There are 18 triangles.
"There are 18 triangles."
How did you get that number?
Sorry, it's just about where you put the parentheses, it's not a problem at all, and it's not very clever. It's just confused.
1
6!/3!=20
1
"6!/3!=20"
Assuming you are using the standard definition the factorial operator:
6!/3! = 6 x 5 x 4 = 120.
Fran (NJ): "And you didn't give the answer for how many triangles"
That's the trouble with these puzzlers -- they imply that there is an "answer". In fact, there is major ambiguity in the problem:
It is not clear whether the figure is supposed be in an infinite plane or if it is bounded by a rectangle that is not actually shown.
If the figure is bounded by a rectangle, then segments of the rectangle actually form one edge of three triangles (by my count).
So a professional mathematician commits the "sin" of ambiguity. Should the rest of us sneer?
Since the lines are not parallel, and since the plane is 2-D, they all intersect eventually. Every triangle is defined by a combination of 3 lines. So you just need to calculate every unique combination of three (lines 1, 2 and 3 make the same triangle as lines 2, 3 and 1). Also you don't want to consider the same line twice in each set of three: line 1 with line 1 with line 2 doesn't make a triangle nor does line 1 with line 1 with line 1). If the number of lines is N, then the number of unique pairs of lines is N(N-1)/2. The number of unique combinations of 3 lines is N(N-1)(N-2)/6. The denominator is actually 3 factorial which is written 3! and is equal to 1*2*3. There are different ways to illustrate why this is the way it is but it would be easier in front of a black board than in a NY times comment. But the key is that the -1 and -2 in the numerator removes contributions where you considered the same line multiple times and the denominator accounts for combinations that were already counted with the lines in a different order.
4
"... the trope that the core of math consists of memorized recipes of calculation."
However, memorization is very important in mathematics. Here is how the mathematician Paul Erdős (1913-1996) explains the importance of memorization:
"There are three signs of senility. The first sign is that a man forgets his theorems. The second sign is that he forgets to zip up. The third sign is that he forgets to zip down."
A slightly different version of that joke can be found here:
"Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning" by Clifford A. Pickover.
2
Assuming the lines extend indefinitely beyond the figure, I count ten triangles. Obviously, if the borders of the figure count as lines, there are many more.
1
@Neil Bruce The borders do not count (imagine the lines going on to infinity), and it's more than 10.
17
@Neil Bruce
Sorry, wrong. Recalculating.
2
I love that you included some actual fun maths at the end! The triangle problem is really fun! I thought of it this way: each line has 5 points where they intersect the other lines. Any two of these points determines only one triangle. There are 10 ways to choose 2 of those 5 points. So each of the 6 lines has 10 triangles "hanging" off them, giving us 60 triangles. BUT each of these 60 triangles has been counted 3 times in this manner, once for each side of the triangle, so I have over counted by a factor of 3, so the number of unique triangles should be 60/3 = 20.
6
Those are ring-tailed lemurs in the photo. I had to do a web search to find an actually informative caption:
"Ring-tailed lemurs help with the counting during the Whipsnade Zoo annual stocktake in Dunstable, Bedfordshire, UK. Photograph: Carl Court/AFP/Getty Images" (theguardian dot com)
BTW, that photo would be a great one for a write-the-caption contest.
Why does such a simple arithmetical expression evoke such emotional responses from professional mathematicians?
Wilkinson: "I HATE this." "Secretly, it enraged me ..."
Ribet: "irksome"
Kuperberg: "I didn’t care. I wasn’t interested." (ennui being the putative emotion)
'“It implies that the point of mathematics [is] to trip up other people with stupid rules,” Dr. Wilkinson said.'
If anything, the lesson here is that what people learn in elementary school is not what professional mathematicians learn.
Perhaps the problem here is that math professors are failing to explain how mathematical expressions are written and interpreted by highly educated professors who seem to have forgotten what they learned in elementary school.
1
@BWG
I think you might have missed it. The frustration is that even though math professors teach these things all the time, many elementary, middle and high school math teachers don't really know what math is and when asked questions like "why do I need to know this?" they tend to trot out ridiculous answers like, you need math when you go grocery shopping. The real answer to that question is that a large number of jobs and disciplines require math and it's associated logic. Failing to learn math beyond the elementary stuff will limit your opportunities and those limits are getting more and more severe.
3
Paul: "Failing to learn math beyond the elementary stuff will limit your opportunities and those limits are getting more and more severe."
Learning to recognize ambiguity and how to resolve it is the more fundamental skill. Indeed, language is fundamentally ambiguous to one degree or another.
