## 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
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?)
:

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.

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

How did I get the answers of 90410, 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.

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
+ 2 + 4 = 183652
+ 6 + 3 = 482466
+ 4 + 5 = 202541
+ 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:

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

and, finally:

+ 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!

1. G'day Graham.

I was looking for confirmation that 90 was the correct answer to the first puzzle when I found your post.

90 is the correct answer but the way you found the answer does not match with the other 2 puzzles. I found 90 by using the following formula.

N x N + N = answer.
9 x 9 + 9 = 90

If you apply this formula to all of the examples above the question it proves correct as well.

Cheers.

1. Absolutely! Actually, you're approach is quite the same as mine. Since:

N x (N+1) = N x N + N

For the first one, where I use 2x3=6, you can also use 2x2+2=6, and they are both equivalent!

For 9:

9 x 9 + 9 = 9 x (9 + 1) = 9 x 10 = 90

2. For the (7+2+5) question, this also works ((7*5)+(7*2))-2 or ((A*B)+(A*C))-B

2. But the answer could legitimately be 56 or 72 as well. So, for that reason it's not a very good puzzle as it has more than one possible answer.

Mulltiply the first number on the left hand side by 3, then the second number by 4, then the third number by 5 etc. Following that pattern, when you get to 9 on the left hand side, you'd multiply it by 8 e.g.

2x3=6, 3x4=12, 4x5=20, 5x6=30, 6x7=42, 9x8=72

Or you could look at the differences between the numbers on the right hand side and notice that the difference between the one before and the one after increments by 2 each time e.g.

12-6=6
20-12=8,
30-20=10,
42-30=12,
therefore to fit this pattern the next answer should be
56-42=14

I don't particularly like this second solution as it largely ignores the numbers on the left hand side. But even so, the solution could most definitely be 90, 72 or 56.

1. Both of your solutions, though, assume that the final equation follows directly from the first five. I think the point of these problems is usually to recognize the pattern and then apply the pattern to a similar equation, but one that doesn't inherently follow from the last set in the pattern. Since they're skipping 7 and 8 and going straight to 9, it would make sense to do the same in your application of the pattern to find the solution, which is why you would multiply 9 by 10 to get 90, since multiplying it by 8 to get 72 doesn't follow the pattern (it does, as you say, follow it from top to bottom, but the change in format by skipping 7 and 8 on the left already breaks that top-to-bottom structure). Actually, applying the pattern in that manner, then your approach of adding 2 to the solution each time would also yield 90 (assuming you included the missing sets of the pattern - namely the 7 and 8 on the left). I guess it really does depend on how you look at the problem, though I don't fin it as fulfilling to apply to the 9 what would have been applied to the 7.

2. Indeed, my proposed solutions work with the information actually given, rather than assuming there is a 7 and 8 on the lefthand side. Anyway, it's intriguing how people see things differently - some on Facebook gave 81 as an answer, whilst others thought 54. Quite amazing :-)

3. That also depends on how you look at it. The information actually given does skip the 7 and the 8, so you can apply a pattern linearly without regard for the inputs (the left hand numbers) or you can build the pattern to rely on the left hand number. These problems read to me as though you are given the first few objects in a series and then asked to compute another object somewhere further along in the series, but I can see how many people arrive at the solution of computing the next object in the series using the same rule (even though the pattern won't be exactly the same).

4. The problem uses the words "if" and "then", which to me implies you are to "solve" for 9, thus giving 90

5. That's how I read the problem myself (the solution should be 90, since that means that the input is important, and not just the sequence alone), but it is interesting to see the large number of answers people give. Definitely some fun problems!

6. 2(3)=6
3(4)
4(5)
5(6)
6(7)
7(8)
8(9)
9(10)=90

it is 9x10 not 9x8

7. second method
2=6
3=12 =6+6
4=20 =12+8
5=30 =20+10
6=42 =30+12
7=56 =42+14
8=72 =56+16
9=90 =72+18

56 is for the 7 not 9....you miss the rest thats y you got different answer

3. But the Nr 10 is Not given. I dont think you can calculate with an asumption. Using paterns like in the other puzzels. I get 60...

1. I'm admittedly unsure of which problem you're talking about. Where is the number 10 missing for you?

4. f(z) = z * | |z-6| -7|
2 = 6
3 = 12
4 = 20
5 = 30
6 = 42
9 = 36

1. I love it! I'll update the post to include your approach. If you tell me your name, I can give you credit for it.

3. Ha ha, boom nothing. You're taking yourself far too seriously. These types of puzzles are obviously not intended to be pure math problems (I'm guessing you haven't actually read the post). You might be surprised to learn that many IQ tests and tests of "outside of the box" thinking include puzzles that bend or break the rules of math or grammar to see how a person can think. I love mathematics and I respect the need for people to understand how math really works, but I also love these puzzles because they make people think about math in fun ways.

