Friday, September 7, 2012

Where the math is lacking...

My friend and martial arts instructor (I'd call him more of a guru than an instructor, really) posted this image on his Facebook recently (a friendly invite to attempt an answer):





There's a bit of a trick to this puzzle, but not a very difficult one to figure out.  Once the trick is realized the answer must obviously be 30, but is it all that obvious?  Apparently not.  Many people responded with an answer of 1.  Where did they go wrong?

Let's talk about the trick first and then we'll get into the place that seems to be mentally "tangling up" the many people who responded with incorrect answers.



The "Trick"

Mathematical equations are read on a page just like our English: left-to-right and top-to-bottom.  Usually, a number or a term is not split, meaning that additions and subtractions can be split across rows, but terms and numbers usually aren't.

It would be common to see

1+2+3+4
+5+6+7...

But far less common to see 

11+12+1
3+14+15...   

(see how the 13 got split apart there).


We keep numbers and terms together to avoid confusion.  And that's the trick to the problem at hand.  It's just a bit of confusion.  A first look at the problem might suggest the answer is 12.  That's because our minds want to read it as

1+1+1+1+1
+1+1+1+1+1
+1+1x0+1 = ?


But, upon close inspection, you can see that this is not the case.  There are not "+" signs in front of the second and third row.  The string contains two elevens that are split.

1+1+1+1+1
1+1+1+1+1
1+1x0+1=?

could also be read as

1+1+1+1+11+1+1+1+11+1x0+1 = ?


Once someone has figured out the silly little trick of the problem, they should come quickly to the correct answer of 30 (or maybe it's 2.  Or even 12!  See the update at the end of the post).  

But, this is where I'm seeing a lot of trouble...



PEMDAS

Many people are responding with an answer of 1.  It appears that they are all running through the string without considering the order of operations (i.e. 1 plus 1 plus 1... times 0 equals zero, plus one equals one...), but this is incorrect.  Does anyone remember Please Excuse My Dear Aunt Sally (PEMDAS).  Before learning algebra in school, most children are taught about the order of operations in equations.  PEMDAS (or the mnemonic if you prefer) stands for Parentheses then Exponents then Multiplication then Division then Addition then Subtraction.  It's the order by which operations stand when considering an equation.

From PEMDAS, the equation

1+2*3 = ? 

is different from the equation

(1+2)*3 = ?

(the answer to the first is 7, and the answer to the second is 9)


Due to the order of operations, the initial problem posted by my friend should be solved by multiplying the 1 by 0 first (obtaining an answer of zero) and then running through all of the additions to achieve the answer of 30.



Another Way to Think About It  

Seeing that some people don't quite understand PEMDAS and the Order of Operations, I gave another suggestion for looking at the problem using algebra (though this is a bit more complex than using PEMDAS, it may seem more intuitive for some who are used to logic or algebra):



"Let me try to help a little more here. 

Let's replace the zero in the equation with a variable, let's call it "a", and then we can set the answer of the equation to a variable as well, I'll call it "b". Then the equation becomes 1+1+1+1+11+1+1+1+11+1*a+1=b

Since 1*a=a (multiplicative identity), we can re-write the equation
1+1+1+1+11+1+1+1+11+a+1=b

If we solve a little further we get 30+a=b. 

Now go back in and set a=0 and you will see that b=30."

This is really how I personally would look at this problem.  Instead of thinking directly of PEMDAS right away, I would consider 1x0 to be a singular term since it involves a multiplication and then run through all of the additions while adding in the entire term of 1x0=0 when I get to it.  That may be troublesome for some people, though.  



Who Cares What the Answer Is?

I understand that many of us are quite removed from our grade school years, and so I don't begrudge those who have forgotten the little tricks we were taught in arithmetic and basic mathematics (such as PEMDAS).  There's nothing wrong with getting the wrong answer to a simple math question on a Facebook post, although those who get the wrong answer would do themselves a service by figuring out why they were wrong.  

