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The puzzle has already been solved but you can still try it just for fun without looking at the posted correct answer.

What is the missing number in the following sequence?:

0, 4, 10, 32, ?, 317 ...

Hint # 1:

It has nothing to do with differences between given terms such as (4-0 = 4, 10-4 = 6...). Try a different pattern.

Hint # 2:

The 10 should be a clue. We use base 10 numbers so 10 is a special number and is a key part of the pattern. There is a reason it is not 9 or 11 for example.

Hint # 3:

Don't bother with + or -, those wont help. That is, for the function part (see hint 7), F(0), F(4), F(10)... don't use + or -. Take a close look at the pattern in the 5 numbers given, and think about a function of those numbers having some pattern too.

Hint # 4:

Look carefully at the number sequence. Do you see any pattern of how they are increasing in value? I can tell you they will continue to increase.

Hint # 5:

The pattern doesn't involve any complex mathematical expression, rather, it is quite simple. Run some function on each of the 5 numbers given, and analyze the output of that function, then look for a pattern. F(0), F(4), F(10)... will all follow this pattern.

Hint # 6:

The 5th term (after 32) is a 3 digit number in the range 100..316 inclusive so 100, 101, 102... 315, 316. Take your pick, but if you randomly guess, you only have a 1 in 217 chance of getting it right. Also the correct answer, (to get the checkmark), needs an explanation but "I guessed" wont "cut" it. There is a "simple" solution (pattern) to this puzzle.

Hint # 7:

There is no conditional solution, meaning that there is no particular term in the sequence (such as the first one), that is treated differently than the rest. For example, if you found a solution that works with all terms except the 1st one (at least one person submitted an answer like that but it may have been deleted), a solution that states 0 for the first term, and some other pattern for the rest, although technically a valid solution, is not what I am looking for. My solution works for any and all terms in the sequence regardless of position (1st term, 75th term, 1000th term...). However, the pattern has a relation between the function of neighboring terms. For example, suppose we define a function called F. F(0) has a relationship with F(4) and that relationship is preserved the same way between F(4) and F(10). That is, the pattern is invariant between neighboring terms. Lets us call this relationship R. Thus we have R(0,4) = R(4,10) = R(10,32) = R(32, x) = R(x, 317)...

Hint # 8:

The even # terms (2nd, 4th, 6th...), if scaled to 1 digit before the decimal point only, (4.0, 3.2, 3.17...) will become asympototic to a certain irrational number as the term positions get higher and higher (20th, 22nd, 24th...).

I am looking for a particular answer with a particular pattern that applies to all 6 numbers (5 given plus missing one). Please give your answer as a number with an explanation why you think that is the correct answer. I guess this problem is not as simple as I thought. Good luck.

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    $\begingroup$ Given only four digits to work with, splitting them into odd and even positions (or allowing to have more then one consistent rule for the pattern) seems to open up tons of possible solutions. On a side note the closest I got was rounding up pi^n starting at n=0 which yields 1,4,10,32,98 but is obviously wrong due to the 1 :( $\endgroup$ – Zilvarro Apr 10 at 15:24
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    $\begingroup$ Eight hints in four hours, that's gotta be some kind of record. IIRC the usual advice is to wait at least 24 hours before starting to provide hints. $\endgroup$ – F1Krazy Apr 10 at 18:26
  • $\begingroup$ I changed the question slightly to hopefully make it less difficult, however the correct answer and reasoning are identical to the previous version of this question so nothing has really changed as far as previous attempts to solve it. Those attempts are still valid. $\endgroup$ – David James Apr 11 at 0:47
  • $\begingroup$ does it related with binary numbers ? $\endgroup$ – Swati Apr 11 at 12:05
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    $\begingroup$ Thank you to all who reopened it. I imagine someone wants the glory of solving it and many people want to see the correct answer. I can assure you it is well worth the wait when you see how simple the answer is. $\endgroup$ – David James Apr 11 at 20:43
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My guess

100

The sequence may be

$\left \lceil{\sqrt{1}}\right \rceil$, $\left \lceil{\sqrt{10}}\right \rceil$, $\left \lceil{\sqrt{100}}\right \rceil$, $\left \lceil{\sqrt{1000}}\right \rceil$ ...

The n-th number is

$\left \lceil{10^{0.5(n-1)}}\right \rceil$

I'm aware that it cannot explain the first one, but the guess fits the others requirements quite well:

1. It's simple(I think), no complicated + or - involved
2. Related to base 10
3. it'll eventually reach $\sqrt{10}$ (Hint #8)

So I'm not sure how to deal with the first number (as you stated in #7), maybe someone else can slightly tweak this function to make it work?