4. Hey, i say if 3=12 then 9=36, why the complications?

5. Multiplying by 4 fails for every other instance. Your surely free to give any answer you want, but the power of your answer lies in how it fits the puzzle (of which, your approach is an utter failure).

5. 2*3=6, 3*4=12, 4*5=20, 5*6=30, 6*7=42, ... 9*0=zero.

6. it's that?

3|2|=|7|
5|4|=|23|
7|6|=|47|
9|8|=|79|
10|9|=|?|

1. I would say the most apparent answer is 98. 3*2+1=7, 5*4+3=23, 7*6+5=47, etc. Looks like multiplying the first number by the second and then adding one less than the second. Could be written, in general, as: (n)*(n-1)-(n-2), where n is the first number on the left.

2. 10*9=90+9=99 you took 8 instead of 9

3. This was for the problem that Moises Silva presented. I was taking 10*9+8=98 to fit the problem that he wrote, which was of the form (n)*(n-1)+(n-2).

7. Theres no 7 or 8 on the second puzzle so u cant get 42 ud get 54 from 9x6

1. The problem does skip 7 and 8, but if you apply an operation that is consistent then my answer of 9*10=90 works very well. However, if you treat it as though you should just multiply the nine by the next number in a sequence that disregards the input and rather follows a list (which is what I think you're suggesting), you should then get 9*8=72.

8. how about since 2 = 6 we can amuse 2x2+2 = 6 since it's comprised of all 2s and 3x3+3 = 12 and 4x4+4 etc skipping 7x7+7 and 8x8+8 your left with 9x9+9 or 90 or the equation z*z+z :P

1. Absolutely! The operation z*z+z is actually the exact same as z*(z+1) (the distributive property). So, whether you see it as 2*2+2 or 2*(2+1), you get the same answer!

9. the second puzzle the "easy" one, is not algebra, it is Logic the answer is 58.
2+6+3+1=12, etc. 6+42+9+1=58

1. Interesting approach. I've not yet seen anyone do it that way. Your approach definitely works to get a reasonable answer. Though if there were lines for 7 and 8 (which aren't there, obviously) you would still get 90 (8+72+9+1=90). If you do it this way, then it does go back to the algebra (here, the expression would be n+(n*(n+1))+(n+1)+1 or n^2+3n+2, which works to get the answer of 90 for 9). I like that your approach aims to solve the problem as is using a consistent system, though the same can be said for those who think the answer should be 9*8=72 (since 8 would be the next multiplicand in the sequence, as is, if the inputs are not important).

10. The Answer is 99,,,See the equation LHS Numbers are multiplied with each other and then Add 1,3,5,7,9,11(+2)......So on Hence correct answer is 99.

11. 2 = 6
3 = 12
4 = 20
5 = 30
6 = 42
9 = ?? 72

1. It should be 90 not 72.......Since LHS digit is multiplied by +1 Result.

12. Why notre 56 ?
2=6(+6)
3=12(+8)
4=(20+10)
5=(30+12)
6=(42+14)
9=56

13. That would seem to fit the problem, at least on the right-hand side of the equations, but it would lack a good reason for why the first line is 2=6 (since that would suggest that 1=0(+6)) and it fails to include the left-hand side numbers (i.e. the jump from 6 to 9).

14. I came across another suggestion for 2=6 of 9=1350

f (x)=x^2 + x + ((x-2)(x-3)(x-4)(x-5)(x-6))

15. I came across a nice solution on another site for the 2 = 6 puzzle where 9 = 2610.

If f (x) = x^2 + x + ((x-2)(x-3)(x-4)(x-5)(x-6)) when x <= 6 the answer will be the same as x^2 +x but for 9 it would be 9×9 +9 +7×6×5×4×3 or 81 + 9 + 2520

1. It's a very interesting approach, but then I guess you could create any approach that gave the x^2+x or x*(x+1) function for x from 2 to 6 and then gave just about anything else afterward. For instance, if the function was f(x)=x^2+x+(100(x-2)(x-3)(x-4)(x-5)(x-6)) then the answer would be f(9)=252,090.

16. If 3=12, then 9=36

1. Multiplying by 4 fails for every other instance, making that approach rather useless.

2. 2=6 2x3=6
3=12 3x4=12
4=20 4x5=20
5=30 5x6=30
6=42 6x7=42
Here the part you don't see.
7=56 7x8=56
8=72 8x9=72
9=90 9x10=90

17. I did it 2 different ways and hot 90 both times for the 9=?

19. For which problem and why?

20. First one is 79?

1. That's the solution that I found that fits the pattern.

21. Wat is de uitkomst van 3-3×6-2=

1. Het resultaat moet zijn -17, sinds 3-3x6-2 = 3-18-2 = 3-20 = -17

22. I read the first couple of comments. So that's what two geniuses say when they walk up to each other....wow. i was so lost...