See, for example, the problem of circularity in dictionary definitions as exemplified by Irish mathematician and physicist John Lighton Synge's game of Vish:
https://en.wikipedia.org/wiki/Vish_(game)
The whole point of parentheses in an equation is to eliminate ambiguity like that found in the silly viral equation. That is why code is fussy about parentheses and brackets. If you meant to divide 8 by 2 first, you’d write (8/2)x(2+2).
9
"... which has been twisted with intentionally ambiguous notation ..."
That's mind-reading.
Steven Strogatz makes the same blunder when he opines that "someone who must have been a troll off-duty" had posted the expression. And when Strogatz gets to bat a second time he proceeds to strike out again with this doozy:
"The question was not meant to ask anything clearly. Quite the contrary, its obscurity seems almost intentional. It is certainly artfully perverse, as if constructed to cause mischief."
Profession communicators, such as journalists and math professors, should not be ascribing bad "intent" to people who have a perfectly reasonable question.
2
@BWG Given that this whole hullabaloo had made the rounds on the internet once already, with nearly identical kicking and screaming, I find it highly unlikely that was an innocently curious query.
15
@BWG
And the sentence: "The horse raced past the barn fell." was written with the intent of being clear and readable? It's easy to see that both the equation and this sentence are intentionally crafted to be confusing. It's as obvious as the conclusion that Trump is a racist.
8
Saw a nice T-shirt the other day;
"My password is the last 8 digits of pi."
43
"That’s the kind of problem that makes mathematicians smile."
And what about everyone else? Personally, I find that problem to be reminiscent of an SAT question.
1
Nine
@D. Wagner No.
15
D. Wagner: "Nine"
Please explain how you got your result.
Kenneth Chang: "@D. Wagner No."
You should ask the commenter for clarification instead of imposing your own interpretation on the comment.
@D. Wagner To clarify, I am referring to the triangles, not the equations. I printed out the diagram, and now I am seeing 10 rather than 9. I am not sure if that is correct.
I got 16 the first time I did the equation when it hit the news a couple of days ago.
Thank you Kenneth Chang for your brilliant editorial, "Why Mathematicians Hate That Viral Equation." As mathematician and an educator, if you use PEMDAS method, the answer is 1 or 16 if you do not use PEMDAS. ( although when I have taught this type of problem, I would use PEMDAS and thus would state the answer as 1).
By the way, to answer the posed triangle is 20, since this is the number of permutations of 6 lines, chosen 3 at a time or
6! / 3! = (6 * 5 * 4 * 3 * 2* 1) / (3 * 2 * 1) = 120 / 6 = 20
4
20 lines, 4(4)
I agree that there is no serious mathematical issue in the ambiguous equation, and it is very easy to use a fraction bar and parentheses to unambiguously write either of the ambiguous answers.
And there is no ambiguity in the solution (8/2)(2+2) = ? to the waiter problem. The correct use of parentheses yields a correct and unique solution.
And why am I wasting my time on this? There are way more interesting problems in mathematics. Here is one that is easily stated: You are in New Jersey plotting the logarithm , base 10, function. When x=1 foot you put a dot on the blackboard 0 feet high.
At x=10 feet you put a dot 1 unit high. At x=100 feet you put a dot 2 feet high. How far would you have to go in the horizontal x direction to put a dot 28 feet high?
Are any of the following correct answers? x = 300 feet? x= distance between NJ and California (roughly)? x= mean distance between Earth and the Sun?
For a hint to the correct answer look at https://en.wikipedia.org/wiki/Observable_universe
1
My students ask me 'how many significant figures in 120,000'
I say, 'the person reporting the number did not report in in such a way so as to communicate that'
Maybe this is the same.
Exactly! I didn't even read past the opinions expressed at the beginning of this article. Great that the NYT audience was entertained, but it ain't (real) math... Ho Hum...
@PGL There's some cool real math at the bottom of the essay.
4
You're taking me back to trig class! (Or was it calculus?) 20 years later, my calculus 2 classes haven't faired well in my career field of public relations!
In any case, based on the hint in the article, my guess is we're dealing with exponents, yes? 6 to the 3rd power or 6 ×6 × 6. Which means there's 216 triangles. Am I right??
As for the equation in question, my advanced math training says the answer is 1, purely because of how the equation is written. 8 is being divided by the the solution to 2 x 2 + 2. If the questioner wanted 8 to divide by 2 before solving the equation in the parenthesis, then the equation should have been written as such -- (8/2)(2+2).
