalent to (e^2)/4Ï€Îµ_0Ä§c. I only said that implied multiplication should take precedence over division, not over exponentiation. If you do need to raise something to a product or fraction then you have to use brackets with my method, but you also have to use brackets with the other method, so I don’t see how that’s relevant.

Thats rather arbitrary.

Exponentiation is given higher priority than multiplication because doing it that way makes it easier to write polynomials. If multiplication was higher, then you’d have to write ax^2 +bx + c as a(x^2) + bx + c. That’s less convenient, so we give multiplication a lower priority. Also, exponentiation right-distributes over multiplication, so doing exponentiation first lets us express this property as (a*b)^c = a^c * b^c instead of as a*b^c = (a^c) * (b^c). This is analogous to the reason that multiplication takes precedence over addition, since a*(b+c) = a*b + a*c. Earlier, you said that the order of operations was the way it was because of how the operations behaved, so I thought you already understood this.

I was talking about what comes after the ^ sign.I discussed this in the thread below,but to me x^yz is not the same as x^(yz).

Those things aren’t the same to me, either. I don’t know why you’d think that I would think that they were the same.

Multiplication is commutative. You multiply and divide at the same time, in one sense, and in the other sense it doesn’t matter.

6/2*3 gives 18/2=9 or 3*3=9 either way.

The error it looks like you were pointing at is confusion about which numbers are in the divisor and which are in the dividend, and not order of operations at all.

I find that writing it as 220*1/2 , or as 220/2 is far more convenient.

The problem with writing it as a decimal number instead of a division is that most of the time, you need to use the number again in another operation. And decimal numbers are terrible at that, especially if the divider becomes larger or anything that doesn’t divide neatly. Trying to do operations with 1/6 is much easier than with 0.16666666667.

I’ve been out of math for a while; how do you invert a root?

Nth root of X is the same as X^(1/N)

To me, that answer is insanely unfair. Answers should be in the simplest form, while this one used an operator to modify the answer and presented the modified form as correct. Sure, (5!) is right, but so is ((square root of 25)!), 12×10, and 240/2!

It’s not faaaaaaaaaaaaaaaaaaair I tells ya!

Also , they used a mathematical character in the middle of a sentence, and then failed to provide proper punctuation to end that sentence. Even though they did use punctuation to end all the other sentences.

Well, the point is to troll the audience. Fairness is directly contrary to that goal.

I made what I think was the first of these, and in my version I explicitly wanted to avoid the implication of the simplified form, so I said “this equals” instead of “the answer is”.

(https://www.reddit.com/r/math/comments/3mfgar/i_made_this_to_see_how_many_people_i_could/)

Completely irrelevantly, it bugs me that 220Ã—Â½, 220â‹…Â½, and 220(Â½) are all equivalent. It sometimes feels like we’re trolling students as they advance through the grades and are told they need to switch symbols.

Don’t even get me started on math using more than one term to describe the same thing. Apparently high level math feels the need to use different terms for DAMN NEAR EVERYTHING (e.g., calling a function/functional “bijective” instead of “one-to-one”) for no goddamn reason. I spent the first semester in grad school with Wikipedia open to have any idea what my instructors we’re saying

Yeah, I’m American and took the International Baccalaureate “maths” higher level exam in the Queen’s English. You wouldn’t think there was much of a difference in math vocabulary between US and UK, but there is.

Mathematical notation is very much rooted in the past. Many similar or identical objects were called different things by different authors simply because they didn’t know about each other’s work at the time.

And then there’s pi vs. tau. I swear, it’s the worst bit-shift debate in history!

Richard Feynman, the famous physicist, wrote in his autobiography that he didn’t like a lot of mathematical symbology (his example was “sin x” looks like s times i times n times x) so he made up his own, and then when he started teaching he abandoned it because he’d have had to teach all his students a new symbology (and, of course, they’d have to stop using it the next year) because nobody else knew it.

Yeah — I

reallywish I had the authority to rebuild quite a bit of mathematical notation. Inverse functions in particular have always infuriated me.Mandatory relevant XKCD

Seriously, is there anything that guy hasn’t covered yet?

All through my job in the chemistry lab I pushed for abolishing pH, and just expressing the concentration of hydrogen ions with scientific notation. Never got very far.

In fairness, bijectivity also includes surjectivity (e.g., the function maps onto its entire codomain). We usually call one-to-one functions without that property (like a typical exponential function) injective.

