Showing posts with label puzzles. Show all posts
Showing posts with label puzzles. Show all posts
Wednesday, December 18, 2019
Monday, September 2, 2019
Tuesday, September 4, 2018
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!
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:
Sunday, April 8, 2018
Tuesday, September 19, 2017
"Who's The Most Stupid Here?" - a test of your bias in viewing a problem
I've seen the following picture pop up a lot lately in my social media feeds:
What do you think?
It seems like a lot of people instantly answer "4" and move on, but I don't know if the answer is all that simple. Let's consider what might actually be happening to everyone in the picture.
Number 1, the guy in the blue shirt who kind of looks like he's probably the kind of guy who smokes a pipe, is sitting on a branch doing nothing while numbers 2 and 3 are both trying to make him fall. We don't know how high the branch is, so the fall may cause minor injury or it may kill. Either way, number 1 is doing nothing to act against those who are trying to harm him. Is he a pacifist or is he just not paying attention to the world around him? Hard to say. If he's aware of what's happening but does nothing to stop it, then I'd say that's pretty stupid. If he's simply unaware of what's happening then he's also pretty stupid. Inaction, by choice or due to ignorance, is not a strong position to me.
Number 2 has the look of a guy who was once a high school bully but now is the old balding asshole who still takes pleasure in hurting others. We don't know why number 2 is sawing off the branch on number 1. The smile definitely makes it look like he's being a jerk, though it's also possible that, say, number 1 is a pedophile and number 2 is helping society make a hard but righteous decision. Again, it's hard to tell. However, since number 2 is looking at the branch he's sawing, we can say that he's aware that he's about to injure or kill number 1. Without having any justifiable reason for doing this, we can assume that 2 is a jerk. However, 2 also seems unaware that 3 is sawing the branch on which both he and number 1 are seated. Although the thickness of the branch may suggest that 2 will saw off number 1 long before 3 can saw off number 2, it still seems like 2 is oblivious to the fact that 3 is sawing the branch. His ignorance to his own situation while harming number 1 is pretty stupid, if you ask me.
Number 3 is the guy I'm most worried about here. We can't see his face, though he may be wearing a suit (maybe he's a Martin Shkreli executive asshole, kind of guy). Number 3 is sawing the branch of 1 and 2. Number 3 is not in danger himself (as far as we can tell). Number 3's action here will directly cause harm to 1 and 2 (well, as I said above, it may be more likely that 2 harms 1 first and then, after 1 falls, 3 will harm 2). Number 3 is looking at the branch he's sawing. He seems like he might be aware of the harm he's about to cause. What I see here is that 3 is about to injure or kill one or two people by choice. There could be good reason for it. What if 1 and 2 are both CEOs for health insurance companies that have been preying on the weaknesses of the nation while making themselves uber rich? Or, what if they're both serial rapists? However, without knowing more, it seems like 3 is the biggest jerk of them all. He's targeting others for harm.
Number 4, the one a lot of folks seem to think is the dumbest of the lot, is sawing his own branch and appears to be aware of it. He's looking down at the branch while sawing. However, he also has a smile. He could be unaware of the fact that his own action is about to cause him harm, but his smile makes me wonder if he knows what he's doing. Maybe he's committing suicide. Maybe he's harming himself on purpose. Maybe he knows the drop isn't very far and just wants to get away from the other idiots. Again, hard to say. If number 4 is unaware that he's sawing his own branch, then, yeah, he's pretty fracking stupid. However, if 4 is aware of what he's doing, then he may be the least stupid of them all. He's making a choice to do something (action), this something will not harm anyone else (1, 2, and 3 are not in danger through 4's actions), and he appears to be happy with the choice (not necessarily a good thing, but it could mean that he's found resolution in this choice). So, the only way 4 is the dumbest of them all, as most people say, is if he has no idea what he's actually doing. But we can't know that based solely on the picture.
So the choices become a little more difficult then. Which is dumber: inaction in one's own demise (number 1), causing harm to another while being unaware of your own danger (number 2), harming others outright (number 3), or harming yourself (number 4)? A lot of people think that 4 is the dumbest of them all, but I think that's only possibly the case if 4 is unaware of what he's doing. However, if 4 knows what he's doing, then I think 1, 2, and 3 are all far dumber than he. Number 1 is going to be harmed through inaction, number 2 is causing harm while seeming to be unaware of his own danger, and number 3 is about to harm others. But that's just my take on this situation (and making a lot of assumptions). What do you think?