12
@Jasmin It would have been probability or statistics. You're on the right track, but 216 is way too high. Also, once you make the choice of 1 line to be the first side of a triangle, you no longer have 6 choices to the 2nd side, right? Does that help nudge you in the correct direction? Love that you're thinking it through.
8
@Kenneth Chang this sounds all too familiar. The cobwebs are slowly clearing.
I knew I shouldn't have read this article -- I won't be able to get any work done until I recall which theorem this falls under so I can apply the correct formula.
Thanks for giving me some justification for going all the way to calculus two when I was a poli sci major and only needed algebra. Solving this problem will make it all worth it! ;-)
3
If I had been a math instructor during the recent dust up over the "16 or 1 equation" and my phone had been ringing off the hook with friends seeking the one true answer to the equation, I would have had a clarifying message they could access. "Sorry I missed your call. If you are calling about the equation, take two aspirin and call Kenneth Chang in the morning." Problem solved.
18
@richard wiesner I'd forward my voicemail to Steven Strogatz.
7
20 triangles, I think. There are 6 * 5 * 4 = 120 unique *ordered* trios of 6 lines. Each *unordered* trio would have 6 ordered variations, thus six duplicates in the list of ordered trios. Caveat that I know little or nothing of combinatorics.
53
@Adina That's exactly right. The cool thing is once you've figured out the pattern, you can do the exact same calculation when there's 7 lines, 23 lines, 547 lines, or 23 million lines. Compare that to brute force counting.
35
Re: the triangle problem. Using the formula for combinations (six non-parallel lines in combinations of three) the answer would be 20 triangles. I don't "see" that many but as Kenneth Chang noted, it's easy to lose track trying to count each triangle.
6
One of the interesting aspects of the triangle counting problem is that the diagram shows all the points of intersection of one line with another to be unique, in the sense that there is no intersection involving more than two lines. If this is not true, then the problem changes. If, for example, all the lines intersect at one point, we have what looks like a star, and there are no triangles at all.
17
@Larry Krablin Correct. In the diagram, there are no intersections of three lines. Thanks for pointing that out.
10
What I learned from the equation is that I don’t want to be a waiter.
1
@Mamavalveeta03, waiters are too pragmatic to ever write such an confusing formula.
I loved the triangle problem! But it did have an interesting twist: the number of triangles not only depends upon the fact that it is stated that the lines are not parallel, but also upon the manner in which they then must intersect. For example, if the diagram were not drawn for us, then working off of the description alone, the answer could be 0. No matter how many lines were drawn, if all of the lines intersected at a single point, no triangles would be formed. If this problem were to be given to students, this is a cool discovery (ie. don't give them a diagram). Also, do these lines all exist in the same plane? That also changes the answer and would make for an interesting twist.
7
@L.Douglas Yes. If it were merely a description, I would have to have been more precise. Picture is worth a lot of words and a few equations.
9
Love the math and geometry questions, I need the distraction from politics right now.
3
All that equation really proves is that communication without context is ambiguous. The division symbol suggested one context--elementary arithmetic--where the answer is obviously 16. The implicit multiplication in 2(2 + 2) suggests another context--algebra --where the answer equally obviously 1. Since implicit multiplication notation is unknown in elementary arithmetic and the use of the division sign discouraged in algebra, it's just plain a badly written equation. Or at best a mathematical pun. It says nothing about mathematics or a person's knowledge thereof.
"Fruit flies like a banana." How do you read that? Does the ambiguity there say anything profound about the English language?
22
@Adina
I'm not sure about profound but certainly complex or confusing depending on one's point of view. It's as if it's sometimes like a mess.
1
@digitusmedius,
Yet in any conversation about either airborne fruit or drosophila the meaning would be clear. 😊
@Adina, or you could say that how well you communicate is just as important as what you're trying to communicate.
I can see, I think, why mathematicians would distain this expression. They need to take their blinders off. It illustrates deep and profound truths about logic and humans. The order of operations matters. Jump off cliff; hit the ground, open parachute. Jump off cliff, open parachute, hit ground. Not Equal.
Cultural and individual understanding regarding order of operation is not as clear as we may think. Presumptions regarding this can be disastrous. Some of the most amazing and elegant mathematical expression somehow mysteriously describe real world physical truths. Schrödinger's equations come to mind. In a very different but real way this expression does the same.
5
@Michael Maher MD Of course, order of operations matter. So if the expression is misleading, rewrite it so that there isn't confusion. That's trivial to do in this case.