Feel free to ask about how many different ways my officemates and I use the word ‘normal!’

Normal? Let’s see…

Normal vector, related to normal force.

Normalized vector, related to unit vector.

Normalized output, meaning basically calibrated against some reasonable scale.

Normal, related to norms, as in mores.

Normal, as in average (even if running contrary to mores).

(The latter ‘normal’ is supposed to be caused by the previous ‘normal,’ but frequently isn’t.)

Normal, related to that guy called Norm. (Hehe, gotcha again Norm!)

That’s all I have, but I still feel like I missed some.

The whole maddening list…

… of Doom.

I would’ve only included the ‘Mathematics’ section, but why not toss in the rest?

Ooh — there’s also ‘homogeneous,’ if you want to dive deeper into the rabbit hole. We don’t even all pronounce that one the same way!

I’m ashamed to have forgotten about Gaussians.

I think you forgot normal as in perpendicular.

Or that might be a german thing in regards to angles.

That’s “normal as in normal force”, I believe.

…I swear, I meant “injective,” I just second guessed myself and changed it at the last minute of the edit period.

I actually don’t remember what the pleb term is for a function that’s both “one-to-one” (injective) and “onto” (surjective), but I vaguely remember there is one

I usually call it a one-to-one correspondence if I’m trying to avoid the -jections.

I call my functions morphisms, and bijective functions are isomorphisms for me. Category theory is violent with its names. We have a joke: “A monad is just a monoid in the category of endofunctors, what’s the problem?”.

I know what you mean, but technically bijective means one-to-one

andonto. The individual terms are also called injective and surjective, which makes your point about unnecessary multiple terms.*EDIT*

Oops, I should have refreshed the page before checking that nobody had already made this comment.

You dont know how much time it saves to not having to write â‹… in front of a parentheses.Its essential!

I’d certainly take a while finding ‘â‹…’ (as opposed to ‘.’) on my keyboard…

The way you wrote it and the way my RSS feed parsed the ‘ symbols (with one facing right and the other one facing left), I thought you were talking about emoji.

In my case, it’s not so much saving time as minimizing the risk of misinterpreting a dot as a decimal point, or a cross for an x.

Of course, if I’m working in LaTeX, that’s not a real problem.

That’s why we write mathematician x’s. Why change the notation when you can change the alphabet?

It took me eight years of teaching to advance to the stage of putting bars through my z’s to distinguish them from 2’s. Baby steps.

My chalk board hand writing is sloppy enough that I had to resort to this almost immediately.

I usually do not write a multiplication sign.

When typing I will use *. If have to write it by hand i will use a dot unless I am working with vectors in that case I will make sure to use the correct sign as operator. You do not want to switch up your scalar product and your cross product.

Yes, and while we are at it, why do people in different countries have to talk in languages that aren’t English? Its so inconvenient to have to learn multiple languages if you want to communicate effectively with everybody.

(just in case its not obvious, I am joking here…)

However, I think that the root of the “language problem” is the same as the root of the “many symbols mean the same math” problem, which is that different groups get together to talk about the same problems and simply evolve different terminology to describe them.

My immediate thought was that they were actually solving that block of nonsense in the background. Nice to see it’s actually clever.

It looks like the stuff in the background is a linear equation expressed badly, meaning it would merely take a lot of simplifying busywork to solve.

I probably would have been one of the provocateur math majors snickering if I ran into this in the wild. Not out of a feeling of superiority, but rather because that’s what you do when someone makes a good pun: stand by snickering and prodding the victim until they finally GROAN

Per https://xkcd.com/169/, the correct solution is to cut off the right hand of all involved parties.

It also works for annoying modern art! :D

It’s amazing! I just proved 2 – 1 = 0!

I think this is actually the way the joke was told to me the first time I ran into it

I remember having quite a long discussion with a math teacher friend about the best way to teach kids intuitively why 0! == 1, without resorting to the usual appeal to definition. It was a remarkably productive few beers.

…I’m not even sure how you ever

couldmake that intuitive without an appeal to definition.My friend’s argument is based around combinatorics:

You can (non-rigorously) think of N! as the number of ways you can arrange N objects. Then, he argues, you can understand 0! as the number of ways you can arrange 0 objects. i.e. just one way, there is no way of rearranging nothing into a different order. The only permutation of {} is {}.

