## Find the 6th number

For the discussion of math. Duh.

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abeethabsl
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### Find the 6th number

1, 2, 6, 42, 1806, ?

Pathway
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### Re: Find the 6th number

3263442?

If so, 10650056950806, 113423713055421844361000442, 12864938683278671740537145998360961546653259485195806.
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### Re: Find the 6th number

How original do you want to be? Yeah you could go with the obvious answer that Pathway gave, or you could just create a quintic polynomial and have the sixth number be whatever you want.

Of course, you could be even more obstinate and just define any arbitrary sequence.

Say

a_1 = 1
a_2 = 2
a_3 = 6
a_4 = 42
a_5 = 1806
a_n = 0 for all n > 5

Blatm
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### Re: Find the 6th number

Spoiler:
an = an-1(an-1+1)

I know, I'm boring.

Frimble
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### Re: Find the 6th number

It irritates me that this sort of question frequently appears on 11+ tests and GCSE Maths exams. The next term could be any number at all, yet only one specific number is marked as correct.
"Absolute precision buys the freedom to dream meaningfully." - Donal O' Shea: The Poincaré Conjecture.
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Whyareall
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### Re: Find the 6th number

Spoiler:
its 1 180 482. Next is 1 393 538 932 806, and then 1.94195076 × 10^24
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Token
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### Re: Find the 6th number

Frimble wrote:It irritates me that this sort of question frequently appears on 11+ tests and GCSE Maths exams. The next term could be any number at all, yet only one specific number is marked as correct.

It irritates ME that people think "it could be anything" is a clever answer.
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hyperion
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### Re: Find the 6th number

Token wrote:
Frimble wrote:It irritates me that this sort of question frequently appears on 11+ tests and GCSE Maths exams. The next term could be any number at all, yet only one specific number is marked as correct.

It irritates ME that people think "it could be anything" is a clever answer.

You're saying that if someone finds a pattern (which isn't the one they're looking for but is nevertheless correct) and uses it to predict the next number then they are wrong/stupid? Really?
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Frimble
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### Re: Find the 6th number

Token wrote:
Frimble wrote:It irritates me that this sort of question frequently appears on 11+ tests and GCSE Maths exams. The next term could be any number at all, yet only one specific number is marked as correct.

It irritates ME that people think "it could be anything" is a clever answer.

Really it comes down to whether they are testing understanding of mathematics or just how well the candidate understands the examiner.

The question should be: "Write down a formula that will generate this series." or better still they could relate it to properties of shapes, graphs or simple sets.
"Absolute precision buys the freedom to dream meaningfully." - Donal O' Shea: The Poincaré Conjecture.
"We need a reality check here. Roll a D20." - Algernon the Radish
"Should I marry W? Not unless she tells me what the other letters in her name are" Woody Allen.

Buttons
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### Re: Find the 6th number

Token wrote:It irritates ME that people think "it could be anything" is a clever answer.

Maybe, but this one really could be many things. Eleven different sequences in Sloane start with those five numbers.

antonfire
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### Re: Find the 6th number

Huh? There are three, and one of them just ends there, and another is just a weird modification of the expected answer.

There are eleven which contain 1,2,6,42, and a bunch of them are just weird.
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Buttons
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### Re: Find the 6th number

Oops. I miss-copied the terms when searching the database. Um, go me.

Random832
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### Re: Find the 6th number

hyperion wrote:
Token wrote:
Frimble wrote:It irritates me that this sort of question frequently appears on 11+ tests and GCSE Maths exams. The next term could be any number at all, yet only one specific number is marked as correct.

It irritates ME that people think "it could be anything" is a clever answer.

You're saying that if someone finds a pattern (which isn't the one they're looking for but is nevertheless correct) and uses it to predict the next number then they are wrong/stupid? Really?

I think he was assuming Frimble would write "This is a silly question and I'm not going to answer it" in the answer space.
Buttons wrote:Oops. I miss-copied the terms when searching the database. Um, go me.

I did a search and there are two sequences starting with these numbers, and also one consisting only of these numbers (there is no 6th number).