It seems like a lot of people instantly answer "4" and move on, but I don't know if the answer is all that simple. Let's consider what might actually be happening to everyone in the picture.
Number 1, the guy in the blue shirt who kind of looks like he's probably the kind of guy who smokes a pipe, is sitting on a branch doing nothing while numbers 2 and 3 are both trying to make him fall. We don't know how high the branch is, so the fall may cause minor injury or it may kill. Either way, number 1 is doing nothing to act against those who are trying to harm him. Is he a pacifist or is he just not paying attention to the world around him? Hard to say. If he's aware of what's happening but does nothing to stop it, then I'd say that's pretty stupid. If he's simply unaware of what's happening then he's also pretty stupid. Inaction, by choice or due to ignorance, is not a strong position to me.
Number 2 has the look of a guy who was once a high school bully but now is the old balding asshole who still takes pleasure in hurting others. We don't know why number 2 is sawing off the branch on number 1. The smile definitely makes it look like he's being a jerk, though it's also possible that, say, number 1 is a pedophile and number 2 is helping society make a hard but righteous decision. Again, it's hard to tell. However, since number 2 is looking at the branch he's sawing, we can say that he's aware that he's about to injure or kill number 1. Without having any justifiable reason for doing this, we can assume that 2 is a jerk. However, 2 also seems unaware that 3 is sawing the branch on which both he and number 1 are seated. Although the thickness of the branch may suggest that 2 will saw off number 1 long before 3 can saw off number 2, it still seems like 2 is oblivious to the fact that 3 is sawing the branch. His ignorance to his own situation while harming number 1 is pretty stupid, if you ask me.
Number 3 is the guy I'm most worried about here. We can't see his face, though he may be wearing a suit (maybe he's a Martin Shkreli executive asshole, kind of guy). Number 3 is sawing the branch of 1 and 2. Number 3 is not in danger himself (as far as we can tell). Number 3's action here will directly cause harm to 1 and 2 (well, as I said above, it may be more likely that 2 harms 1 first and then, after 1 falls, 3 will harm 2). Number 3 is looking at the branch he's sawing. He seems like he might be aware of the harm he's about to cause. What I see here is that 3 is about to injure or kill one or two people by choice. There could be good reason for it. What if 1 and 2 are both CEOs for health insurance companies that have been preying on the weaknesses of the nation while making themselves uber rich? Or, what if they're both serial rapists? However, without knowing more, it seems like 3 is the biggest jerk of them all. He's targeting others for harm.
Number 4, the one a lot of folks seem to think is the dumbest of the lot, is sawing his own branch and appears to be aware of it. He's looking down at the branch while sawing. However, he also has a smile. He could be unaware of the fact that his own action is about to cause him harm, but his smile makes me wonder if he knows what he's doing. Maybe he's committing suicide. Maybe he's harming himself on purpose. Maybe he knows the drop isn't very far and just wants to get away from the other idiots. Again, hard to say. If number 4 is unaware that he's sawing his own branch, then, yeah, he's pretty fracking stupid. However, if 4 is aware of what he's doing, then he may be the least stupid of them all. He's making a choice to do something (action), this something will not harm anyone else (1, 2, and 3 are not in danger through 4's actions), and he appears to be happy with the choice (not necessarily a good thing, but it could mean that he's found resolution in this choice). So, the only way 4 is the dumbest of them all, as most people say, is if he has no idea what he's actually doing. But we can't know that based solely on the picture.
So the choices become a little more difficult then. Which is dumber: inaction in one's own demise (number 1), causing harm to another while being unaware of your own danger (number 2), harming others outright (number 3), or harming yourself (number 4)? A lot of people think that 4 is the dumbest of them all, but I think that's only possibly the case if 4 is unaware of what he's doing. However, if 4 knows what he's doing, then I think 1, 2, and 3 are all far dumber than he. Number 1 is going to be harmed through inaction, number 2 is causing harm while seeming to be unaware of his own danger, and number 3 is about to harm others. But that's just my take on this situation (and making a lot of assumptions). What do you think?
Saturday, March 19, 2016
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?
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.
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:
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:
Now moving on to the Exponents part of PEMDAS, we see that we have to solve the six squared term. That's pretty simple:
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,
We have to solve the division of 36 by 2 first and then we can apply the multiplication by 3. Yielding:
And now we can take this thing home by solving the final step, Addition and Subtraction.
Bump bump baaaaaaah! The answer to this problem is 58.