A bit fluffy, but reasonably pleasing.

My preferred is to simply write it out the result of the relation, n! = n*(n-1)! for 1!:

1! = 1*0!

We already know 1! = 1*1 == 1, so we can substitute that on the left,

1 = 1*0!

then the only thing 0! can be is 1.

Neither is rigorous, but there are bigger fish to fry at high-school level maths that pure rigour.

Personally, I think it’s the mathematicians that need teaching. You could spend the effort inventing a new rule to allow 0! = 1, and then further time trying to explain to people how it works and why, and look at all these clever patterns…or you could just say that factorial is only defined for whole numbers greater or equal to 1.

Fair warning: You’re speaking to a physicist, expect the abuse of mathematics in the following.

I don’t agree with your assertion that the factorial operation is only defined for integers greater than 0. The definition of 0! = 1 has important uses.

Consider a Taylor series, defined as a summation (https://wikimedia.org/api/rest_v1/media/math/render/svg/9e4e3f2a8f914c3a03ad35da7cb877d700a614ea):

Sum_{n=0}^{âˆž} (d^n f(a)/ dx^n) / (n!) * (x – a)^n

For this we need 0! = 1 for the first term to be well defined.

It’s worth the bit of intellectual pain whilst we get used to it.

Trolling the math majors is one of the major perks of doing physics.

OMG I came to my math instructor asking why a function didn’t have a weak derivative when you could use Dirac delta to define one and I thought he was going to have an aneurysm! XD

Weak derivatives have to be functions, unfortunately. And the Dirac-Delta cannot be a function since that would mean its integral would be zero.

However, every

distribution(and, therefore, any locally integrable function, including every continuous, bounded and/or monotone function on IR^d), has a distributional derivative f’, which also has to fulfillintegral(f’ phi) = -integral(f phi’)

for every test function phi, and does not have to be a function in the classical sense. So, you probably found a valid distributional derivative. However, you’ll have to be careful whether whatever theory you’re discussing actually addresses distributional solutions.

Yeah, we went through that whole discussion when I brought it up. It was just funny to see the look on his face in response to my even bringing up Dirac delta–apparently that’s one of those great trolls physicists have inflicted on mathematicians.

If you allow neither rigorous definition nor intuitive pattern recognition to improve your understanding of mathematical objects, what’s even left? “Everything that makes intuitive sense without explanation” is quickly going to ram the iceberg that is statistics and sink into the ocean of fallacies that are caused by poor understanding of statistics and stochastical processes.

That said, I have yet to see the factorial operator appear in a real-world application where it cannot be extended to the 0! case without breaking horribly.

Am I understanding your double negatives correctly, you’re saying that all the applications of the factorial you’ve seen have broken at 0! ? Though maybe I’m not reading that right?

If so, I’ll point to the Taylor expansion of the exponential excitation operator in quantum mechanics. This requires the 0! == 1 relation to be defined, though it is frequently omitted from the written equations.

I’d say most series involving a factorial handle 0! gracefully.

I not very much English. Me sorry!

Is no sorry need. Make bad of words like many person. This thing continues its confusion.

Pattern matching and definition both require effort, which needs to be justified. All of the material I found before posting earlier, simply provided the proofs or the patterns, without any justification whatsoever. A few sentences explaining that 0! has uses in solving other problems is all that was needed, which Functional Theory readily provided.

Well, we came up with this thing to generalize factorials to more than positive integers.

That thing can remain confined to the domain of wikipedia until I have a problem that could be conveniently solved by its misuse.

It’s pretty integral to the Gamma distribution, which shows up all over the place if you’re doing statistics on event/time measurements and Poisson doesn’t cut it, or when you’re doing Bayesian voodoo.

I know what a factorial is, but it never occurred to me that the exclamation point was intended as the factorial operator. I interpreted it as punctuation, as the sentence otherwise lacks a proper punctuation mark at the end. So rather than being impressed by their math cleverness I am instead somewhat irked by the creators’ poor writing skills. What they

shouldhave done was put two exclamation marks at the end, one for the operator and the other for punctuation. I still would have missed the joke, since redundant exclamation marks are all too common, but I would have been much more impressed once somebody explained it to me.You can have double factorials, though, so…

So it

doeshave a punctuation typo after all…That’s my argument, too: for the statement to be correct, the grammar must be wrong! What they ought to have done — though perhaps this would have made it too easy and therefore less of a good troll — is to say, “Most people won’t believe it, but the answer is 5!.” Still, pretty clever troll.