GreedyAlgorithm
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### Re: Find the 6th number

hyperion wrote:
Token wrote:
Frimble wrote:It irritates me that this sort of question frequently appears on 11+ tests and GCSE Maths exams. The next term could be any number at all, yet only one specific number is marked as correct.

It irritates ME that people think "it could be anything" is a clever answer.

You're saying that if someone finds a pattern (which isn't the one they're looking for but is nevertheless correct) and uses it to predict the next number then they are wrong/stupid? Really?

Yes. If the question is done well, then definitely yes. It's called Occam's Razor. Or just science, for that matter.

If you constructed a polynomial of ginormous degree that perfectly described all of the past and extrapolated it to guess that tomorrow the sun would turn into a sunflower, you would be wrong/stupid. The idea is to find the simplest/most elegant/etc explanation for the facts. Sometimes questions are set up badly and you can find a simpler explanation than used to get the given answer, but that means it's a bad question, not a bad type of question. Here's an example with a very informal measure of complexity:

1, 2, 3, 4, _

One guess is 5, generated by f(n) = n. The RHS is 1 character.
Or you could say 6, f(n) = 1 - 1.083*x + 1.4583*x^2 - 0.416*x^3 + 0.0416*x^4. Here the RHS is 41 characters.

Where do you think mathematicians get ideas about which theorems to try to prove? Do they pick something completely random from theoryspace? No, of course not. They see a pattern somewhere and think "hey, what if this is really a pattern" (well, that's one way). Figuring out which elegant patterns fit data is an extraordinarily useful skill. Claiming that all theories are created equal leads to letting intelligent design into the science classroom.
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Blatm
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### Re: Find the 6th number

Though I'll admit that the majority of "find the next term in this series" questions are probably poorly worded (I've never actually had one on an exam), generally sequences given have one answer that is much simpler than the rest. Given the series 0,0,0,0,0,?, the "right" answer is simply 0. The "I don't like this question" answer is f(6) where f(x) = (x-1)(x-2)(x-3)(x-4)(x-5)(x-6349562398), but that's much more arbitrary. The question is really "I've used a formula to generate a series, and these are the first couple of terms. What is the next term that the formula I used will give?", and with enough terms, it's pretty easy to figure out which formula he was using, considering the assumed shared knowledge that tests are supposed to be doable by someone who understands the subject matter.

Ninja'd. Pretty impressive.

Random832
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### Re: Find the 6th number

There is a middle ground, though, between "stupid arbitrary polynomial" and "just a different answer than they wanted". There are hundreds of sequences containing "1, 2, 3, 4". (Though, granted, a lot of them do indeed follow it with 5), and none of them that I saw browsing the results were quite at the level of "stupid arbitrary polynomial"

GreedyAlgorithm
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### Re: Find the 6th number

Random832 wrote:There is a middle ground, though, between "stupid arbitrary polynomial" and "just a different answer than they wanted". There are hundreds of sequences containing "1, 2, 3, 4". (Though, granted, a lot of them do indeed follow it with 5), and none of them that I saw browsing the results were quite at the level of "stupid arbitrary polynomial"

Quite. And if you give an answer other than 5, you're wrong. The evidence you have should not be strong enough to make you believe that 6 is correct rather than 5, or 7, or 11. Now if the test looked like this:
1, 2, 4, 8, 9 (union of nonzero squares and twice squares)
1, 1, 1, 1, 2 (Pascal's Triangle written out)
1, 4, 9, 16, 26 (a(n+1) = a(n)-th composite number, with a(0) = 1)
1, 2, 3, 4, ?

Here you would be justified in giving 6 as your answer, because you have very good evidence that the missing number should not fit the simplest pattern. But unless you have some reason to believe, you have no reason to believe.
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### Re: Find the 6th number

1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1,

Now what comes next?
"Absolute precision buys the freedom to dream meaningfully." - Donal O' Shea: The Poincaré Conjecture.
"We need a reality check here. Roll a D20." - Algernon the Radish
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Random832
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### Re: Find the 6th number

GreedyAlgorithm wrote:
Random832 wrote:There is a middle ground, though, between "stupid arbitrary polynomial" and "just a different answer than they wanted". There are hundreds of sequences containing "1, 2, 3, 4". (Though, granted, a lot of them do indeed follow it with 5), and none of them that I saw browsing the results were quite at the level of "stupid arbitrary polynomial"

Quite. And if you give an answer other than 5, you're wrong. The evidence you have should not be strong enough to make you believe that 6 is correct rather than 5, or 7, or 11. Now if the test looked like this:
1, 2, 4, 8, 9 (union of nonzero squares and twice squares)
1, 1, 1, 1, 2 (Pascal's Triangle written out)
1, 4, 9, 16, 26 (a(n+1) = a(n)-th composite number, with a(0) = 1)
1, 2, 3, 4, ?