The main reason I cared enough to write about this problem has less to do with people getting the wrong answer and more to do with some comments that were posted by those who have the wrong answer.  There were some not-so-bothersome comments, such as:

"The answer is 1 simple math"

"OKAY ANYTHING X 0 IS 0 + 1=1... ANSWER IS 1 ...FINAL ANSWER"

These weren't too bad, admittedly.  Their answers are incorrect so the math is not as simple or direct as they assume.  They could probably just use a friendly hand to help them to find the correct answer.  

There was a slightly more bothersome response:

"Let's forget this Alghabra or whatever crap and turn
This into a real math problem , if
Johnny has
1 apple and finds an
Apple
How many does
He have. Now that's valuable
Math the highest math I had explained parentheses first , my kid told me
Some
Pendas thing
But didn't make
Sense
To 
Me."

The lack of good grammar and the weird formatting are far less bothersome than calling algebra "Alghabra or whatever crap".  That saddens me.  Algebra has been a great mathematical tool in human history.  Without algebra, we wouldn't have gotten to the point of having an internet and a computer for this person to have responded to a post on Facebook.  

But, it gets worse, someone who posted an incorrect answer also posted this:   

"Obviously too many people are relying upon their union backed public education. Makes no difference what was before the x 0 because at that point mathematically it all becomes 0, then the final part of the equations is 0 + 1 = and unless your IQ is an equivalent digit you must come up with 1 as the result. And there was no annotation indicating this equation was part of some computer language programming so it should be read as a simple mathematic equation. DuH! No wonder the alien won't talk to us when we have this many dumbasses on the planet."

"...union backed public education."  Unfortunately, this person appears to have missed out on such an education.  Not only is this person's answer incorrect, but they go even further in an attempt to intellectually insult anyone with a different answer.  It might seem petty for me to be bothered by this, but I think one reason we have so many poorly educated people in this nation who are lacking in scientific and mathematical literacy is because people such as the person who posted this response make others feel bad for attempting to learn.  Learning can be hard.  It can be embarrassing.  People should never be meant to feel like they are less intelligent or less capable as humans if they get a wrong answer.  We should go out of our way to share our collective knowledge with our fellow people (when they are willing to listen), but we should never go as far as to insult them along the way.  

One final thing that "erked" me was this part of that poster's comment:


"No wonder the alien won't talk to us when we have this many dumbasses on the planet."

As a cosmobiologist, I think about the "what ifs" of intelligent alien life and potential interactions with other species.  Is it possible that there are intelligent aliens out there who know about us but choose not to communicate with us?  Yes, this is one of the potential solutions to the Fermi Paradox (Wikipedia: Fermi Paradox).  

Could it be that there are intelligent aliens out there who won't talk to us simply because there are too many "dumbasses" in our populations?  Considering the last comment I shared on this puzzle post one might assume that may just be the case.  We have a species full of intelligent people who are lacking in their critical thinking and reasoning skills and yet who are fully certain of themselves in their ignorance.  However, it's pretty doubtful that an extraterrestrial civilization wouldn't talk to us solely because we have some "bad apples".  

We have problems in our world that are far more important, and far harder to reason through to an answer, than the simple math puzzle at the top of this post.  I think we would do ourselves and our fellow humans a great service to avoid treating others like less for getting different answers (even if we are certain our answers are right).  I, for one, think that a greater focus on education and teaching is necessary to make our world a better place.



An Update (February, 2015): 

Should the Answer Actually be 2?  Or 12?
This post has long been my most-viewed post.  As of February 13th, 2015, this post has received over 10,000 views!  I decided to share this post once again (maybe I'm sentimental about it now). My friend, Anthony Rasca (a man who knows far more about mathematics than I do!) averred that the answer to this problem should have been 2.  Or maybe it's 12.  This is what Anthony says:  

"You cannot know for sure that all three lines are intended to be one unbroken string unless you know the text formatting rules used. Of course, that same argument can be used to claim uncertainty with the answer is 2. Assuming it is all one statement, but trying to look at it different again, the answer could also be 12"

Maybe this problem is more "problematic" than some of us have thought!  Is your answer 2?  12?  30?  Infinity plus one?  

This tricky little problem leads to some ambiguous solutions, depending upon which way you choose to look at it.