Meme grammar is slightly different though, as in meme’s you never add a period at the end of a sentence. You only use ‘special punctuation’ (Like exclamation marks, question marks, that kinda stuff)

Which they did in the previous sentences. So the writer had to deliberately sabotage their own grammar in order to make the troll work.

I absolutely agree that you can’t dual purpose the exclamation point as both a sentence terminator and a factorial indicator. If you’re going to play that game, the equation is only true if x=1.

8-)

But I also, being an editor by profession, need to mention that, with only a very few exceptions, you do not make plurals with apostrophe-‘s’. The word “meme” is not one of those exceptions.

(Sorry, this sort of discussion just demands pedantry. It’s not you; it’s me.)

Thank you!

This is an addicting problem for me. Many people’s methodology for writing is just wrong!

Well, that’s the troll. I use factorials all the time, and I still failed to notice it in the equation above. Clarity would defeat the purpose of the problem.

Hmm, I missed the explanation with the factorials in the first go through, because I was mistakenly thinking that “But the answer is !5”.

Which would also be correct.

Objection! The top of the image has exclamation points, which are clearly not a factorial symbol, preceded by the words only and carefully. So the problem itself says we are EXCITED not careful math people. This means in order to see the answer as correct you need to assume that the last statement is categorically different than all that came before.

So really what we have hear is

https://xkcd.com/169/

Indeed, while the

bottompart needs ! to mean “factorial” in order to be correct, thetoppart sets up the given expectation so that even people who know factorials exist aren’t expecting one in this context. Sneaky, and I was fooled for a few seconds, but I did manage to figure it out before scrolling far enough to see why it was right.The classic examples involve tricks of wording where something is like 5 / 10(8+12), which is caused by using the short-hand of skipping the multiple sign when it’s right next to a parenthesis. If properly articulated, it would be 5 / 10 * 20, which = 10.

Some people tend to argue incorrectly that multiplication happens before division, and addition before subtraction, when they’re supposed to be simultaneous. That’s why pre-algebra classes will go into exercises that show that dividing by a number is just multiplying by a reciprocal, and subtracting a number is the same as adding its negative.

What’s interesting is that I recently heard a mathematician explain that in the above example, many people would use 10(calculation) in the same way that others would use 10c, which is to say an established product, which would therefore occur before the original division. 5 divided by the *product* of 10 times 20.

In the end, it’s as Shamus and XKCD state, it’s an issue of poor writing deliberately done to cause confusion and be a jackass.

Goddamn internet and its overuse of the exclamation mark.

I’m now virtually impervious to it to the point I don’t even see it anymore. So when you said the phrase should be expressed “The answer is (5!), but most people won't believe it!” I was thinking: “bullshit, where did this factorial come from?”

Then I looked back at the image and was aghast to see the exclamation mark was indeed there.

I am studying engineering, so I tend to do those problems as maths practice, and also enjoyed the 5! question. I came across another one which actually had backing to the X% of people get this wrong, as it was done in a national survey.

Here is the question:

Jack is looking at Anne, but Anne is looking at George. Jack is married, but George is not. Is a married person looking at an unmarried person?

A: Yes

B: No

C: Cannot be determined

Here is the original article:

https://www.theguardian.com/science/2016/mar/28/did-you-solve-it-the-logic-question-almost-everyone-gets-wrong

I got the right answer, but I had to check it a couple of times because it seemed WAY too obvious an answer. However, my first instinct answer was the one that most people thought wrong, and it was only when I tried to think of it as an actual logic problem that I got the right answer.

Yeah, the reason why so many people get it wrong is because there is the ‘easy answer’, with only two options, the correct answer rate increases massively, because then people are forced to think.