Here you would be justified in giving 6 as your answer, because you have very good evidence that the missing number should not fit the simplest pattern. But unless you have some reason to believe, you have no reason to believe.

Right, but that only works when there is a simple pattern. But for the purpose of this topic, I'm not really seeing how 3263442 is better than 3270666.

GreedyAlgorithm
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### Re: Find the 6th number

Frimble wrote:It irritates me that this sort of question frequently appears on 11+ tests and GCSE Maths exams. The next term could be any number at all, yet only one specific number is marked as correct.

Frimble wrote:Really it comes down to whether they are testing understanding of mathematics or just how well the candidate understands the examiner.

The question should be: "Write down a formula that will generate this series." or better still they could relate it to properties of shapes, graphs or simple sets.

Frimble wrote:1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1,

Now what comes next?

With no context, the correct answer is 2. That doesn't necessarily mean it's right, but it is the answer most likely to be right given the information we have. We have a little bit of context, though - your two posts, which indicate you are irritated by this sort of question and think it reflects more on the examiner than properties of mathematics. A little bit of OEIS says there are four sequences that contain your sequence explicitly as a subsequence. Three of them have a 2 next and one has a 1 next. Obviously it could be any number, but I'd say determine your credence that Frimble is trying to make a point (i.e. the answer is not 2), call that p, and we should assign:

Probability 1-p the answer is 2.
Probability 0.8*p the answer is 1.
Probability 0.2*p the answer is something else.

I pulled the 0.8 from the aether. If Frimble is playing a deep game then he might be trying to trick us into thinking he's trying to make a point. Without looking too far down the rabbit hole that lets me say p is about 0.5, and 1-0.5=0.5 is greater than 0.8*0.5=0.4, so I'm still writing down 2.
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kgrizzly
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### Re: Find the 6th number

Random832 wrote:Where do you think mathematicians get ideas about which theorems to try to prove? Do they pick something completely random from theoryspace? No, of course not. They see a pattern somewhere and think "hey, what if this is really a pattern" (well, that's one way). Figuring out which elegant patterns fit data is an extraordinarily useful skill.

Agreed.

GreedyAlgorithm wrote:Claiming that all theories are created equal leads to letting intelligent design into the science classroom.

Huhnnnn???? Non-sequitur. Weren't we talking about mathematics? Or at least about guessing sequences?

The "what's next in the sequence?" type of question, when asked without a context, is just a parlour game. It is a dressed-up version of "what number am I thinking about?" It can be a enjoyable pastime between friends. But guessing wrong in such a situation carries nothing like the sort of objective assessment that you seem inclined to attach to it.

Your comment also makes me wonder: Do you have access to some sort of higher knowledge or criteria which would dictate what sort of mathematics a mathematician should (morally) be pursuing??
Last edited by gmalivuk on Tue Aug 12, 2008 3:30 am UTC, edited 1 time in total.
Reason: Fixed the second quote tag

GreedyAlgorithm
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### Re: Find the 6th number

Random832 wrote:
GreedyAlgorithm wrote:
Random832 wrote:There is a middle ground, though, between "stupid arbitrary polynomial" and "just a different answer than they wanted". There are hundreds of sequences containing "1, 2, 3, 4". (Though, granted, a lot of them do indeed follow it with 5), and none of them that I saw browsing the results were quite at the level of "stupid arbitrary polynomial"

Quite. And if you give an answer other than 5, you're wrong. The evidence you have should not be strong enough to make you believe that 6 is correct rather than 5, or 7, or 11. Now if the test looked like this:
1, 2, 4, 8, 9 (union of nonzero squares and twice squares)
1, 1, 1, 1, 2 (Pascal's Triangle written out)
1, 4, 9, 16, 26 (a(n+1) = a(n)-th composite number, with a(0) = 1)
1, 2, 3, 4, ?