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:
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:
We could take away the multiplication symbol since the parentheses will already tell us to multiply, yielding:
Now it should be clear that we solve the parentheses first and then multiply, giving us:
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:
Then it becomes far more apparent that the answer won't be 10. We can then see the right answer will be 58:
 |
| Find this puzzle and more awesome stuff at Curiosity |
If your answer is 10, you likely made a mistake in using the order of operations. Let's talk about what that mistake was.
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.
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
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
Wednesday, January 20, 2016
Ten Fish in a Tank Riddle
I just saw a post on God's Facebook page that had a title with this fairly bold claim:
"NO ONE Can Solve This Riddle. Are You Able To Outsmart The Rest"
I couldn't help but click on the link, which then took me to a CrowdSocial page which had the following easy riddle on it:
Well, what do you think? It hopefully won't take you too long to figure out the answer. This riddle most certainly wasn't difficult enough to warrant the claim that I saw on the post that God shared. In fact, this kind of riddle really isn't difficult at all.
There are 10 fish in a tank. 2 have drowned, 4 swim away, and 3 have died. How many fish are left?
It's weird that only two fish have drowned since drowning fish means a lack of oxygen in the water, which then suggests that all of the fish should drown. I guess maybe the two fish that drowned just had a much higher oxygen requirement than the others. Maybe I'm overthinking this...
You've probably already figured out that there are 10 fish left in the tank. The 2 that drowned are part of the 3 that died, but we're not told anything about them being removed from the tank after death. 4 of them might swim away, but they won't get far (they're in a tank). So, all 10 fish will still be in the tank, even if dead or swimming in circles.
Of course, you have to allow that this is a fish tank we're talking about. The tank could be just small enough to hold 10 fish or it could be bigger than all of the oceans of Earth combined, but it's still a fish tank. Some guy posted in the comments on God's post that it could be a combat tank, but even then the answer still holds.
If you like an easy riddle like the one above, then here are some more that you might like from Distractify:
Before I leave you, here's another fairly simple riddle that might tickle your fancy. What do you think?
"NO ONE Can Solve This Riddle. Are You Able To Outsmart The Rest"
I couldn't help but click on the link, which then took me to a CrowdSocial page which had the following easy riddle on it:
Well, what do you think? It hopefully won't take you too long to figure out the answer. This riddle most certainly wasn't difficult enough to warrant the claim that I saw on the post that God shared. In fact, this kind of riddle really isn't difficult at all.
There are 10 fish in a tank. 2 have drowned, 4 swim away, and 3 have died. How many fish are left?
It's weird that only two fish have drowned since drowning fish means a lack of oxygen in the water, which then suggests that all of the fish should drown. I guess maybe the two fish that drowned just had a much higher oxygen requirement than the others. Maybe I'm overthinking this...
You've probably already figured out that there are 10 fish left in the tank. The 2 that drowned are part of the 3 that died, but we're not told anything about them being removed from the tank after death. 4 of them might swim away, but they won't get far (they're in a tank). So, all 10 fish will still be in the tank, even if dead or swimming in circles.
Of course, you have to allow that this is a fish tank we're talking about. The tank could be just small enough to hold 10 fish or it could be bigger than all of the oceans of Earth combined, but it's still a fish tank. Some guy posted in the comments on God's post that it could be a combat tank, but even then the answer still holds.
Pretty simple riddle, but still fun.
If you like an easy riddle like the one above, then here are some more that you might like from Distractify:
(Again, the title makes it sound like these riddles will actually be hard to solve, but that's really not the case. They're simple.)
Before I leave you, here's another fairly simple riddle that might tickle your fancy. What do you think?
Wednesday, April 8, 2015
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:
Which can then be simplified by taking care of the division and multiplication to yield:
or, removing parentheses:
Which one can then easily see is equal to 50:
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:
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:
Wednesday, March 4, 2015
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
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?):
(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!
Sunday, November 16, 2014
Plutarch Dies at the End
Wallowing in my own self-pitty last night due to the continued presence of my runny nose, cough, and stuffed-up sinuses from this damned sickness I've had, I sought out a horror film to watch alone and in the dark. I came upon John Dies in the End while searching through my Netflix queue. I had added said film because it sounded promising, though I couldn't recall having ever seen a trailer or read any reviews. So I jumped to the ol' Google and found the film's trailer to be enticing. The film is definitely worth a watch for anyone who enjoys humorous comedy, but I'm not offering a review of the film here. Rather, I'm writing this because of one interesting part of the film: the prologue.