Well, I can admit my first instinct was the wrong answer… I even “tried” to think more deeply into it in case I was missing something, especially as it is step up to suggest it’s non-obvious. But I got focused on the wrong details…

~~I was annoying wondering what ridiculous forced assumptions are being made in this problem? What is Anne somehow “obviously” married to George due to some B/S social expectation?~~~~You don’t know Anne’s marital status so you can’t know if Anne is unmarried to know if the condition is true for Jack-> Anne, or if Anne is married for the condition from Anne -> George is true.~~~~Then saw the solution and, it breaks down nicely… no forced assumptions… just exploring all possible conditions, helps that the only variable in the problem has a binary output. And showing they are in fact consistent in relation to the wording of the question… since it doesn’t matter if it’s Jack -> Anne or Anne -> George. The reader tends gets too focused on solving who the relationship might be true for rather than it is true regardless. And then takes the “easy way” of “cannot be determined”~~Not sure I think it’s a

goodquestion…[sarcasm](I got it wrong therefore it’s automatically “bad”)[/sarcasm]

…but it does let the reader get too focused on the wrong details…

If you want to talk troll maths problems, how about something like 6Ã·2(1+2). This is particularly nasty because there isn’t actually an unambiguously correct answer. The way order of operations is taught in school, it would be 9 (brackets first, which gives 6Ã·2(3), then go left to right for division and multiplication, giving 6Ã·2(3) = 3(3) = 9), but the way some actual mathematicians interpret the order of operations is to take implied multiplication by juxtaposition (i.e. interepreting 2(1+2) as 2Ã—(1+2)) as having a higher precedence than division, which then gives you 1 (again, brackets first, which gives 6Ã·2(3), but then the implied multiplication comes next, giving 6/6 = 1). There’s no consensus on which order of operations should be used.

Personally, I think that implied multiplication should take precedence over division. The reason for this is pretty simple: if I write 1/ab, it’s clear that what I

meanis 1/(ab), not (1/a)b. Being able to chain products together like this while having them still be part of the denominator of the fraction makes it more convenient to write more complicated expressions, which is the reason that we have the order of operations we do now (the order of operations makes it really convenient to write polynomials. Changing the order would mean you’d have to introduce brackets into every term).There is a consensus.You just write the equation a bit differently.Instead of Ã·,you write *1/2.With 6*1/2(1+2) there is no ambiguity in the final answer.Whether you multiply first and get 6*3/2 or go left to right and get 3*3,the final answer will always be 9.

No, if you use the convention of multiplication by juxtaposition taking higher precedence, then you still get that 6*1/2(1+2) = 1. Brackets first gives 6*1/2(3), then multiplication by juxtaposition gives 6*1/6, then going left to right gives 1.

The key point here is that it distinguishes between multiplication by juxtaposition and multiplication by other methods, so there is a distinction between 1/ab and 1/a*b (the first is 1/(ab) while the second is equal to b/a).

Thats not how you operate with fractions.You multiply the numerator,not the denominator.Of course,the way computers treat forward slashes as both the symbol for division and as a fraction line can muddle the issue,but if you write the fractions properly with the fraction line being horizontal,even that little bit of confusion disappears.

Yes, if you can use a horizontal divider, then it’s always clear what is meant, but if you can’t, say if you’re writing in plain text as we are, or if you’re using an inline expression that would be inconvenient to break into multiple lines, then letting multiplication by juxtaposition take precedence over division makes a great deal of sense.

If you write in plain text,then you can always encompass the whole number in brackets.So instead of 1/2 you write (1/2).

Yes, but if you take multiplication by juxtaposition to have a higher precedence than division, then you don’t need so many brackets. Example: say I want to write down the expression for the fine structure constant. If I let juxtaposition take precedence over division, I can do it like this: e^2/4Ï€Îµ_0Ä§c. If I don’t let that be the case, though, I need brackets: e^2/(4Ï€Îµ_0Ä§c). Clearly the option that lets me write it without brackets is superior.

The superior version is not the one that lets you write the least amount of text,but the one that lets you pack in the most amount of information in the least ambiguous package.Thats why we are all writing clear english here and not txt spk,evn doh txt spk hs fwr chrs.

And this whole conversation is the perfect example of that.Instead of writing two extra brackets,I went for brevity and writing 1/2 in the middle of the equation,which clearly does not work that well in plain text.

Both versions convey the same amount of information and both are equally unambiguous (the ambiguity in the troll question is not in any particular order of operations, but rather in which order of operations to apply). And brevity is important. If the order of operations was PASMDE instead of PEMDAS, then you’d have to write polynomials like (a(x^2)) + (bx) + c instead of like ax^2 + bx + c. (a(x^2)) + (bx) + c

isunambiguous using the PASMDE ordering, and conveys exactly as much information as ax^2 + bx + c does using PEMDAS, but it’s still clearly an inferior choice of order of operations.Er…I don’t follow. 6*1/2(1+2) has the same number of brackets as 6Ã·2(1+2). All you did was replace the obelus with a slash. With the additional 1 needed, that actually uses more characters. How is that an example of brevity?