23. Hi.. Graham Lau

if 2=6 3=12 4=20 5=30 6=42 then 9=?

i discus this puzzle @here : http://math.stackexchange.com/users/80469/etienne

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

and for this puzzle is very ill formatted in every aspect and the answer could be anything you want

and you can see the discussion here : http://www.cregox.com/blog/2014/08/only-for-geniuses-and-not-for-stack.html

1. I don't see much need to take these problems too seriously. They surely are ambiguous (as I mention and you can see in the comments), but still fun. They force you to break some of the "rules" of math and instead to just think about a possible answer. I think as long as an answer can be logically justified for the problem, then it's all good.

24. I've seen this posted as IF: 2=6 3=12 4=20 5=30 6=42 THEN: 9=?

In a math binary system we can say 2=6 3=12 4=20 5=30 and 6=42 :

in binary system the value only have two symbol 0 and 1

0=0,2,4,6,8,11,13,15,17,19,20,22,24,26,28,31,33,35,37,39,40,42,44,46,48
1=1,3,5,7,9,10,12,14,16,18,21,23,25,27,29,30,32,34,36,38,41,43,45,47,49

Notice 2=6=0; 3=12=1; 4=20=0; 5=30=1; 6=42=0 and 9=10=1 or 9=1.

1. Ewww, I love that answer! That's a fun way to think about the problem.

2. lol.. math is always fun :)

and my friend use ABS formula like this

x*(abs(abs(x-8)-9))

9=72

then if that only one correct answer, what is should be?

1, 9, 56, 72, or 90 ?

3. Ya, that's a good approach as well. I honestly hadn't thought of using absolute values when I first looked at that puzzle last year. I like that approach.

25. If the numerals in the problem are hexadecimal:

f(0x2) = 0x6 = 6
f(0x3) = 0x12 = 18
f(0x4) = 0x20 = 32
f(0x5) = 0x30 = 48
f(0x6) = 0x42 = 66

then a solution is f(n) = x² + 7x - 12

which when used to evaluate f(9) = 81 + 63 - 12 = 132

which converted back to hexadecimal is 0x84

26. I got all 4 on my first try, and I am 12. No joke.

1. Awesome, Zachary! That's fantastic!

27. I know the answer given above for the following columns of numbers is 79, but I offer this as an alternative:

111 = 13
112 = 24
113 = 35
114 = 46
115 = 57
117 = 69

The first number in the answer represents the first number in the left column added to the numbers above. The second number in the answer represents the sum of all the numbers in the row. Therefore, the last answer would be 69 representing 6 ones for the first number and two ones and a seven for the second.

1. Oh, for sure, that works great as long as you consider the jump from 115 to 117 to be continuous and not skipping a line (116). If that weren't the case, then by that approach you'd still have 116=68 and 117=79. It just depends on whether you choose to see it as missing a line on purpose or not. Thanks!

1. How so. Let me know how I can help. These problems aren't straight-forward and are more just for fun, especially since they require that you break the rules of math a bit to solve them.

28. 99 IMHO and how my brain works... 2=6, (2x3) 3=12 (3x4) 4=20 (4x5) 5=30 (5x6) 6=42 (6x7) 7=56 (7x8) now since the 8 doesn't appear in the list and 9 is next in the sequence its now jumping up by 2 digits next in line after 9 would be 11 so 9=99 (9x11)

1. I can see how you got there, but if you look at your approach you'll see that you were using n*(n+1) for each line until you skipped the 8 and went to 9. If your solution is 9=99, then you're changing the approach to n*(n+2)

29. Just wanted to let you know I just enjoyed figuring out the problems! What is it with everyone trying to challenge the way you worked the problem? Loved the challenge and working my brain a little!!!

1. Thank you, Jean. I appreciate that. There are some of us who are math nerds who can take some math problems a little too personally, but I also think the way that our society frames math, as though it's something only the super smart people can understand, makes a lot of people very defensive about their approach to solving math problems. I think puzzles that use math but also break the rules of math and ask for some creativity are more likely to cause some blowback from people. However, it's all good. These problems are super fun to tinker with and they provide fun moments for talking about how we solve problems.

3,3,5,4,4,3,5, ?

1. Not sure to which problem you are referring.

31. The first I think that the last is 69.

32. 2=6
3=12
4=20
5=30
6=42
9=?

? is 60, on the left it is increasing with 1 on all but from 6 to 9 here it is 3*1. On the right it increases with 2 (6 -> 8 -> 10 -> 12) then the last line should increase with 3*2=6 wich give 12 -> 18 and 42+18=60

33. 2=6 - 3=12 4=20 5=30 6=42
7=56 - 8=72 (not listed in equation)
9x10=90

34. For the last problem, I got the same answer of 143547 but came up with it a little differently and works for all of them. The first four digits were calculated the same way as you stated above, but the last two digits were calculated a little differently:
(B+C) x (A-1) = ____ + C = the last 2 digits
(2+5) x (7-1) = 42 42+5 =47