The opening of the film presents a simple thought experiment in a not-so-simple and enjoyably quirky way:
What do you think? If you're anything like me, the first answer that comes to your mind is an obvious "no". The axe has been completely re-constructed, so the original parts that were used to behead the now-rotting, corpsified zombie-dude are no longer in your possession and are most likely just adding to the mass of waste at some local landfill.
However, that's not the reason that I think the answer of "no" is astoundingly obvious (you might not have caught the primary reason for the answer being "no" on your first watch of that video; if so, watch it again. Good to go? Awesome). Hopefully you saw that the primary reason that the answer to the question is "no" is because what slew Swastika-Tongue in the first place was one, some, or all of the eight bullets that you had shot him with before using the axe to remove his head (like I said, it's a simple riddle). However, if you take away the obvious answer and just allow yourself to assume that the real question of the riddle is whether or not the axe you now hold in your hand in the presence of Zombie-Swastika-Tongue is the same one that you had used the previous winter to remove the head of his former self, then you have another riddle that is really a re-hash of a much older thought experiment: The Ship of Theseus.
The Ship of Theseus is a thought experiment proposed by Plutarch in the first century C.E. It goes something like this: the ship in which the hero Theseus and the young Athenian men returned from Crete (see the myth of Theseus and the Minotaur) was honored by Athenians and kept in good repair in the harbor of Athens for many centuries. Over time, as parts of the ship would degrade, they were slowly replaced, so eventually there were not many of the original parts of the ship remaining. The question then became "Is the ship, after replacing part for part over time, still the Ship of Theseus?"
The thought experiment, as it stands, really questions the value an object has based upon its parts. It's a question of philosophical identity.
There are other versions of this thought experiment. Some of them predate Plutarch's Ship of Theseus. For instance, there is a version in which Socrates and Plato each slowly exchanged the parts of their carriages such that the parts that once were in Plato's carriage have been completely replaced with parts from Socrates' carriage and vice versa and then the question is posited as to whether Plato is now using Socrates' carriage or if he's still in his own.
Other variants of the thought experiment have come since the time of Plutarch (with some interesting additions). The version that appears most similar to the prologue from John Dies at the End is the one known as "My Grandfather's Axe": my grandfather had an axe which he gave to my father. My father replaced the haft before giving the axe to me. I had to replace the head. Do I still have my grandfather's axe?
One of the more interesting variants of this thought experiment was proposed by Thomas Hobbes, the 1700's English philosopher and author of Leviathan. Hobbes' addition to the thought experiment works this way: you have the Ship of Theseus. You slowly take one piece of the ship off and replace it with a new piece. The old piece you keep. You continue in this manner, replacing pieces of the ship and saving the removed pieces. As this is happening over time, you take the pieces that had been removed and use those pieces to build a new ship, of exactly the same structure and design. By the time you have replaced the final original piece of the ship, you now have two identical ships. Which one is the Ship of Theseus?
Here's a fantastic breakdown of the original Ship of Theseus thought experiment and the Hobbes version from Wireless Philosophy:
The thesis of Joseph Butler, as reviewed in that video and suggesting that "objects persist in only a loose and popular sense", seems like a nice way to shrug off the problem as not being a problem in the first place. This is usually a fun approach to a lot of philosophy problems since a lot of the time it seems like there's no resolution to a lot of philosophy problems.
The reason I like this thought experiment, be it after replacing axe parts following your unexplained need to slay and behead some dude with a swastika tattooed on his tongue or replacing pieces of Theseus' ship, is because it questions identity. We are constantly shedding cells and gaining new ones, so are we ever identical with who we were previously? Darth Vader was almost fully replaced by mechanical parts, so was he still Anakin Skywalker? The philosopher Wittgenstein might have thought these questions were balderdash ("Roughly speaking: to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing") and, if that were the case, he might have been right.
It would be interesting if we could just say that something is such because people agree to call it such. Maybe the Ship of Theseus is really just whatever anyone decides to call "the Ship of Theseus". Maybe Darth Vader is Anakin Skywalker because someone calls him so. If that were the case, then the answer to the thought experiment as proposed in John Dies at the End might be that the axe you're holding in your hand is the same axe as the one that beheaded Swastika-Tongue because his zombie has now said that it is the same axe (it might be a good conclusion since chances are you should be more worried about dealing with said zombie before considering philosophical puzzles anyway).