Finally, and this is petty, but you do realise you’re supposed to put spaces after punctuation in English, right?

The order of operations isnt like it is because of brevity but because of how the operations are defined.

No,I didnt.I replaced “division by two” with “multiplication by one half”.And if I wasnt stingy with the brackets this wouldve been clear.

What I find funny is that you focus on that,which isnt a requirment,instead of my omission of apostrophes,which actually is a requirement of english grammar.

Apparently this comment thread has run out of nesting, but whatever.

The order of operations can be whatever we say it is, but we make the choice we do because it makes things easier to write. It makes things easier to write because of how the operations work, yes, but that’s not the fundamental reason for its use. It’s a notational device we use for convenience and precision, not a mathematical truth. Besides, the alteration I’m suggesting doesn’t affect any mathematical truth. All it does is make denominators with products in more convenient to write.

Er, yes you did. You literally wrote “Instead of Ã·,you write *1/2.With 6*1/2(1+2) there is no ambiguity in the final answer”. All you did was take the obelus and replace it with “*1/”. Unless you meant to indicate that you would replace the division by two with a multiplication by 1/2

where 1/2 is written out with a horizontal divider, like this, in which case, I agree, that evaluates to 9, but that’s not the formula we’re talking about. I am explicitly talking about a formula like this or this, and I am arguing that this should evaluate to 1 because you should do 2(1+2) before you do any division.Since you bring it up, yes, use apostrophes too. That would be nice.

Yes it does.The operations themselves can be whatever we wish them to be,but once those are established the order of operations flows naturally from their properties.The reason multiplication goes before addition is because the definition of those operations on the set of numbers is the way it is.In order to change it,you either have to change the set the operations are used on(which does happen in certain fields of maths),or change the definition of those operations(which again does happen in certain maths).

That was my intention,yes.And after reading djws response I see now that the way I see / and how others see it is different.To me,only the first number after the / is the denominator,and if there are multiple ones the whole thing needs to go in brackets.Same goes for exponents,trigonometric functions,ln,etc,etc.To me lnx(y+z) is not the same as ln(x(y+z)).I can see the reasoning why you see it differently though,so in the future Ill be more careful when typing out formulae.

You’ve almost got it, but you’re still missing my point. The way that you interpret the /, where you just do any multiplication or division from left to right so 1/ab means (1/a)b is a totally valid way of doing the order of operations. Indeed, it’s the way that you’ll probably find being taught in schools. You’re not unique in interpreting it that way. My way, where any implied product is done before any other division or multiplication, and then you go left to right for everything else, is also a valid way, that many mathematicians use. The ambiguity in the expression 6/2(1+2) is in the fact that it’s unclear which method is intended. Your method and my method give different answers to this. That’s why it’s ambiguous.

As an aside, if you write this in Python you get 9 (although you do have to type a * in between the 2 and the ( or you get an error).

6*1/2*(1+2) = 6*1/2*3 = 6/2*3 = 3*3

This one bites me in the ass fairly often, since my brain uses the convention that everything after the / is in the denominator.

Python is right, you should get 9 if you write it like that. I’m saying that if you leave out that asterisk, you should get 1. Doing this lets us distinguish between 1/ab and 1/a*b. That’s a useful distinction to be able to make when you’re writing in plain text like we are now and can’t use the proper fraction notation, or if you want to fit your expressions on a single line.

If I were writing in plain text (which I do often in correspondence with students) I would write 1/(ab) if I want the product of ab to be computed before the division, and (1/a)b if I want the division to happen first. Both of those are less ambiguous than 1/ab.

The fact that I have to “waste” two keystrokes to print the parenthesis is not a big deal compared to the reduction in ambiguity. Maybe I would think differently if I ever bothered to type math in Twitter, but I have no plans to ever do that.

The Monty Hall problem is the one I hate. The answer can be anything from 0% to 100% depending on the host’s strategy, and describing a single instance is not enough to determine which strategy he is using.

Well, of course it will depend only on the host and you. And since for you, every door is essentially the same at the start, it only really depends on the host in the first two steps.

