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When setting a number sequence puzzle, one normally gives a finite sample and asks for the rule that specifies the members.

The sample has to be finite because it's impossible to uniquely specify an infinite series without giving away the answer. (Note: Is this true?)

Of course the problem is that any finite string of numbers can have an infinite number of rules that specify it.

Example

What is the next number in this series?

2 3 4 7 9 ?

To which someone could say: These are members of the set {2 3 4 7 9 73} so the answer is 73.

or

This is the union of the sets {2, 3, 4, 7, 9} and {10, 11, 12, 13, ...} so the answer is 10.

This is taking things to an extreme but how is it possible to counter such suggestions?

Note: The real-life example that caused me to pose the question is here Can you generate the next number in this integer series and describe the rule? and the first answer proposed was some arbitrary union of sets.

Question

Are there any recognised ways, or can you suggest a way, of designing number sequence puzzles that can tie down the generating rule exactly without actually specifying the answer? If not, is the number sequence puzzle genre dead?

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  • $\begingroup$ While that answer made me laugh, I think that such answers are pretty much like the polynomial ones. It is up to you to mark the intended one as correct and up to the community to tolerate (or not) such answers or questions. It's hard to protect yourself from such "hacks" and IMHO you shouldn't try or care to. $\endgroup$ – dmg Oct 18 '15 at 10:53
  • $\begingroup$ I don't know 'the polynomial ones'. Do you have a link to what this means? $\endgroup$ – chasly from UK Oct 18 '15 at 10:59
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    $\begingroup$ I can't find an example, but the approach is presented here. $\endgroup$ – dmg Oct 18 '15 at 11:23
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    $\begingroup$ Thanks - that's a great link and it addresses my question directly. It even gives an answer, i.e. 'What they really mean is, "What is the next number in the sequence a test writer or puzzle maker is most likely to intend by the following?" In a way it's more psychology than math: what kind of sequence would a reasonably normal person think was "nice" enough to ask about?' $\endgroup$ – chasly from UK Oct 18 '15 at 11:31
  • $\begingroup$ Personally, I find answers that use polynomials or the set example you posted as simply "not nice" (as in "disrespectful to the OP"). Puzzling is not simply about being right. There are counter examples of course, like this lovely answer to an unfortunately deleted question. However, note that the question is tagged as "lateral-thinking". $\endgroup$ – dmg Oct 19 '15 at 6:29
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1."Are there any recognised ways, or can you suggest a way, of designing number sequence puzzles that can tie down the generating rule exactly without actually specifying the answer?"

No, there is not. The prove to this statement is in your answer:
Any statement like "These are members of the set {2 3 4 7 9 73} so the answer is 73." is an appropriate answer, until you write specifically in your problem conditions ether "73 is not correct answer" or "f(i) is correct answer, where f is function defined as ...".

2."If not, is the number sequence puzzle genre dead?".

I'm not sure that I understand this question exactly, but number sequence puzzle genre is not dead! But not because those puzzle can be made correct and objective, but because puzzles are created for people and all people love subjective things! By definition of "love" :).

3.The really good sequence puzzles have:
a) a very short and easy to understand pattern. b) are tested (with many people) to be sure that all other appropriate patterns are quite bulky.

For example Fibonachi sequence: 1 1 2 3 5 8 13. The pattern a_{i+2} = a_{i+1} + a_{i} is short and easy to remember and understand, on the other hand the pattern "it is part of sequence 1 1 2 3 5 8 13 75 87 98" is not short and very hard to remember.

4."The sample has to be finite because it's impossible to uniquely specify an infinite series without giving away the answer. (Note: Is this true?)"

I see no reason why this should be true. "Giving away" means "to make obvious", but concept of obvious depends on mans mind and not only subjective, but includes some finite things by it's own, for example if you write down infinite number of objects in a row, a man still can read only finite number of them during his life.

Also consider this example, which took me 30 sec to come up with:

04, 087, 03484, 091276, 15996464
122540867, 10960141044, 2828916977616

I would imagine that this sequence is similarly hard to crack with 5 and with 8 (20,100) samples. Isn't it? This is the case simply because the complexity of samples after 5th is just too big to be of any use if you have no idea about the pattern.

If you want to know, the pattern is:

Take the following sequence: a_i = 1024^i and exclude from each number digits with odd position in the number.

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