However, I still feel like the answer would be "no". Even if you had killed the dude with an axe in the first place (and not with one, some, or all of those eight bullets), the original axe has been completely replaced. The answer feels like "no" because none of the original axe remains and there are only two major parts of the axe to replace. When the problem is introduced as in the case of the Ship of Theseus, where the object is replaced a small amount at a time, that's when it gets harder to decide when to even consider the ship to no longer be the original.
Maybe one of the more interesting answers comes from those who like to add the temporal dimension to the consideration, such as in the Worm Theory as presented in the video above. When we question the temporal aspect of an object along with it's identity, we start hinting at a possible answer to the question (see Temporal Parts at Stanford Encyclopedia of Philosophy). We can say that any one thing is only ever fully identical to itself at one point in time, but then at other points in time it can only be similar to itself. Then using a name to define something falls back to the "loose and popular" context that Butler suggested. That sounds just about right, honestly. The answer that would suggest then is that the Ship of Theseus was only the same ship in the sense that it bore similarity to itself over time and that people still called it the Ship of Theseus is the only thing that made it the Ship of Theseus. In that case, the axe that hewed the head laden with a swastika-marked tongue and the axe that you now have to defend yourself against the zombie at the door are only similar, and maybe you would call it a different axe since you know you've replaced the parts but the zombie calls it the same axe since it looks similar to the original. Not a very rewarding answer, but an answer nonetheless (and now you can get on with hacking down the zombie as he is more than likely about to come at you).
There's rich food for thought there. Maybe Wittgenstein is right and it's nonsense to even worry about two things being identical. That seems to fit well with the answer that considers the temporal aspect to mean that an object has unique temporal parts during its existence (look up perdurantism). Whatever anyone's consideration of this little thought experiment may be, I think we can all agree that it's a lucky thing we don't truly live in a world where a guy can get shot 8 times and have his head chopped off with an axe but then still find a way to come back from death and then sew his head back on before coming to find us with the likely intent of exacting revenge.
Update (22 October 2017): I re-shared this recently and have had several people ask if I've read the book John Dies at the End. Happily, I can say, yes, I have, and I've also read the sequel, This Book is Full of Spiders. A third book in the series, What the Hell Did I Just Read: A Novel of Cosmic Horror, just came out this month. Looking forward to reading that as well!
The opening of the film presents a simple thought experiment in a not-so-simple and enjoyably quirky way:
What do you think? If you're anything like me, the first answer that comes to your mind is an obvious "no". The axe has been completely re-constructed, so the original parts that were used to behead the now-rotting, corpsified zombie-dude are no longer in your possession and are most likely just adding to the mass of waste at some local landfill.
However, that's not the reason that I think the answer of "no" is astoundingly obvious (you might not have caught the primary reason for the answer being "no" on your first watch of that video; if so, watch it again. Good to go? Awesome). Hopefully you saw that the primary reason that the answer to the question is "no" is because what slew Swastika-Tongue in the first place was one, some, or all of the eight bullets that you had shot him with before using the axe to remove his head (like I said, it's a simple riddle). However, if you take away the obvious answer and just allow yourself to assume that the real question of the riddle is whether or not the axe you now hold in your hand in the presence of Zombie-Swastika-Tongue is the same one that you had used the previous winter to remove the head of his former self, then you have another riddle that is really a re-hash of a much older thought experiment: The Ship of Theseus.
The Ship of Theseus is a thought experiment proposed by Plutarch in the first century C.E. It goes something like this: the ship in which the hero Theseus and the young Athenian men returned from Crete (see the myth of Theseus and the Minotaur) was honored by Athenians and kept in good repair in the harbor of Athens for many centuries. Over time, as parts of the ship would degrade, they were slowly replaced, so eventually there were not many of the original parts of the ship remaining. The question then became "Is the ship, after replacing part for part over time, still the Ship of Theseus?"
The thought experiment, as it stands, really questions the value an object has based upon its parts. It's a question of philosophical identity.
There are other versions of this thought experiment. Some of them predate Plutarch's Ship of Theseus. For instance, there is a version in which Socrates and Plato each slowly exchanged the parts of their carriages such that the parts that once were in Plato's carriage have been completely replaced with parts from Socrates' carriage and vice versa and then the question is posited as to whether Plato is now using Socrates' carriage or if he's still in his own.
Other variants of the thought experiment have come since the time of Plutarch (with some interesting additions). The version that appears most similar to the prologue from John Dies at the End is the one known as "My Grandfather's Axe": my grandfather had an axe which he gave to my father. My father replaced the haft before giving the axe to me. I had to replace the head. Do I still have my grandfather's axe?
