Find the Perimeter!

Note: the lines on the drawing are not to scale.

Calculus Made Easy, 1910: All the little bits

Calculus Made Easy is a book on introductory calculus originally written in 1910 by Silvanus P. Thompson. It was such a well-received text for teaching calculus, that it can still be found in print today (it's now in the public domain and also freely available online).

The first bit of text from the first chapter is fairly intriguing. So much so, that I thought you might like to read it here:

Thinking of the integral of dx as an easy way of saying "the sum of all the little bits of x" is a really intriguing approach to explaining some of the most basic ideas of calculus. 

If you've struggled at all with learning calculus, then you may want to give Calculus Made Easy a read. It may be over 100 years old, but much of the writing is still of great value for the interested student.

An emoji puzzle that takes a little integral calculus to answer

You may have seen some of the emoji puzzles that make their way around Facebook and other social media sites. Usually they require that you do some simple algebra (like adding 3 monkey emojis equal the number 15, so what number is one emoji monkey representative of?), but I just saw this little gem that actually required a little bit of fun ol' integral calculus. Give it a look-see:

What do you think? Does it look like fun? Give it a go and then I'll post the answer below.






















Okay, let's talk about the answer to this fun little puzzle. It starts off pretty easy, just using a little bit of algebra. The first part gives us three bottles of beer added together to equal 30. Easy enough, right? Each bottle must represent the number 10:

After that, we get to mix our beer bottle variable with a new variable, a cheeseburger! Again, pretty easy math. The cheeseburger must represent the number 5:

And then, again, we get to use the variable from the last bit to workout the next part of the problem, where we find that two glasses of foaming ginger beer (hey, it can be whatever you want it to be, really) will represent the number 2:

But after that, things get a little harder. Now we have the following integral:

We can start by plugging in the stuff we already have (in this case, our beer bottle, cheeseburger, and glasses of foaming ginger beer variables). That yields:

Which then can be rewritten as:

If you haven't had much experience with integral calculus, that expression above probably still looks pretty confusing. If this is the case, then you might want to check out Khan Academy's lessons on integral calculus, since that'll give you a good leg up on how this type of math works. But, assuming you already have some experience with integral calculus, you might notice that the above expression is very similar to the improper integral of the sinc function over the positive real numbers. This kind of function actually has a specific name and a well known solution. It's called a Dirichlet Integral, and, in this case, has a solution of pi over 2:

So, if we solve the same way using our previous expression from the problem at hand, we get:

So the answer to the original problem is numerically 5pi/2. But we started off with a mix of emojis and numbers, so why not go back to emojis. We already have emojis for 5 and 2 (cheeseburger and two frosty glasses of ginger beer, respectively), but we need one for the number pi (which is usually represented by the Greek lower case letter). Why not use pie?! We then get a final answer of:

And that makes the problem even more fun! Now I think I'll go enjoy a cheeseburger and some ginger beer and follow up with a little pie. Cheers!

The blue lines in this image are parallel!

Whoa, Fractals!

A section of a Mandelbrot Set

Fractals are sets of information that exhibit self-similar patterns at many scales. They've become rather famous because they can be expressed as beautiful geometric designs which are quite appealing to explore.

With our current era of digital video and graphic design, there's a lot that can be done with fractal visuals these days. I've recently been following artist Julius Horsthuis who renders some gorgeous fractal visualizations. Here's one of them, called Fractalicious 4:  

If you enjoyed the video, then definitely jump over to Horsthuis' website and watch some of the others there. There's definitely some awesome stuff that can be done with fractal designs.

Happy Thanksgiving. Enjoy the Pumpkin Pi

For those of you who celebrate Thanksgiving, I hope it's a very merry day for you (and that you get to eat some pumpkin pie!). Everyone else, I hope your year's harvests have been grand. Be they in foods that you worked through the year to cultivate, or maybe if your harvest this year comes in a bounty of happiness, health, and life, I hope you have some time to reflect on what life has provided for you and what you've managed to provide for yourself and others. Cheers all! 

Another order of operations adventure

Here's a gem I've seen floating about in Facebookland lately. A lot of people are answering with either 58 or 10. What do you think?