Here you would be justified in giving 6 as your answer, because you have very good evidence that the missing number should not fit the simplest pattern. But unless you have some reason to believe, you have no reason to believe.

Right, but that only works when there is a simple pattern. But for the purpose of this topic, I'm not really seeing how 3263442 is better than 3270666.

Yep, no problem. If I asked:
1, 2, 6, 42, 1806, ?
and you had no other context (like "I'm taking a course in combinatorics" in which case you should answer 3263442, or "I'm taking a course in number theory" in which case you should answer 3270666), then you'd probably end up with close to 0.5 credence for both answers. So go to your situation to find out what to do:

a) I got the list of numbers from somewhere and I can generate the next one and I'm looking for the formula: do science! Find the next one. Falsify one of your theories.
b) I'm taking a standardized test with multiple choice answers and cannot clarify, etc. and don't have the aforementioned context: Pick randomly! It's the best you can do, the question was badly designed, sorry. Actually pick 3263442 since A007018 was entered before A100016 (I think?).
c) I'm taking a test and can write questions/explanations/etc: Write 3263442 as your answer and explain why your credence is near 0.5 for 3263442 and near 0.5 for 3270666.
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### Re: Find the 6th number

Random832 wrote:
GreedyAlgorithm wrote:
Random832 wrote:There is a middle ground, though, between "stupid arbitrary polynomial" and "just a different answer than they wanted". There are hundreds of sequences containing "1, 2, 3, 4". (Though, granted, a lot of them do indeed follow it with 5), and none of them that I saw browsing the results were quite at the level of "stupid arbitrary polynomial"

Quite. And if you give an answer other than 5, you're wrong. The evidence you have should not be strong enough to make you believe that 6 is correct rather than 5, or 7, or 11. Now if the test looked like this:
1, 2, 4, 8, 9 (union of nonzero squares and twice squares)
1, 1, 1, 1, 2 (Pascal's Triangle written out)
1, 4, 9, 16, 26 (a(n+1) = a(n)-th composite number, with a(0) = 1)
1, 2, 3, 4, ?

Here you would be justified in giving 6 as your answer, because you have very good evidence that the missing number should not fit the simplest pattern. But unless you have some reason to believe, you have no reason to believe.

Right, but that only works when there is a simple pattern. But for the purpose of this topic, I'm not really seeing how 3263442 is better than 3270666.

I wold say that (n+1) is a hell of a lot more simple than the (next prime greater than n) especially when n=1806. Hence one of the two patterns is clearly simpler to calculate, and is therefore the logical one to apply first to get the next number in the sequence.

GreedyAlgorithm
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### Re: Find the 6th number

kgrizzly wrote:
GreedyAlgorithm wrote:Claiming that all theories are created equal leads to letting intelligent design into the science classroom.

Huhnnnn???? Non-sequitur. Weren't we talking about mathematics? Or at least about guessing sequences?

The "what's next in the sequence?" type of question, when asked without a context, is just a parlour game. It is a dressed-up version of "what number am I thinking about?" It can be a enjoyable pastime between friends. But guessing wrong in such a situation carries nothing like the sort of objective assessment that you seem inclined to attach to it.

Why not? If someone can get the answer right 90% of the time and you can only get it right 40% of the time, clearly they're doing something far better than you are. Objectively better. Perhaps the consequences are trivial, but that doesn't mean suddenly there's not an objectively right answer.

kgrizzly wrote:Your comment also makes me wonder: Do you have access to some sort of higher knowledge or criteria which would dictate what sort of mathematics a mathematician should (morally) be pursuing??

Yes. Yes I do, and we all do, but it's rarely spelled out. For a moment I will ignore the issues of terminal/instrumental values, etc. because I think they aren't relevant for the answer I want to give. A mathematician should be more likely to pursue mathematics that is likely to be fruitful than mathematics that is not likely to be fruitful. In practice, what does this mean?

