Question

Hurry up!! Fan!!!!! Medallllllllllll !!!!!!! :))

What's the next number?
889669886 , 61636617 , 61272 , 861 , ?

Answer 1

Technically every number would work given the right polynomial.

Answer 2

\[
\begin{align*}
861&=3 \cdot 7 \cdot 41\\
61272&=2^3 \cdot 3^2 \cdot 23 \cdot 37\\
61636617&=3^2 \cdot 7 \cdot 978359\\
899699886&= 3^4 \cdot 29 \cdot 191507
\end{align*}
\]

Answer 3

Can't see anything.

Answer 4

me neither

Answer 5

.

Answer 6

@Yavar do u know the answer?

Answer 7

i found somewhere that the answer is 84,
but i don't know what's the reason.

Answer 8

It's an IQ test. You must break the frames...

Answer 9

@imqwerty @pooja195 @ParthKohli @nincompoop @Mehek14 @mayankdevnani @Nnesha
as a hint:
Do not see any number as a unique number. separate the digits of each one and do something to separated digits.

Answer 10

by as a unique number i meant "as a whole". sometimes my english doesn't help me well.

Answer 11

IQ test question? No wonder it looks so pointless.

Answer 12

its pretty annoying

Answer 13

*will look at this later*

Answer 14

Ok,
IQ tests do not need you to do only some basic or even complicated calculations. One needs to be able to look at everything in a different way. And this is not called "Pointless"

Anyhow, I already told u the answer. I wanna tell you how the answer is calculated 84.
Is there anyone who wants to think more or I should tell you why the answer is 84?

Answer 15

Ok,

\[889669886 \rightarrow 8\times8\times9\times6\times6\times9\times8\times8\times6=71663616\]

do you see any relationship between 71663616 (which is the result of multiplications above) and 61636617 (which is the second number in the series) ?

Answer 16

The point I am trying to make is that every number is a equally valid answer mathematically. Correctly deducing the answer that the examiners want does not make you a more intelligent person. Judging the intelligence unfortunately is the point of an IQ test.

Answer 17

yes read it in reverse

Answer 18

the next numbers should be 23 and 6, isn't it ?

Answer 19

84, 23, 6
nice problem :)

Answer 20

thomas has a valid point, these IQ sequences are not mathematically well defined,
there are an infinite number of polynomials that satisfy the initial terms,

Answer 21

The tests are made in a way that there's only one possible answer not two different possible ones.

Answer 22

889669886 , 61636617 , 61272 , 861
you can fit the terms to a third degree polynomial, a fourth degree polynomial, a fifth degree polynomial, etc.
i will produce examples

Answer 23

yes and nest numbers can only be 23 and 6 and then 6 and then 6 ...
889669886 , 61636617 , 61272 , 861 , 84 , 23 , 6 , 6 , 6 , 6 , ....

Answer 24

here is one example
third degree polynomial that satisfies the four initial terms
$$f(n) =
\\ \frac{-352471495 n^3}{3}+1088171952n^2- \frac{9810346910 n}{3} +3189104069$$

Answer 25

I know. Give me a polynomial that satisfies
889669886 , 61636617 , 61272 , 861 , 84 , 23 , 6 , 6 , 6 , 6

Answer 26

yes but the sequence is ambiguous

Answer 27

here is an example
of an ambiguous sequence
2, 4, ...
the next term could be 6, or 8
so in your case the next terms could be 84, 23, 6,6,...
or one of those interpolating polynomials

Answer 28

also there are other tests which can't be solved using polynomials.
for example this very simple one:
What's the next number?
011111011111 , 100111100111 , 110001100011 , ...

Answer 29

strictly speaking there is more than one answer
*if* you have in mind the sequence that ends in 6,6,6...
thats one sequence.
if you have in mind the sequence that fits an interpolating polynomial
you would get a different sequence

Answer 30

you can assume the test makers do not have an interpolating polynomial in mind, as that would involve tedious calculations and the test is timed

Answer 31

I agree!
So you say there must be more numbers in the question to make it unambiguous and lead the person to a unique answer.

Answer 32

Most good IQ tests are not timed. for example The Nemesis Test, The Marathon Test, etc...

Answer 33

you can find most of good IQ tests here on the webpage of the Giga Society:
http://gigasociety.com/qualification.html

Answer 34

here is another example of an ambiguous sequence
what are the next two numbers for the sequence
2,3,5,

Most people would say the next numbers are 7,11, etc. the nth prime.