Find this puzzle and more awesome stuff at Curiosity

What do you think? Did you get 58? Did you maybe get 10? 

If your answer is 10, you likely made a mistake in using the order of operations. Let's talk about what that mistake was.

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The Order of Operations

As you may know from my previous posts (like Where the Math is Lacking), I love a good math problem. Problems like the one above are a great way to see whether people remember the Order of Operations for solving mathematical equations. 

In case you have forgotten the order of operations, then you might recall the mnemonic acronym "Please Excuse My Dear Aunt Sally", which stands for PEMDAS.

PEMDAS is an acronym for the order of operations and it stands for Parentheses-Exponents-Multiplication/Division-Addition/Subtraction:

This order of operations allows us to make sure that we're all following the same rules for solving or simplifying problems, or, at least, that's what it's supposed to do. Unfortunately, a lot of people have forgotten how to use the order of operations. Let's talk about that problem above again and how the order of operations applies to it.

We have the following problem:

6² ÷ 2(3) + 4 = ?

Let's step through each part of PEMDAS and see how the order of operations applies to this problem.

We have to start with the Parentheses part of the order. This is actually the step where most people get the wrong answer to this problem. The Parentheses step means that we have to solve terms or sets of terms within parentheses and brackets before solving the rest of the problem. 

For instance, if I had 2+(3+4), then I would solve the stuff inside of the parentheses first to get 2+(7). At that point, the parentheses are no longer necessary, since 2+(7)=2+7=9.

In our problem above, what's inside of the parentheses (the number 3) doesn't need to be operated on since it's already simplified. So, following the parentheses step, we still have the problem:

6² ÷ 2(3) + 4 = ?

Now moving on to the Exponents part of PEMDAS, we see that we have to solve the six squared term. That's pretty simple:

6² ÷ 2(3) + 4 = 36 ÷ 2(3) + 4 = ?

Then, the Multiplication and Division step tells us to solve all multiplications and divisions in the problem, but there's also a caveat there as well. Since many western languages read from left-to-right, we've also developed our modern mathematics to read left-to-right. This means that we have to sequentially solve multiplications and divisions reading from left to right in a problem. So for out current version of the problem,

36 ÷ 2(3) + 4 = ?

We have to solve the division of 36 by 2 first and then we can apply the multiplication by 3. Yielding:

36 ÷ 2(3) + 4 = 18(3) + 4 = ?

and then

18(3) + 4 = 54 + 4 = ?

And now we can take this thing home by solving the final step, Addition and Subtraction.

54 + 4 = 58

Bump bump baaaaaaah! The answer to this problem is 58. 

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So how did some people get 10? 

Like I said, most people who had trouble with this problem made their mistake in the first step. When solving for the parentheses part of PEMDAS, you only have to sequentially solve the parts that are within parentheses and/or brackets. However, if you apply the multiplication outside of the parentheses first, then you'd be solving the problem like this:

6² ÷ 2(3) + 4 = 

6² ÷ 6 + 4 = 

36 ÷ 6 + 4 = 

6 + 4 = 10

However, this is actually the wrong way to solve this problem. If you apply a multiplication outside of the parentheses first, you're basically then solving the operations out of order (MPEMDAS, in this case). The parentheses step in PEMDAS doesn't apply to operations outside of parentheses or brackets.

Here's another example I can throw at you, say we have this problem:

6 × (5 + 3) + 4 = ?

We could take away the multiplication symbol since the parentheses will already tell us to multiply, yielding:

6(5 + 3) + 4 = ?

Now it should be clear that we solve the parentheses first and then multiply, giving us:

6(5 + 3) + 4 = 6(8) + 4 = 48 + 4 = 52

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The same way that we removed the multiplication symbol in that problem could be applied to the original problem above. When you look at the problem and see it like this:

6² ÷ 2 × (3) + 4 = ?

