really hard puzzle, noone has solved yet...

[MidGe]

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I simply claimed there was no unique solution to those sorts of sequence puzzles

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True, true. Often missed by people who set them. [img]/images/graemlins/smile.gif[/img]

[billygrippo]

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For there to be a unique solution there would have to be one and only one rule that predicted the first n numbers in the sequence correctly. But there is never only one such rule, there are always an indefinite number of rules that can fit or 'predict' the pattern, and which may all yield distinct answers (all equally 'correct') for what the next number is in the sequence.

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occam's [censored] razor!!! jesus [censored]!

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Those sorts of puzzles have nothing to do with Occam's razor. Occam's razor (say, adopt the theory that is most ontologically parsimonious) has to do with choosing between scientific theories that are equally empirically adequate. You simply gave a puzzle and asked for a solution.

Now we could invoke some principle like Occam's razor in our instructions for answering that sort of puzzle--like saying "give the simplest rule/formula that solves the problem," but that still does not mean there is a unique solution to the original puzzle--just one that we favor over others on grounds other than that it's an accurate solution.

I simply claimed there was no unique solution to those sorts of sequence puzzles, and that is true whether or not we decide to adopt some standard for choosing among equally adequate solutions in terms of simplicity of formula, or some such standard. But even in such a case you might be surprised at how difficult it is to provide an adequate philosophical defense of 'simplicity criteria.'

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very well put. however, try to solve the original puzzle. I assure you there is a "simple" answer, that stands out from the other possible soultions.

[thijsr]

terrible example,
obviously when the sequence is presented to us by its inventor, there is a hidden assumption that the property of the m^th-rule used to generated the m^th entry (x_m) can't be dependent on m (the structure of the rule that is, it might of course contain m as a variable).
also there is sth of the form (razor or no razor): 'this rule is not ridiculously hard to formulate' in there.

in almost all examples the rule will be rigid and only contain the variables x_{m-1}... x_0 and m.
in more complicated examples the rule itself might change, but only in such a way that that pattern in which it does can be observed in the initial data.

these assumptions are there, b/c it was presented as a puzzle. it's understood as part of what we all agree to be a 'puzzle'.

without the assumptions any random continuation is fine (i guess i should add to the assumption: there is a rule). in a different context (like when you are a scientist looking at nature) you can't rely on such matters and your (well...) point makes some more sense (although i'm sure you're counting on the sun coming up tomorrow). you'll have to find somebody else to debate that though, i've had my share.

anyways... you say they are infinite, i'd say suprise me by getting 1 (alternative one with the assumption that is), no way you can get 2.

(it's obvious your example doesn't work with these assumptions, saying sth to the effect of: use a diffent formula when n>M (and thus referring to M))

cheers,
matt

[_TKO_]

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these assumptions are there, b/c it was presented as a puzzle. it's understood as part of what we all agree to be a 'puzzle'.

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It was likely only brought up because nobody can answer the original puzzle.

[KipBond]

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these assumptions are there, b/c it was presented as a puzzle. it's understood as part of what we all agree to be a 'puzzle'.

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It was likely only brought up because nobody can answer the original puzzle.

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I got close. :P

[bluesbassman]

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philo, if there is a puzzle that starts like this:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59


how is the next number ANYTHING BUT 61?!?!?!? this must be the only unique answer to this puzzle. if you can come up with some math or logic that will yeild a different number, please show us now.

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Let F(n): R==>R be the function defined by:

F(n) = a0*(n^17) + a1*(n^16) + ... + a16*n + a17

where n is any real number, and {a0, a1, ... , a17} is a set of 18 real constant coefficients. It is easy to show by construction that there exist particular values of {a0, a1, ... , a17} which have the property:

F(1) = 2, F(2) = 3, F(3) = 5, ... , F(17) = 59, but that
F(18) <> 61. (i.e. F(18) does not equal 61.)

As Philo stated, there is no reason to favor the number 61 over F(18) as the next number in the sequence.

[billygrippo]

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As Philo stated, there is no reason to favor the number 61 over F(18) as the next number in the sequence.

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there is plenty of reason. when "61" is the answer, you can explain the result in like 5 words. Id bet some children could get 61.

[madnak]

So if a child can undertand it, that makes it the correct answer? I understand the aesthetic reasons why 61 is better (it's more intuitive), and I assume the answer to your problem is also more intuitive than other answers. But that doesn't necessarily make it more correct, does it?

[bluesbassman]

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As Philo stated, there is no reason to favor the number 61 over F(18) as the next number in the sequence.

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there is plenty of reason. when "61" is the answer, you can explain the result in like 5 words. Id bet some children could get 61.

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Of course I meant there is no mathematical reason to choose one solution over another. You claimed 61 "must" be the next number in the sequence, and challenged someone to construct an alternative rule (i.e. function) that gives a different solution, which is exactly what I did. (And my solution can be understood with grade school level math.)

Obviously, given how our pattern-seeking brains understand mathematics, the sequence of primes is the most intuitive or natural solution. But that solution is not unique, which was Philo's original (and correct) point.

[billygrippo]

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So if a child can undertand it, that makes it the correct answer? I understand the aesthetic reasons why 61 is better (it's more intuitive), and I assume the answer to your problem is also more intuitive than other answers. But that doesn't necessarily make it more correct, does it?

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what makes it correct is that its the simplest logical solution.