30
@Michael Maher MD, the problem here is not order of operation. It's one of effective communication. Whether you use words or symbols, it's just as important to express yourself clearly in math as it is in English prose.
9
Hi y'all,
If you're interested in the triangles problem, Dr. Loh's hint at the end of the essay is key starting point, which gives away about 30 percent of the solution. If you've studied any combinatorics, you'll know the right equation and get the answer almost instantly. If so, great! but I'd like to encourage everyone else is try it out for a while. I'll give hints and help.
2
I decided that I would try it myself when I got the Times's newsletter. So I used PEMDAS. But before I start, I wrote the equation 8/2 x (2+2).
First, parentheses. 2+2=4. Easy. Now we have 8/2 x 4.
We don't have any exponents, so let's move on to multiplication and division, left to right.
To the left is 8/2, which is 4. Know we are left with 4 x 4, which is 16.
The reason this is so confusing is that on the internet there are mostly adults, some of which forgot PEMDAS and other basic math skills. I am in school, so I remember my math skills.
@Aruna
The order of operations would be multiplication before division so the next operation after resolving the parenthesis would be 2 × 4, not 8 ÷ 2.
This was quite uncontroversial to anyone who's ever used a programming language.
The expression "8 / 2 ( 2 + 2 )" will always evaluate to 16.
End of story.
2
@asdfj that is exactly wrong! if you write the equation the way that you did, which is correct, then you can't perform the division until you have evaluated 2(2+2) which is 8. So you end up with 8/8=1. It's simple.
7
@asdfj
Unfortunately, that isn't the end of story asdfj. The reason there are rules about the order in which you evaluate a multi-operational expression is because when there is no context, there is no logic as to why one operation should be done prior to another--so rules were established. The way you have written the problem, the answer IS 1 not 16! Using a bar instead of the arithmetic symbol for division (as was done in the original problem) implies not only division but also grouping (think of it as a fraction): 8 is divided by the product of 2(2+2). 8 divided by 8 is 1.
8
@asdfj The (silly) question boils down to:
Does 8 ÷ 2(2+2) mean
8
-- (2 + 2)
2
or
8
--------
2(2+2)
?
Both of these are unambiguous expressions, so why use something ambiguous that is intentionally formatted to trick one into the latter?
If you meant the former, that would be a really weird way to write down that expression.
20
So, how many triangles are there? I think I counted 14.
@Sarah Mangini If you're trying to count, you're pretty much guaranteed to get it wrong. It's really hard to keep track.
9
Poorly written article. And you didn't give the answer for how many triangles
3
@Fran Did you figure it out? Happy to provide hints and help.
22
The answer is there!
1
@Kenneth Chang you're taking me back to trig class! (Or was it calculus?) 20 years later, my calculus 2 classes haven't faired well in my career field of public relations!
In any case, based on the hint in the article, my guess is we're dealing with exponents, yes? 6 to the 3rd power or 6 ×6 × 6. Which means there's 216 triangles. Am I right??
1
The expression in parentheses is always evaluated first. Then multiplication and division are done before addition and subtraction. Expressions are evaluated with left to right associativity so the division is done first. DUH! Everyone that ever programmed a computer knows this. Are mathematicians really as innumerate (like illiteracy with numbers) as this? This seems like an article about general ignorance, not math.
8
@Billy Bob Gascan
I totally agree. The rules apply. Expressions in parentheses are calculated first.
4
@Billy Bob Gascan In computer languages, there is (always? almost always?) an explicit multiplication symbol.
If it were written as 8 ÷ 2 x (2+2) = ? then (almost) everyone would agree the answer is 16. The implicit multiplication is in there deliberately to add confusion.
Suppose you had the expression
8 ÷ 2a, where a = 4
In algebra, you would regard 2a as one factor so that would be
8
---
2a
Subbing in a = 4, you'd get 1 in this case. Unless you insist this is actually 8 ÷ 2a must mean (8 ÷ 2)a.
20
@Billy Bob Gascan except that this is not a computer program. no programming language or very few will let you write that expression without an explicit multiplication operator *. This changes the world. Without a * you are implicitly saying that the 2(2+2) is a single cohesive unit. With a * in the middle, 2 * (2+2) become two separate units 2 * (2+2)
5
"There’s no aha moment, just confusion and discord."
Sure there is. It teaches people to avoid ambiguity whenever possible. The question isn't "how do we solve this?" It's a rhetorical "shouldn't this be written in a better way?"
So no, maybe it's not math, but there's still a lesson underneath the conundrum. Classify it as simple philosophy, perhaps? A debate on the ethical value of clarity in the digital realm?
4