Then it becomes far more apparent that the answer won't be 10. We can then see the right answer will be 58:

6² ÷ 2 × (3) + 4 = 58

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Silly online math puzzles: Here are three more for April 2015

I've been having a pretty fun time with sharing some of these puzzles that come across my Facebook feed, so here's another installment of some of the silly puzzles that have been posted by my friends and my solutions to the problems.

The puzzle above should be an easy one if you follow the order of operations. The answer should be C: 50. If you wanted to rewrite the equation to make it a little easier to understand, you could add some parentheses to make it:

7 + (7/7) + (7*7) - 7

Which can then be simplified by taking care of the division and multiplication to yield:

7 + (1) + (49) - 7

or, removing parentheses:

7 + 1 + 49 - 7

Which one can then easily see is equal to 50:

7 + 1 + 49 - 7 = 50

Trova la Soluzione!

As I mentioned in a recent post, those bits of text people keep adding to these puzzles suggesting that they're "only for geniuses" or that only some percentage of people get them right are absolutely bogus. Anyone can get the right answer if they understand what the puzzle is looking for and are willing to spend enough time working for it. Also, it's highly unlikely for most of these problems that there has been a significant portion of the population who've been tested with these problems to be able to make statistical statements about the likelihood of finding the answer. Still, I love puzzles and math problems, so I can't help myself. Take this one, for instance:

Trova la soluzione is Italian for "find the solution". I don't speak Italian, but in these glorious days of online information accessibility, we pretty much have Star Trek's universal translator at our fingertips whenever we're online.

Were you able to figure out the answer for the problem above? It's another that's pretty easy once you figure out the operation that the problem wants you to apply to each line. In this case, the final answer should be 126. If you start by adding the numbers on the left side of each equation, you should quickly see that there's a connection between that sum and the number on the right. In the first line, 2 plus 3 gives you 5 which can be multiplied by 2 to get 10. In the second line, 8 plus 4 gives you 12 which can then be multiplied by 8 to get 96. So, it pops out pretty quickly that you have to add the two numbers on the left and then multiply that sum by the first number on the left. Pretty simple, right?

Okay, well here's a slightly different type of silly online puzzle. This one isn't built from numbers, but rather asks you to think about what you're seeing. Take a gander at this one, trova la soluzione, and then tell me what you think in the comments:

Only for geniuses? No, finding the answer doesn't mean you're a genius, but these puzzles from Facebook can still be fun

I just came across another one of these puzzles on Facebook the other day. You may already know that I love a good puzzle. Ones like you see here are usually quick and fun. In the puzzle above we have this problem:

111 = 13
112 = 24

113 = 35

114 = 46

115 = 57

117 = ??

In this case, the solution could be 79. But you knew that, right?

These kinds of problems can often have multiple solutions. Some of them are easy and some of them require a bit of work.

The above problem has two little tricks. Firstly, you have to notice the pattern for each operation (the equality symbol in this case does not imply that the left and right are equal, but that there is some operation that can be done to one side to make it equal to the other). To get my answer, the last digit on the left of the equality symbol can be taken as the first digit on the right. The second digit on the right is then the sum of all of the digits on the left. However, the second little trick in this problem is to then skip 116 = 68 and go straight to 117 = 79. Ya, not a big shocker there at all.  Like I said, quick and fun.  

Here are some more of these types of puzzles
(can you figure out answers for all three?):

So, what'd you get for your answers? My answers are, in order:

90, 410, and 143547 (though there are other answers!)

I posted those three problems with an order of increasing difficulty, but they're all still quite manageable since they all follow the same idea as the problem that started this post: namely, assume that the equality sign implies an operation is necessary to form one of the numbers from the other. If you're curious to now how I got my answers to those three problems, then you can cruise down to the bottom of this post. 

Before I get to that, however, I want to point out that answering these questions correctly does not make you a genius, nor should you believe the statistics that claim that only some number of people can get the answer correct.

Only for Geniuses?

Something that really bothers me about these types of puzzles on Facebook is that so many of them say something like "Only for Geniuses" or "Only geniuses will get the answer". What's with that? It may seem a bit silly to think that anyone would feel like they are somehow a certified genius just for answering a very simple problem just because some text on the problem said so, but I wonder how many people really do. How many people out there are swindled by such claims? 