But it could also be
2 3, 5, 14, 69,
the rule here is: f(n) = f(n-1) * f(n-2) -1
3*2 -1 = 5
5*3 -1 = 14
14*5 -1 = 69

and there are more

Answer 35

this is a problem for the testmakers, how to make a sequence that does not have multiple valid solutions.
Any sequence with the elipsis \( '...'\) which means "and so on", has this problem. Unless its a trivial sequence 1,1,1,1,1...

Answer 36

what we are doing with this sequence is technically called extrapolating, and this uses induction. using induction carries risks, it is not a valid deductive method.

Answer 37

I am pretty sure for all finite sequences, there exist at least one interpolating polynomial that fit the sequences. For a infinite sequence, it highly depends on the nature of that sequence.

Furthermore, why do IQ tests put such a heavy emphasis on "thinking outside the box" but require the candidates to think "in the box" that the examiners want? And why does the sequence has to be calculated like this? You can technically chain a bunch of operations on the individual digits to fit the first 4 terms in the sequence and have different answers after the 4th term. How do you prove the uniqueness of this method? If this method is not unique, why is this method more "valid" than others?

And why do we want such esoteric answers? Can't we just accept the ordinary is not necessarily bad and is in fact good for many situations?

Answer 38

I know ambiguous examples jayzdd:
your first example showed it clearly 2, 4, ...
next numbers can be 6 , 8 , 10 , ... (2n)
or
8 , 16 , 32 , ... (2^n)
or
20 , 72 , 232 , ... (3^n - 4n+3)
or
7 , 12 , 21 , ... (n+2^(n-1))

Answer 39

the following interpolating polynomial goes through all the points
of the finite sequence
889669886 , 61636617 , 61272 , 861 , 84, 23, 6
http://www.wolframalpha.com/input/?i=curve+fit+ {1%2C889669886}+{2%2C61636617}+{3%2C61272}+{4%2C861}+{5%2C84}+{6%2C23}+{7%2C+6}

once the 6 starts repeating, the interpolating polynomial method breaks down,
but you can define it as a constant function at that point
so a piecewise defined sequence is a valid solution to this problem

Answer 40

the wolfram link shortened
http://goo.gl/ulrPnK

Answer 41

@thomas5267 IQ is a quality of mind that makes a person able to decode the codes behind a context.
It's clearly essential where you do not have any idea what is the nature of a phenomenon or a secret. It has helped Human through our history and made our science.

Answer 42

@thomas5267 That's a spark in the mind...
It's exactly like when Archimedes went naked to the street saying "Eureka! Eureka!" crying. It's a type of satisfaction for the mind. That is all.
And our current knowledge is enough in many areas but still not enough for human's mind.
That's why human should think out of the box, like he/she used to for millenniums ...

Answer 43

this is pedantic but this is a perfectly valid solution to the problem
$$ f(n) = \begin{cases}
(130188293 n^6)/180-(240472741 n^5)/12\\+(2035308169 n^4)/9 -(15954326905 n^3)/12\\+(772668074267 n^2)/180-(43294012493 n)/6 \\+4935436239 ~~~~~~~~~~~~~~~~~~~~~ if ~~ 1\leq n \leq 7
\\~\\ 6 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ if ~~n > 7
\end{cases}
$$

Answer 44

http://prntscr.com/9jh2fj

Answer 45

:)
@jayzdd
If we wanted to think this way we could say something like this for the weight of a mass:

for M=1 Kg --> W=9.81N
for M=2 Kg --> W= 19.62 N
.
.
.
And there were no need for Newton to find that W=mg (g=9.81 N/Kg)

Simplicity is the most important factor for the best answer. It should be simple enough, but not simpler.

Answer 46

@IrishBoy123

Answer 47

simplicity is something that humans seek, mathematically its not relevant .
it helps us conceptualize results

Answer 48

from a purely formal point of view, simplicity is not a requirement

Answer 49

Formal point of view --> In the box --> Doesn't answer to all of our questions.
please watch the movie " The Beautiful Mind ". It can help us understand better what it means thinking out of the box. If u've already watched it, u know what I'm saying.

Answer 50

this is not relevant to the problem you posted above. it is interesting in its own right, there are limitations to formal methods (thats a topic of intense debate).