Suppose you are trying to decide between two different things to study. How do you decide which? Randomly? Unlikely. Which you anticipate will be more fun? Possible. Which you anticipate you are likely to make progress on? Sounds fine. Which will get you more grant money? Okay. These are just pursuing the mathematics that leads to outcomes you desire - more fruitful mathematics.

And here's the tie-in: Figuring out which sorts of mathematics will be more fruitful before the fact often uses at least in part skills very similar to those used for figuring out which number comes next in the sequence.
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Frimble
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### Re: Find the 6th number

GreedyAlgorithm wrote:
Frimble wrote:It irritates me that this sort of question frequently appears on 11+ tests and GCSE Maths exams. The next term could be any number at all, yet only one specific number is marked as correct.

Frimble wrote:Really it comes down to whether they are testing understanding of mathematics or just how well the candidate understands the examiner.

The question should be: "Write down a formula that will generate this series." or better still they could relate it to properties of shapes, graphs or simple sets.

Frimble wrote:1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1,

Now what comes next?

With no context, the correct answer is 2. That doesn't necessarily mean it's right, but it is the answer most likely to be right given the information we have. We have a little bit of context, though - your two posts, which indicate you are irritated by this sort of question and think it reflects more on the examiner than properties of mathematics. A little bit of OEIS says there are four sequences that contain your sequence explicitly as a subsequence. Three of them have a 2 next and one has a 1 next. Obviously it could be any number, but I'd say determine your credence that Frimble is trying to make a point (i.e. the answer is not 2), call that p, and we should assign:

Probability 1-p the answer is 2.
Probability 0.8*p the answer is 1.
Probability 0.2*p the answer is something else.

I pulled the 0.8 from the aether. If Frimble is playing a deep game then he might be trying to trick us into thinking he's trying to make a point. Without looking too far down the rabbit hole that lets me say p is about 0.5, and 1-0.5=0.5 is greater than 0.8*0.5=0.4, so I'm still writing down 2.

My point really is that in order to answer my question you have to guess what I'm thinking rather than using mathematical ability alone.

A sequence is nothing more than a way of choosing elements from a set and then arranging them (obviously each element can be selected more than once). Given any finite number of terms of a sequence there is an infinite number of other sequences that also begin with those numbers (the next term could even be "blue" or "velociraptor" or "the set of all sets which are not members of themselves").

To know which number (or random object/concept) comes next in a sequence requires more information than just the first n terms, perhaps that the sequence is repeating, that it is an arithmetic progression (or an arithmetic progression modulo n) or that the set of which the terms are members is isomorphic to some properties of platonic solids.

In the case of many questions from SATS, 11+ and GCSE papers the only information a candidate will know other than the first n terms of the sequence is what kinds of sequence are on the syllabus. It's not hard to guess the answer, I just don't think this is the kind of reasoning that should be encouraged.
"Absolute precision buys the freedom to dream meaningfully." - Donal O' Shea: The Poincaré Conjecture.
"We need a reality check here. Roll a D20." - Algernon the Radish
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Random832
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### Re: Find the 6th number

floyd4one wrote:I wold say that (n+1) is a hell of a lot more simple than the (next prime greater than n) especially when n=1806. Hence one of the two patterns is clearly simpler to calculate, and is therefore the logical one to apply first to get the next number in the sequence.

Yeah, but (n+1) is your 1,2,3,4 sequence, not 1, 2, 6, 42, 1806; 1806 certainly isn't 42+1. Rather, it's a choice between n*(next prime greater than n) and n^2+n; and the second is arguably more complex.

Plus, that only works when you already have both formulas and are choosing which one is better - sucks if someone figures out the prime number based one first when the person writing the test was thinking of the n^2+n one.

Nitrodon
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### Re: Find the 6th number

Random832, are you really trying to say that n*(n+1) is more complex than n*(next prime greater than n)?

Random832
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### Re: Find the 6th number

Nitrodon wrote:Random832, are you really trying to say that n*(n+1) is more complex than n*(next prime greater than n)?

no, I fail at factoring polynomials. However, my other point still stands.