And if you know nothing about the host’s stategy, it’s less of a maths problem and more of a guesswork problem, like “which finger am I holding up?”

You can deduce which strategy the host would use though, based on what would make for an interesting game show. The game always starts with the contestant guessing one of three doors. If they guess the prize, the host has two doors to show off which are duds, so it doesn’t matter which door they show off. If the contestant guesses one of the duds, the host can either show the other dud, which means the contestant can still guess for the prize, or the host can either show the prize, which would not work as a game show, because the contestant already knows what door has the prize. This still works even if it’s more complicated with a “best” prize, “medium prize”, and “worst prize”. The host will never show the prize that’s highest value, because it’s the thing the contestant must try to guess for.

But “most interesting” is not an objective metric. If the contestant guesses the right door you could argue that not letting him second-guess his decision is the most interesting, if the contestant winning is more interesting than the contestant second-guessing himself and losing.

The host never reveals what’s behind the first-guessed door, though. If that were the case, it would be a different game-show / problem to solve.

The Monty Hall problem used as an example assumes that the host is required to open one door that does not have the prize behind it after you make your initial selection.

When properly presented, the Monty Hall dilemma makes perfect sense. It basically boils down would you rather pick one door out of three or two doors out of three?

Most descriptions I’ve seen do not mention this.

It’s usually considered to be implied he won’t open the car door and doesn’t have discretion to not open a door, I think. Anyways, those are the rules of the classical version where you should always switch because it gives you a 66% chance of getting the car. The actual presenter has admitted that on TV he didn’t play by those rules, and usually didn’t offer the switch if the guest picked a goat door.

I’ve never heard the problem presented that way, but if that is what he actually did then sticking with your first guess is the appropriate strategy when offered a switch.

If the host follows the rule: open door to weak (or no prize) and then allow switch then you have the classic problem and should always switch.

I’d love for you to describe a host strategy that makes the chance of winning when you switch 100% or 0%. That sounds impossible to me, assuming the basic premise of 1) you pick a door, 2) Monty opens another door to reveal a goat, and 3) he asks if you want to stick or switch.

Alex started from a different assumption,where you witness just one case of monty opening a door.With this you dont know if monty is opening a door at random or if he knows in advance whats behind every door.If he does not know,then there is a chance for him opening a door with a car,thus reducing your chances for win to 0%.

As for 100%,the only thing I can think of is that you dont know in advance that there are two goats,so that you can think there are two cars.But thats a bit iffy.

You can hardly complain about the Monty Hall problem being ambiguous if you go and change the parameters of the question. If you start from different assumptions, then you aren’t talking about the actual Monty Hall problem.

There

aredifferent strategies that Monty can apply that result in different odds (for instance, if Monty picks the door randomly, then you have a 50% chance of winning by switching when he reveals a goat), but I’m certain that 0% and 100% are impossible if the problem behaves as I described (and additionally, in most descriptions I’ve seen of the problem, it’s mentioned that Monty deliberately reveals a goat, which means the chance of winning by switching is unambiguously 2/3).Small addendum: as described below, 0% is a possible outcome if Monty only reveals a goat when you pick the car. This does satisfy my criteria, since you can make a choice, then have Monty reveal a goat, then switch, and you will always lose. The probability P(switching wins | Monty reveals a goat) is 0.

Edit: Double damn, you can actually get a 100% chance of winning by switching if Monty only offers the choice to switch if you choose wrong. Very well, I retract my remark.

If Monty Hall’s system was “If the contestant picks the right door, reveal a random goat. If the contestant picks the wrong door, do not give him a chance to change his mind,” then changing your selection has 0% chance to win and keeping your selection has 100% chance to win. (Though in this case, 2/3 of the time he will not give you the choice.)

Dang, you’re right. That technically counts, since Monty does reveal a goat, and you do get the choice to switch. You just have P(switching wins | Monty reveals a goat) = 0.

Edit: Thinking about it some more, you can get a 100% chance of winning by switching if Monty only offers the choice to switch when you choose wrong the first time.

The host may know which door has the car and which have the goats, and his decision-making process may be informed by that knowledge.

The 0% strategy is “The contestant picked the car door, therefore show him one of the goat doors and ask him if he wants to switch (to the other goat door).”

The 100% strategy is “The contestant picked a goat door, therefore show him the other goat door and ask him if he wants to switch (to the car door).”

The 67% strategy – the intended answer – is both of these strategies at the same time.