I guess the important question is what exactly is a genius? A quick search online will reveal that there is no well-constrained definition of genius. Dictionaries may give some definition for genius, such as "an exceptional natural capacity of intellect, especially as shown in creative and original work in science, art, music, etc." from Dictionary.com. You might also see definitions claiming that genius implies having a high Intelligence Quotient (IQ). But scoring high marks on a standardized test is probably not the best way to judge. 

It's seems fairly well agreed upon that all people who are recognized as geniuses can score fairly high on tests of logic and reason, but they don't necessarily have to have the highest scores. Richard Feynman, one of the greatest physicists and thinkers of the past century, self-reported that his IQ was only ever measured as high as 125. 

The word genius should be used to imply respect for someone who has shown their mental prowess with mathematics, art, science, creativity, oratory, or teaching, but surely it's not something that can be determined from a simple puzzle on Facebook. That's absurd. Maybe the people who add the "only geniuses" text to those Facebook puzzles are just misled about what "genius" implies, but it's more likely that they're pandering to others in the hopes to increase the "likes" and "shares" of their images. I think the latter seems more likely. Indeed, it's become pretty common for people to use pandering techniques on social media to try to improve their internet presence.

So, what's the point? Why do I care so much? 

I think the "only for geniuses" crap bothers me because I love puzzles, but I don't like having my intellect attacked by pandering. Finding the answers to these puzzles on Facebook doesn't make you a genius, but the people who actually believe that are most likely not geniuses. Even a genius can be suckered into something stupid once in a while, but I think a measure of genius is having the ability to recognize when you're being cajoled.

There's a quote that goes "Everybody is a Genius. But If You Judge a Fish by Its Ability to Climb a Tree, It Will Live Its Whole Life Believing that It is Stupid".  This quote is often misattributed to Albert Einstein though it was likely written in this form first by someone named Matthew Kelly. Is everyone a genius in their own way? I'd like to think so, but the problem with being a genius is that it's only a quality in a person that can be recognized by others. If you want to be a genius, don't worry about what some pandering asshole wrote on a Facebook image, but rather find what you're good at and use that to make the world a better place.

We need a society of people who are literate in statistics

One other crappy thing about some of these Facebook puzzles is when they throw out some ridiculous faux statistic about how few people get these things correct. For instance, the claim that "Only 20% will give right answer on 1st attempt" on the first problem in this post is laughable. Where did they derive such information? Nowhere, of course. They made it up because it sounded impressive and because, once again, they're pandering to people who can be won over through emotional wheedling.

Todd Snider has a song called "Statisticians Blues" where he points out that "...64% of all the world's statistics are made up right there on the spot. 85.4% of people believe them whether they're even accurate statistics or not." Check out that tune here:

What do you think? Are 72% of all statistics really made up right there on the spot? Are you one of the 80.4% who will believe those made up statistics whether they're even accurate statistics or not?

You might have noticed that none of those values matched, and you might have also noticed many times in your life when someone tries to throw some numbers at an opinion to make it sound stronger without having the ability to say where those numbers came from. So what's up with that? 

Statistics is a field of study that focuses on how to qualify and quantify data in a rigorous manner. It's about learning from measurable data and making predictions from known evidence. Statistics is very much a scientific approach to considering data, yet it's constantly misused, especially in our era of data overload.

Misusing statistics is just another form of pandering on these Facebook puzzles, just like the "only for geniuses" claims. These fake stats are an attack on your intellect and your reason, but also on your emotions. Don't fall victim to the wheedling of the people who don't understand statistics! We should all take a little time in our lives to be sure we understand the "writing on the wall"; statistics literacy should be a common goal of all peoples' education in our modern world.

Okay, okay, enough ranting

Here's how I found my answers to those three problems earlier in the post:

How did I get the answers of 90, 410, and 143547?

Let's step through each one and see if you agree with me (again, there are usually many ways to solve these problems, and they often include breaking some of the "rules" of mathematics to do so):

Here's the easy one

This one is very simple. The problem is:

2 = 6 

3 = 12

4 = 20

5 = 30

6 = 42

9 = ??