One of the more interesting variants of this thought experiment was proposed by Thomas Hobbes, the 1700's English philosopher and author of Leviathan. Hobbes' addition to the thought experiment works this way: you have the Ship of Theseus. You slowly take one piece of the ship off and replace it with a new piece. The old piece you keep. You continue in this manner, replacing pieces of the ship and saving the removed pieces. As this is happening over time, you take the pieces that had been removed and use those pieces to build a new ship, of exactly the same structure and design. By the time you have replaced the final original piece of the ship, you now have two identical ships. Which one is the Ship of Theseus?
Here's a fantastic breakdown of the original Ship of Theseus thought experiment and the Hobbes version from Wireless Philosophy:
The thesis of Joseph Butler, as reviewed in that video and suggesting that "objects persist in only a loose and popular sense", seems like a nice way to shrug off the problem as not being a problem in the first place. This is usually a fun approach to a lot of philosophy problems since a lot of the time it seems like there's no resolution to a lot of philosophy problems.
The reason I like this thought experiment, be it after replacing axe parts following your unexplained need to slay and behead some dude with a swastika tattooed on his tongue or replacing pieces of Theseus' ship, is because it questions identity. We are constantly shedding cells and gaining new ones, so are we ever identical with who we were previously? Darth Vader was almost fully replaced by mechanical parts, so was he still Anakin Skywalker? The philosopher Wittgenstein might have thought these questions were balderdash ("Roughly speaking: to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing") and, if that were the case, he might have been right.
It would be interesting if we could just say that something is such because people agree to call it such. Maybe the Ship of Theseus is really just whatever anyone decides to call "the Ship of Theseus". Maybe Darth Vader is Anakin Skywalker because someone calls him so. If that were the case, then the answer to the thought experiment as proposed in John Dies at the End might be that the axe you're holding in your hand is the same axe as the one that beheaded Swastika-Tongue because his zombie has now said that it is the same axe (it might be a good conclusion since chances are you should be more worried about dealing with said zombie before considering philosophical puzzles anyway).
However, I still feel like the answer would be "no". Even if you had killed the dude with an axe in the first place (and not with one, some, or all of those eight bullets), the original axe has been completely replaced. The answer feels like "no" because none of the original axe remains and there are only two major parts of the axe to replace. When the problem is introduced as in the case of the Ship of Theseus, where the object is replaced a small amount at a time, that's when it gets harder to decide when to even consider the ship to no longer be the original.
Maybe one of the more interesting answers comes from those who like to add the temporal dimension to the consideration, such as in the Worm Theory as presented in the video above. When we question the temporal aspect of an object along with it's identity, we start hinting at a possible answer to the question (see Temporal Parts at Stanford Encyclopedia of Philosophy). We can say that any one thing is only ever fully identical to itself at one point in time, but then at other points in time it can only be similar to itself. Then using a name to define something falls back to the "loose and popular" context that Butler suggested. That sounds just about right, honestly. The answer that would suggest then is that the Ship of Theseus was only the same ship in the sense that it bore similarity to itself over time and that people still called it the Ship of Theseus is the only thing that made it the Ship of Theseus. In that case, the axe that hewed the head laden with a swastika-marked tongue and the axe that you now have to defend yourself against the zombie at the door are only similar, and maybe you would call it a different axe since you know you've replaced the parts but the zombie calls it the same axe since it looks similar to the original. Not a very rewarding answer, but an answer nonetheless (and now you can get on with hacking down the zombie as he is more than likely about to come at you).
There's rich food for thought there. Maybe Wittgenstein is right and it's nonsense to even worry about two things being identical. That seems to fit well with the answer that considers the temporal aspect to mean that an object has unique temporal parts during its existence (look up perdurantism). Whatever anyone's consideration of this little thought experiment may be, I think we can all agree that it's a lucky thing we don't truly live in a world where a guy can get shot 8 times and have his head chopped off with an axe but then still find a way to come back from death and then sew his head back on before coming to find us with the likely intent of exacting revenge.
Update (22 October 2017): I re-shared this recently and have had several people ask if I've read the book John Dies at the End. Happily, I can say, yes, I have, and I've also read the sequel, This Book is Full of Spiders. A third book in the series, What the Hell Did I Just Read: A Novel of Cosmic Horror, just came out this month. Looking forward to reading that as well!
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 = ?
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...
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
But, this is where I'm seeing a lot of trouble...
PEMDAS
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.
"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.
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.
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