The 50% strategy is “The contestant picked a door, now show him another door at random. If that door contains a goat, ask him if he wants to switch to the other unopened door.”

Unless you know that the host

alwaysopens a goat door you did not select, you cannot determine which strategy he is using – or whether it’s some combination of several strategies.The mathematical example problem assumes that “the host opens a door to reveal a goat” is part of the rules of the game, because a lot of people do make the mistake of assuming that even with that stipulation each door has 50% odds of having the car behind it and switching gives no benefit. This might not be clearly expressed in a specific formulation, but the “Monty Hall Problem” is for the case where the host has no discretion to not open a door.

Haha got me :D

My favourite “troll math” puzzle is this fruit puzzle:

https://imgur.com/a/DPhAk

With the clarification that you’re looking for positive whole numbers. Though it’s in a very different league of “troll math” to any of the other usual suspects.

There’s a good writeup here, but you might want to try solving it without spoiling yourself first ;)

Ahh, the picture did not specify “positive whole numbers” so I solved it by setting banana to zero and pineapple to 1, then solve for apple (sqrt(3) is the result)).

Positive whole numbers puts a different spin on it though.

…and I have already used up my quota of “waste time on internet instead of doing work” for the next few hours at least.

That is both awesome and evil. I had a go at it, and managed to prove to myself that only the ratio of the answers mattered, and therefor I could just assume c=1. And I really felt like I was getting somewhere, but I really needed to get back to the work they’re supposedly paying me to do, so I gave up and read the linked article.

I’m _so_ glad I had to go back to work. I wouldn’t have made it any further in a million years.

See, this is why I always add extra parentheses when writing math in my code even when the order of operations is correct and the compiler can figure out what I meant without parentheses. The gratuitous () just makes it easier for me to go back and read what I’ve written.

Most of those things are, I believe, designed to harvest contacts. So they deliberately rile up controversy so that the outraged will comment, and then collect data about all the commenters. I’ll grant you that this example is kind of clever, but it works just as well to simply post a question and the wrong answer.

Love it, and shared it on Facebook. Needed to do it manually though, the “share” buttons didn’t work for me. Might want to test them out.

To me, the most mysterious thing about this article is why the girl in the title image has so much eye shadow on.

And for the non math portion of the story:

Pushes glasses up while enunciating nasally, “I believe you mean “poor grammar”, unless you personally find improper grammar to be morally repugnant.”

Interestingly, this problem wouldn’t work in Spanish unless you troll harder. See, as you probably know, for exclamation sentences, the spanish language has two exclamation signs, one for the start of the sentence (Â¡) and one for the ending (!), so having only one would be a giveaway…

… unless you were to troll harder by deliberately writing the entire problem with poor grammar. That way once you finish with a sentence lacking a sign no one would question it.

In my head, “x!” isn’t “x factorial”, it’s just “x” but shouted with great enthusiasm.

I KNOW WHAT YOU MEAN, IT’S SO STRANGE, ISN’T IT?

I sometimes wish there was a mathematical use for the ? symbol just so we could make more little jokes like this. I’d say we could use it for subfactorials but that’s already got the notation !n for the subfactorial of n.

https://en.wikipedia.org/wiki/Minkowski%27s_question_mark_function

But that’s the wrong way round. I want a meaning for something like “n?”.

If you use the Windows calculator then remember it has “Standard” and “Scientific” mods.

Standard is more or less in the order of appearance, while Scientific uses the “standard” operator order.

As a protip, always use brackets, both in the programming language you use as well as in the calculator, that way you avoid any and all ambiguity. You really do not want wrong order of operators to cause your program to malfunction without failing.

Operator order bugs can be nearly impossible to hunt down.

Since there are so many of them, I can’t find the one I was looking for, but someone made a fruit-math problem out of an actually simple looking–but practically impossible for anyone without a doctorate in math degree–problem. It was something like:

apple/cherry + apple/pineapple + pineapple/cherry = 4.

I wonder if how devious this is was lost on people that actually like solving the regular fruit-math problems.

In case you were curious where this came from, it was started by me 2.5 years ago with a less clever equation here:

https://www.reddit.com/r/math/comments/3mfgar/i_made_this_to_see_how_many_people_i_could/

and improved on by a fellow math geek here:

https://solvemymaths.com/2015/10/09/fun-with-factorials/