Just as with the first problem in this post, the idea is to forget what you know of the equality symbol (it doesn't imply equality here, but rather what must be done to the numbers on one side to make them the numbers on the other side). You should hopefully see rather quickly that the operation is to take the number on the left and multiply it by the integer that is one greater than itself (so multiply 2 by 3 to get 6, 3 by 4 to get 12, and so on...). Once you see that you should find that the final answer (??) is 9 multiplied by 10, or 90. 

It's nice to get the right answer, but fun puzzles are definitely not only for geniuses.


Update (13 June 2015): 

Someone left a comment on this post today. They found an answer for this problem of 36, and I love their approach. Here's the comment they left:

f(z) = z * | |z-6| -7|
2 = 6
3 = 12
4 = 20
5 = 30
6 = 42

9 = 36

As you can see, this approach works perfectly well for this problem. Although this one isn't as easy to "see" by looking at the problem, it's still completely valid.

How about this one?

This one takes a little more thought, but it's not terrible. The problem is:

5 + 3 = 28  

9 + 1 = 810

8 + 6 = 214

5 + 4 = 19  

7 + 3 = ???

Remember my answer was 410? Beyond having to think beyond the normal use of an equality symbol, this problem requires you to break your usual thinking of how numbers are arranged. You need to perform some operation to the numbers on the left to make the numbers on the right, but you don't have to follow the normal rules of mathematics and you don't have to consider a string of numbers to be one unique number. What if I told you to think of these numbers like each one is separate? How does this look?

<5> <3> = <2> <8>

Any better? What if I just said "you have to do something with a 5 and a 3 that can give you a 2 and an 8"?

Did you think that the difference of 5 and 3 is 2 and the sum of 5 and 3 is 8? If so, then you're on the right track. 

Applying that thinking to the rest of the problem shows you:

The difference of 9 and 1 is 8 and the sum of 9 and 1 is 10:

9 + 1 = 810

The difference of 8 and 6 is 2 and the sum is 14:

8 + 6 = 214

Following that convention to the unknown gives you, "the difference of 7 and 3 is 4 and the sum of 7 and 3 is 10", hence my answer of 410:

7 + 3 = 410

Don't worry if you didn't get it. That whole "99% People Failed to Solve this Question" malarky is just a bunch of bunk!

Now for the tricky one:

This one may look pretty intimidating, and, admittedly, it took me a couple minutes to figure out.  Here's the problem:

5 + 3 + 2 = 151022

9 + 2 + 4 = 183652

8 + 6 + 3 = 482466

5 + 4 + 5 = 202541

7 + 2 + 5 = ??????

My answer for this problem is 143547. Notice how it has 6 digits? Did you notice how all the other answers have six digits? That's a pretty important feature of this problem.

This is one where you have to look at the numbers available on the left and start trying to figure out the potential associations to single numbers, strings of numbers, and the whole numbers on the right. Doing this, one thing I noticed right away was that the first line has 5, 3, 2 on the left and it has 15 (the product of 5 and 3) as well as 10 (the product of 5 and 2). If you take a look at the rest of the lines, this relationship holds; product of first and third numbers followed by the product of the fist and third numbers. Let's make some notation to explain this.  Let's set a line up as:

A + B + C = xxyyzz

Then, from what I've said so far:

A*B = xx

and

A*C = yy

That just leaves "zz". You may want to think about that one for a moment. Try combinations of A, B, and C using general arithmetic. I'll post the final answer below, but here's a fun cartoon first:

Get more giggles at Explosm.net

Still hanging around? This is a pretty long post, but, if you love puzzles and games as much as I do, then I hope you've found it worth your while. So here's how I found the final answer:

In the first line we get the product of 5 and 3 (15) and the product of 5 and 2 (10) and, if we're looking close, we have 22 yet to figure out, which is pretty close to 25 (the sum of 15 and 10). In fact, 22 is 3 less than 25, which led me to my answer:

For A + B + C = xxyyzz

where A*B = xx

and A*C = yy,

we can find that zz = A*(B+C) - B

or, in other words, zz = xx + yy - B

Now using the numbers from the problem:

7 + 2 + 5 = (7*2) / (7*5) / (7*(2+5) - 2) = 14 / 35 / 47

and, finally:

7 + 2 + 5 = 143547

Finding this answer will not make you a genius and no one has any idea about how many people have successfully solved this problem (or have even run any tests on an adequate sample to statistically say roughly how likely you are to be capable of solving it compared to the general public). Still, if you're like me, then you've probably at least had some fun working on the answers. 