Answer 51

the answer ultimately depends on what the IQ testmaker had in mind, because he/she is grading it. that's the simple answer :)
personally i dislike this question because you have to consider the psychology of the testmaker, which should not be relevant

Answer 52

or we just take out sequences from IQ tests? lol
i wonder if the spatial reasoning questions also suffers from a similar flaw, can there be more than one answer

Answer 53

Ok,
so to sum up, you believe IQ tests are not that useful and it's not necessarily good to think out of box. Formal point of view can solve majority of our daily life problems. And I agree with the latter.

Answer 54

nope i never said iq tests are not useful
i only said there is more than one valid answer.

Answer 55

if you scroll up i said that's a nice solution, multiplying the digits and reading them in reverse, that was my first post.

i doubt someone would come up with this solution unless they are adept at multiply multiple digits in their head, then make this realization you can reverse the digits

Answer 56

thats why i considered interpolating polynomial,
*that* is thinking outside the box , just as much as multiplying the digits

Answer 57

which is standard practice in industry, i might add
extrapolate weather data, etc

Answer 58

ill delete these comments , sorry didnt mean to spam

Answer 59

I didn't sum up only with you @jayzdd :)
It was a conclusion of our talks : me, @jayzdd and @thomas5267
He doesn't like it so much I guess. ;)

Answer 60

ok no problem :)
i felt thomas had some good points and brought out the big guns to defend it

Answer 61

and you shot me to death, but i'm still alive!!!!!!!!!!!!
and POWERFULLLL!!!!!
ahahahaha

Answer 62

:)

Answer 63

there is even more problems with the problem
what base is it in?
you could be in base 12

Answer 64

so you have make background assumptions when you solve IQ test problems

Answer 65

I honestly enjoyed reading this debate over iq tests.

Answer 66

Yes! I absolutely hate IQ tests since they claim to do things that they don't do IMO.

My problem is that why is the examiners' box more "correct" than all other boxes in the world, including the ordinary ones. How is this thinking outside of the box? Sure, outside of the ordinary box, but is this truly outside of the box?

I would argue such IQ tests are not even indicative of the candidates general intelligence. Almost every year or two we hear someone scored higher than Newton/Einstein/Feynman/(put your favourite famous mathematician/physicists here) in an IQ test. We have yet to see someone revolutionising the field of mathematics/physics like Newton/Einstein/Feynman/(put your favourite famous mathematician/physicists here).

In both mathematics and physics, things are simultaneously way harder and way easier than this. In both subjects, the concepts are much harder yet much more orderly. They seem to follow some patterns which are applicable in a more general sense. In mathematics, if a coincidence seems too amazing, then there is probably something that we don't understand that explains the coincidence. The difficulty lies in the the proof of such coincidence. In physics, if a coincidence seems too amazing, you have to make sure that you did the experiment right and it is not a statistical fluke. The difficulty of physics lies in the uncertainty of such coincidence.

I don't see such generalisability in this question.

The existence of a solution satisfying the sequence is proven by finding a solution. The uniqueness of the solution seems highly improbable if we only consider arithmetic operations on individual digits and the solution is not unique if we allow polynomial extrapolations on finite (sub)sequences. If arithmetic operations on individual digits yield more than one solution, how could you claim this solution is more "correct" than the others when (to me) they seem equally esoteric?

Answer 67

Now tell me one thing.

Fill the blanks with two natural numbers. :D

691 , 961 , 441 , 121 , 1 , 18 , ... , 94 , 63 , 52 , .... , 9

Answer 68

I liked @thomas5267 's point you have to think inside someone else's box. Why is their box important enough to determine the iq of the general population? Why not someone else's box?

Answer 69

to be fair and consider the IQ testmakers side of this discussion,
i don't think one IQ test question is sufficient to indicate a higher IQ.
Its ok to have a bad IQ question, because there are 29 other questions
which include spatial and verbal reasoning, and high scorers tend to correlate with pretty smart individuals
That is, the higher you score you get, you can solve logical puzzles faster, learn new topics faster, etc
but this is debateable if this is the definition of smartness, i suppose

Answer 70

691 , 961 , 441 , 121 , 1 , 18 , ... , 94 , 63 , 52 , .... , 9
14^2 = 196 , read it in reverse
13^2 = 169 , read it in reverse
12^2 = 144 read it in reverse

Answer 71

@freckles It's not that how a test maker solves a problem is the best way. The simplest way is the best way. If you provide a simpler reason for you answer to an IQ test, it's completely acceptable. But the point is: Almost always the simplest way to answer a question is known by the test maker. That's why anybody out there can not be an IQ test maker.