Woxor
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### Re: Find the 6th number

The worst example of this sort of thing that I've ever seen was worked out as the sample question in an SAT prep book or something. It was an analogy: "2 is to 4 as 8 is to ___." The "answer" was 16, but that struck me as pretty arbitrary. I mean, yeah, the fractional ratio seems like the most reasonable relationship in an analogy, but there's really nothing to say that the answer shouldn't have been 10 (2+2=4, 8+2=10) or 64 (2^2=4, 8^2=64) or 16777216 (2^2=4, 8^8=16777216) or any number of more complicated things.

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### Re: Find the 6th number

Frimble wrote:To know which number (or random object/concept) comes next in a sequence requires more information than just the first n terms, perhaps that the sequence is repeating, that it is an arithmetic progression (or an arithmetic progression modulo n) or that the set of which the terms are members is isomorphic to some properties of platonic solids.

In the case of many questions from SATS, 11+ and GCSE papers the only information a candidate will know other than the first n terms of the sequence is what kinds of sequence are on the syllabus. It's not hard to guess the answer, I just don't think this is the kind of reasoning that should be encouraged.

Inductive reasoning is something we don't want to encourage? Aside from being essential for science and everyday life, a very similar thing comes up all the time in math. Haven't you ever seen something like this?

exp(x) = 1 + x + x2/2! + x3/3! + x4/4! + ...

Yes, there are more explicit ways to specify this series, but this way is common and intuitive, and I don't see anything wrong with it.

qbg
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### Re: Find the 6th number

abeethabsl wrote:1, 2, 6, 42, 1806, ?

6979 assuming the last difference that can be calculated from the fragment is consant.

antonfire
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### Re: Find the 6th number

The fact is, extrapolating the rest of a sequence from a finite list of numbers is a very useful skill in mathematics. Its development should be encouraged and tested. (And, by the way, jumping straight to OEIS should be discouraged.)

If you construct your problems well, it's a perfectly good way to assign problems. This particular problem is pretty well constructed. In particular, nobody is going to think of "next prime number bigger than n" before they think of "n+1". We wouldn't be talking about this other sequence if it didn't happen to also be (for whatever reason) on OEIS.

Anyway, if you're claiming that this sort of problem is bad, you might as well claim that problems like "draw a best-fit curve to this set of data points" are bad. Would it be correct to just draw the polynomial that goes through them? It's fun to be a smartass in the classroom, but when you have an actual set of data that you want to analyze, nature won't care how funny you are.

Frimble wrote:In the case of many questions from SATS, 11+ and GCSE papers the only information a candidate will know other than the first n terms of the sequence is what kinds of sequence are on the syllabus. It's not hard to guess the answer, I just don't think this is the kind of reasoning that should be encouraged.
And when you're playing around with combinatorics and you come up with a sequence of your own, you could have very little useful information other than the first n terms in the sequence and what kinds of sequences tend to actually come up in mathematics and in your field. This is exactly the sort of reasoning that should be encouraged.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

Random832
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### Re: Find the 6th number

Cosmologicon wrote:Aside from being essential for science and everyday life, a very similar thing comes up all the time in math. Haven't you ever seen something like this?

exp(x) = 1 + x + x2/2! + x3/3! + x4/4! + ...

Yes, there are more explicit ways to specify this series, but this way is common and intuitive, and I don't see anything wrong with it.

Once it gets beyond the really obvious sequences like 1, 2, 3, 4 (the kind of sequence which you WOULD NEVER FIND ON SUCH A QUESTION) though, no.

And there's a reason for doing .../2! .../3! instead of defining it as "1 + x + x2/2 + x3/6 + x4/24 ..."

I'd say that the integers, steps by a specific number (evens, odds, multiples of 3, etc), and maybe alternating signs, are just about all that's acceptable in a context like that (so, + ...2 - ...4 + ...6); anything else you need to be explicit about. I don't think you ever see a series written-out the way you did above that has "2 ....4 ....8 ....16" or "1 ....4 ....9 ....16" in it rather than 21 22 23 24 or 12 22 32 42

GreedyAlgorithm
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### Re: Find the 6th number