19 comments:

  1. Hey, nice post but here's another approach.

    Why assume its wrapping, why can't it be just 3 different lines and the third line is the actual problem in which case the answer is 2

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    Replies
    1. I apologize for missing this previously. Blogger hadn't been telling me when people were commenting. I like the way you're thinking about the problem. It's true, it could be that there are just some fun digits and cross symbols in the first two lines, and then your solution would be correct. Fun fun.

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    2. The answer is 2. Unless you write eleven like this:
      1
      1

      But that has never happened. Ever.

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    3. I'm guessing you haven't taken the time to read the post. If you had, you'd see I mention the answer of 2. As for you way of writing 11, it also would seem fair for me to guess you've never used a TI-86 or TI-89 calculator (as you would then know that they wrap numbers across lines after 21 digits).

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  2. What if X was not times but an unknowne in a equation like 1x+2=2x-2

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  3. 1x + 2 = 2x -2
    x+ 2 = 2x -2 (subtract x from both sides)
    2 = x -2 (add 2 to both )
    4 = x
    So x = 4

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  4. Indeed. We could look at it that way. If "x" were a variable in the original question, then that term would read as "1*x*0" which is still zero, regardless of the value of "x". Unless you meant to take it a more abstract way and would use the term "1x0" to imply some three digit number with a variable as the second digit. That doesn't exactly follow with the common ways to write mathematical equations, but would make the problem a bit more interesting. That would make the answer something like "130+10*x".

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  5. according to what you say the answer should be 3, because for every line there are only 9 characters then there is no space for 0

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    Replies
    1. I think the character spacing probably has more to do with the aesthetics of appearance than with the content of the problem

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  6. No.
    .this is not logical. Under no circumstances do we ever separate a 2 digit number on 2 separate lines. We don't do it, calculators don't do it and computers don't do it. This makes the top lines an error and invalid to the equation. If 30 is the answer the creator of this problem came up with then this is the dumbest and most pointless equation I've ever seen!

    ReplyDelete
    Replies
    1. I'm guessing you've never used a TI-86 or TI-93 calculator. They have smaller character limits on their screens and so they split numbers across lines. As the post now concludes, it depends upon the way you look at it.

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  7. It is a nice post, but unfortunately people do forget that Math has Rules to follow by. A number is NEVER allowed to split itself from the real number. So if it IS 11, then WHY didn't the number move to either of the rows. It may be for fun, but math rules apply first to any puzzle. So either you have THREE separate numbers or the equation is LABELED, in which is allowed in math rules to be named improper. Remember the rules and laws of math. They always aplly no matter what game is played.

    ReplyDelete
    Replies
    1. I do love math. However, one of the "tricks" of these Facebook puzzles is to bend and break the rules in ways that are fun (and require us to think a little outside of the rules). For instance, you might like my other post on Facebook math puzzles of the kind where they use an equality to imply an operation as well as an equality: http://cosmobiologist.blogspot.com/2015/03/only-for-geniuses-no-finding-answer.html

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  8. I was struck by the unrelated comment about aliens not wanting to contact us.
    Likely as not, humans could be the most intelligent, peace loving race of beings in the universe.
    It's reasonable to assume that most if not all technological races evolved from similar beginnings.
    In spite of "Hollywood science," our history and current problems are not at all likely to be unique in the universe. No matter where you are, you're probably in the middle.

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    Replies
    1. Agreed. I think that comment tends to be used mostly as a joke about our collective stupidity when we as a society do stupid things (I've seen that on lots of different kinds of topics on Facebook posts).

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  9. "unless you know the text formatting rules used." LOL!!! Like it's a computer question or something. The answer is OBVIOUSLY not 12.

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    Replies
    1. Ya, I'm not a huge fan of 12 for an answer. Although I still like 30, I can see why a lot of people prefer 2.

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  10. I agree there are few different ways to sve this. Answers of 30 and 2 have been explained. Now a different idea, when you have letters next to each other it's multiplied ex e=mc...
    So what if we take the same rule for the equation
    (1+1+1+1+1)
    (1+1+1+1+1)
    (1+1×0+1)=
    5×5×2=50
    Or
    1+1+1+1+(1×1)+1+1+1+(1×1)+(1×0)+1=10

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    Replies
    1. Oh, that's an interesting approach to it. I love it!

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