Feel free to comment on these problems, my approach to solving them, or suggest some more fun puzzles!

Element 73

Sheldon: I made tea. 
Leonard: I don't want tea. 
Sheldon: I didn't make tea for you. This is my tea. 
Leonard: Then why are you telling me? 
Sheldon: It's a conversation starter. 
Leonard: That's a lousy conversation starter. 
Sheldon: Oh, is it? We're conversing. Checkmate.

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Just watched the 5th season of The Big Bang theory.  In an early episode of this season, I noticed Sheldon Cooper (played by Jim Parsons) wearing a shirt with the number 73 imprinted on it:

I recalled there being an earlier episode (Season 4, Episode 10, "The Alien Parasite Hypothesis") where Sheldon avowed his love of the number 73, but I couldn't remember why.  Then I did a quick internet search (my how the world has changed) and found this quote from said episode: 

"The best number is 73. Why? 73 is the 21st prime number. Its mirror, 37, is the 12th and its mirror, 21, is the product of multiplying 7 and 3.  In binary, 73 is a palindrome, 1001001, which backwards is 1001001."

73: The number, the myth, the legend...

Ah hah!  Behold the power of the number 73.  If you check out the wikipedia entry for "73 (number)" you'll find all of this as well:

- Seventy-three is the 21st prime number. The previous is seventy-one, with which it comprises the 8th twin prime. It is also a permutable prime with thirty-seven. 73 is a star number.

- 73 is the largest minimal Primitive root in the first 100000 primes. In other words, if p is one of the first 100000 primes, then at least one of the primes 3, 5, 7, 11, 13, 17, ..., 73 is a primitive root modulo p.

- 73 is the smallest prime congruent to 1 modulo 24.

- 73 is an emirp.

- The mirror of 73, the 21st prime number, 37, is the 12th prime number. The number 21 includes factors 7 and 3. The number 21 in binary is 10101 and Seventy-three in binary, 1001001, both are a palindrome. In addition, of the 7 binary digits representing 73, there are 3 ones. Also, 37+12=49 (seven squared) and 73+21=94=47*2, 47+2 also being equal to seven squared. Additionally, both 73 and its mirror, 37, are sexy primes twice over, as 31, 43, 67 and 79 are all prime numbers (sexy primes are primes that differ from their next prime number by a value of 6).

Tantalum: the 73rd Element

Pretty cool, huh?!  73 is pretty awesome.  

But what about the 73rd element on the periodic table?  Is there anything cool about Tantalum? 

Tantalum (symbol: Ta) is the 73rd element on the periodic table.  It was discovered in1802 by Swedish chemist Anders Gustaf Ekeberg.  It has an atomic weight of 180.9479 and it's most abundant isotope has 108 neutrons in the nucleus.  

One could do an internet search and find their fill of chemical and physical information about Tantalum, but here are a few of the most interesting aspects (at least, the ones that I find most interesting!):

- Tantalum has a melting point of 3290 K!  Only tungsten and rhenium have higher melting points

- Ekeberg named Tantalum comes from King Tantalus, father of Niobe, in Greek mythology. Tantalum is almost always found with Niobium in nature and Niobium was named after Niobe.

- Almost all of our tech devices (computers, smart phones, HD TVs, etc) have capacitors containing small amounts of Tantalum

- Tantalum is commonly used in the production of surgical tools, metal sutures, and rods and plates for mending broken bones and other injuries (since Tantalum tends to resist chemical reaction with other agents)

So there you have it! Some interesting things about the number 73 and the element 73. Perhaps 73 really is the best number. Or, at least, maybe just one of the best.

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.