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  • $\begingroup$ The 100 is correct, however the pattern is not the one I was looking for. I had to give the 8th hint because nobody was even close to solving it prior to that. So now that everyone knows the sequence is 0, 4, 10, 32, 100, 317..., they can concentrate on the pattern alone. $\endgroup$ – David James Apr 12 at 14:58
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    $\begingroup$ Before taking the square root, subtract an increasingly negative power of a positive integer. E.g. the first could be 10 to the power of 0 minus 10 to the power of 0. The next would be 10 to the power of 1 minus 10 to the power of -1. Etc. $\endgroup$ – Jens Apr 12 at 15:07
  • $\begingroup$ Although OP did say that no subtraction was necessary... $\endgroup$ – El-Guest Apr 12 at 15:17
  • $\begingroup$ The one thing this logic is missing is you are supposed to be taking a single function and applying it to each given number so that you have F(0), F(4), F(10)... and then detecting the pattern from that. You seem to be doing something else which is coming up with a formula to derive the sequence. The hints say to apply some function to each of the given numbers and determine what the invariant relation is between them. I appears you are trying to turn this into a purely mathematical problem but it is really more of a simple mathematical function and then pattern identification. $\endgroup$ – David James Apr 12 at 15:18
  • $\begingroup$ No additional or subtraction in the function applied F(0), F(4), F(10)... however, the final pattern is relative to its neighbors, so technically if you wanted to generalize the pattern, you would have to reference A(n-1) as the left neighbor of A(n) for example. I said no + and - so people would not come up with a complex arithmetic formula for deriving the sequence when it reality, a $very$ simple function (on the numbers given) will suffice, then just pattern recognition on the results of F(0), F(4), F(10).... $\endgroup$ – David James Apr 12 at 15:25
4
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This anwer is based on the answer of user68244. Feel free to give them the checkmark if this happens to be the intended pattern.

The n-th number of the sequence (starting at n=1) is the first natural number which

produces an n-digit number when squared

Unlike the other attempts, this explains the 0 at the start of the sequence, because

technically 0=0^2 is written using exactly one digit

Therefore, as stated before, the missing fifth number is

100

because

10000=100^2 is a five digit number while 9801=99^2 has only four digits.

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  • $\begingroup$ This answer is correct! The pattern is simply you square the given numbers, and the result keep growing by length 1 (or you could start with the 6th given number and say the neighbor to the left, when squared, shrinks by 1 digit). $0^2$ = 0 (length 1), $4^2$ = 16 (Length 2).... And yes the numbers given are the minimum non negative integers with this property. Nicely done. I wish I would each give you half credit like 75 points each. I can try drawing attention from the moderator to see if that is possible to be more fair. $\endgroup$ – David James Apr 12 at 15:34
  • $\begingroup$ Although technically, user68244 did not follow the instructions of using F(0), F(4), F(10)... then deriving the pattern, so if I was a teacher in school, I would deduct some major points for that, plus the 0 term was not accounted for that way. So I will give him the checkmark for a useful answer (since it was before your answer and may have inspired you to get the correct one), and give you the bounty. That way you both get points. Well done! I think without any hints it would have been MUCH harder. Hint 8 was a gift, making it MUCH easier. $\endgroup$ – David James Apr 12 at 15:39
  • $\begingroup$ And yes I chose 0 as the first given number so that people could not force-fit the square root of 10 function to match the pattern. I think it is a rather clever puzzle because the solution only involves using the square function (and yes it is a function using the numbers given and the missing number), then realizing that the result is growing just barely enough to become 1 digit longer. This illustrates how even a very simple puzzle can stump many people, because their thinking is biased towards an arithmetic approach which this puzzle did NOT require that (and it was stated many times). $\endgroup$ – David James Apr 12 at 15:48
  • $\begingroup$ I cannot award the bounty for another 21 hours but when I can, Zilvarro will get it all, unless I hear back from a moderator allowing me to split the points between them, although I would prefer giving at least 100 to Zilvarro and the remaining 50 to user68244 but my guess is that is not possible. $\endgroup$ – David James Apr 12 at 15:52
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I know I'm not close to the answer, but I want to give my opinion. 4 = 1^2 + 3,

10 = 3^2 + 1,

32 = 5^2 + 7,

Therefore, 7^2 + 5 = 54 is my guess. It doesn't relate to 0, but I can't seem to use the odd numbers to include it in the pattern.

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  • $\begingroup$ Not the pattern I was looking for but kudos for trying. I added a 3rd clue to help people from going down the wrong path. Don't use + or - cuz the pattern doesn't involve those. If you look carefully at the sequence given, a pattern should emerge. Someone will get it I am certain. It may take more than an hour for them to get it but it is ok. $\endgroup$ – David James Apr 10 at 13:37
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    $\begingroup$ It looks like the four numbers 0,1,2,3 are written like in the reverse order, taken in pairs, (0 and 1) as10 then (2 and 3) as 32, but even so, 4 and 5 could be written as 54... $\endgroup$ – Srinath Prasad Apr 10 at 13:45
  • $\begingroup$ That is a reasonable guess, however that pattern doesn't match the 0 and the 4, the first 2 given numbers in the sequence. I will add a 4th clue to help. $\endgroup$ – David James Apr 10 at 14:01
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Here is my guess: the 5th number is 130.

The a(k) term being:

a(k) = (a(k-1) + a(k+1))/k - (k-1)

So...

  • a(2) = (0 + 10)/2 - 1 = 4
  • a(3) = (4 + 32)/3 - 2 = 10
  • a(4) = (10 + a(5))/4 - 3 = 32 -> a(5) = 35*4 - 10 = 130

However, it doesn't quite work for a(1) = 0 as there are no numbers on its left.

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    $\begingroup$ The hint says NOT to use + or - cuz the "correct" solution doesn't use those. I purposely put those in there so people wouldn't go down the wrong path. Also, the pattern must apply to ALL terms, including the 0. When you see my solution, you wont believe how simple it is. Someone has to get it soon, it has been over 3 hours since I posted this puzzle. $\endgroup$ – David James Apr 10 at 16:42
  • $\begingroup$ I appreciate your effort in trying, I give credit for all attempted answers, however the person that solves it will have much glory because it has been over 4 hours now with 100 views and STILL no correct solution! Even with 7 hints! $\endgroup$ – David James Apr 10 at 16:56

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