Answer 72

How did you determine which of the ways mentioned above was simpler for your problem? @Yavar

Answer 73

@jayzdd yes!
I sent it here only cuz one should use the same point to solve it: INVERSE the digits and find the pattern.

Answer 74

How about writers, musicians and painters? How do you measure the intelligence of those people when IQ tests are so biased on pattern recognition? Shakespeare is clearly very intelligent. His use of words are brilliant. How about Michael Jackson? His choreography and music are top notch. Why not Harper Lee? For her really famous To Kill a Mockingbird. Van Gogh? J.K Rowling?

Answer 75

Simple? I would argue a degree 3 polynomial extrapolation is more simple. It uses standard Gaussian elimination, and it is the *unique* polynomial fitting the first four terms of the sequence. It has wide applicability in many areas as well.

Answer 76

Firstly: You should review the answers above and see which one seems simpler.
Secondly: You should send an email to the test maker and ask him/her why he/she thinks the simplest answer is that.
Thirdly: If you find any simpler way to answer it, your answer will be correct! (not 84)

Answer 77

691 , 961 , 441 , 121 , 1 , 18 , ... , 94 , 63 , 52 , .... , 9
14^2 = 196 , read it in reverse 691
13^2 = 169 , read it in reverse 961
12^2 = 144 read it in reverse 441
11^2 = 121 read it in reverse 121
10^2 = 100 read it in reverse 001 = 1
9^2 = 81 read it in reverse 18
8^2 = 64 read it in reverse 46
7^2 = 49 read it in reverse 94
6^2 = 36 read it in reverse 63
5^2 = 25 read it in reverse 52
4^2 = 16 read it in reverse 61
3^2 = 9 stop

Answer 78

well now that i did the last iq problem, i was 'primed' to use the same trick

Answer 79

and those numbers look suspiciously similar to square numbers
the 144 , 169, stuck out

Answer 80

I don't think "simple" should answer a question all the time. Simple is a vague word here. Both ways seem "simple" to me.

Answer 81

very nice. In my opinion this solution @jayzdd just sent is much simpler than a polynomial of a degree of n.

Answer 82

if the IQ test makers are explicit and say you can only solve the IQ test using 8th grade math or 'arithmetic' methods, then we can rule out the polynomial fitting methods

Answer 83

@freckles it's ok. So both ways are correct in ur opinion. Agreed.

Answer 84

But yeah both ways would probably not be considered "simple" to an 8th grader. I guess we are to assume IQ tests do not assume higher mathematics in their test takers, maybe?

Answer 85

if the testmaker says
do not use algebra, just arithmetic (plus, minus, subtract, divide, multiply and exponentiate), thats fair?

Answer 86

another problem with the IQ test, a mathematician will get skewed results since he has mathematical training to solve 'mathematical' type IQ questions.
that is, a mathematician may not have a high IQ but have learned (slowly)
the techniques used to solve a similar class of problems

but thats why iq tests mix it up with spatial reasoning and verbal reasoning

Answer 87

There are different types of IQ tests. Read the information of The Nemesis Test ,attached here, for example:

Answer 88

but there are people who 'study' for IQ tests, so are they cheating?

Answer 89

Apparent Knowledge required is Moderate here. So I guess it's not a good test for an 8th grade student

Answer 90

lol

Answer 91

there are books that help prepare people for IQ type questions, so they get an edge over people who are not prepared (but they dont necessarily have a higher IQ).

a genius is someone who has not prepared for IQ type questions, and still scores nearly perfect. which makes me wonder if there are fake geniuses out there?

if i did not learn algebra i could get the following 'type' of question wrong:
"if john is twice as old as sarah and three times as old as ... hold is john"
with algebra its very easy to solve
also train problems , if they have them

Answer 92

IQ tests are not related to studying I guess. As you see, the time limitation is set as UNLIMITED. so you have time until your last second of your life.

Answer 93

but i solved the second problem you posted much faster than the first , after seeing the trick

Answer 94

Oh, I guess you mean GMAT or such tests? Yes those tests are timed but i don't think that are a good scale of Human Intelligence.

Answer 95

there might be mathematical tricks to recognize spatial reasoning, too.

the IQ testmaker has to adapt and make the test different
for people who have been exposed to such tricks.