Enough theory, here's a better problem. There's a function f:NxN->N that you have found several values for already, but any additional values will take lots of computation time. Determine the function. If you need more values, request them in the form f(a,b) (e.g. f(3,4)). Internet points are awarded for requesting few if any things, and for anything requested being more like f(3,4) than f(52,106). Why is f(2,2) missing? Who knows. The computer processing that value got unplugged overnight or something. Is 0 in N? I don't care which religion you follow, it is for this problem. Do I know whether this is too easy? No, but it's not too hard. Do I think it has one definitely best answer? Yes, but if you find a second that'd be pretty neat too.

f(a,b) = ?
f(0,b) = 0
f(a,0) = 0
f(a,1) = a
f(1,2) = 4
f(1,3) = 10
f(2,3) = 25
f(3,2) = 15
f(3,3) = 46
GENERATION 1-i: The first time you see this, copy it into your sig on any forum. Square it, and then add i to the generation.

Cauchy
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### Re: Find the 6th number

antonfire wrote:If you construct your problems well, it's a perfectly good way to assign problems. This particular problem is pretty well constructed. In particular, nobody is going to think of "next prime number bigger than n" before they think of "n+1". We wouldn't be talking about this other sequence if it didn't happen to also be (for whatever reason) on OEIS.

That's clearly not the case, since I thought of "next prime" before I thought of "one larger". This is because I noted each term was a multiple of the previous one, and so I looked at the multipliers to see if I could notice a common trait. 2, 3, 7, 43. "Prime" jumped out at me first, definitely before "one more than the number they're multiplying". I noticed then that each prime was the next prime after the number it was multiplying, before noting that each one was one more than the number it was multiplying. But by then, it was too late, and the "next prime" idea was stuck in my head.
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
Thanks, skeptical scientist, for knowing symbols and giving them to me.

antonfire
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### Re: Find the 6th number

Huh. I stand corrected, then.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

scarletmanuka
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### Re: Find the 6th number

Random832 wrote:
Cosmologicon wrote:Aside from being essential for science and everyday life, a very similar thing comes up all the time in math. Haven't you ever seen something like this?

exp(x) = 1 + x + x2/2! + x3/3! + x4/4! + ...

Yes, there are more explicit ways to specify this series, but this way is common and intuitive, and I don't see anything wrong with it.

Once it gets beyond the really obvious sequences like 1, 2, 3, 4 (the kind of sequence which you WOULD NEVER FIND ON SUCH A QUESTION) though, no.

And there's a reason for doing .../2! .../3! instead of defining it as "1 + x + x2/2 + x3/6 + x4/24 ..."

I'd say that the integers, steps by a specific number (evens, odds, multiples of 3, etc), and maybe alternating signs, are just about all that's acceptable in a context like that (so, + ...2 - ...4 + ...6); anything else you need to be explicit about. I don't think you ever see a series written-out the way you did above that has "2 ....4 ....8 ....16" or "1 ....4 ....9 ....16" in it rather than 21 22 23 24 or 12 22 32 42

Not to mention that anything beyond the immediately obvious usually has the general term written out as well, to remove any possible ambiguity: "... + (nr) ar bn-r + ..." (not that this is a good example, but I couldn't be bothered looking one up).

This space intentionally left non-blank so that the descenders on the previous line don't get chopped off.

Doraki
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### Re: Find the 6th number

GreedyAlgorithm wrote:Enough theory, here's a better problem. There's a function f:NxN->N that you have found several values for already, but any additional values will take lots of computation time. Determine the function. If you need more values, request them in the form f(a,b) (e.g. f(3,4)). Internet points are awarded for requesting few if any things, and for anything requested being more like f(3,4) than f(52,106).

would f(52,106) be 17817521456210923584397430585901605649528622 ?

GreedyAlgorithm
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### Re: Find the 6th number

Doraki wrote:would f(52,106) be 17817521456210923584397430585901605649528622 ?

Quite so, nice job.

Care to go through your solving process so we can see how much it relied on guessing my intent?
GENERATION 1-i: The first time you see this, copy it into your sig on any forum. Square it, and then add i to the generation.

Doraki
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### Re: Find the 6th number

Looking at g(a,b) = f(a,b) - f(a-1,b) - f(a,b-1) just happens to be one of the 1st thing I do when I have to guess a function of 2 variables,
just like looking at the differences when I have a simple sequence.
And it just happened that there was a very nice value to pick for f(2,2).

I'm nowhere near a simple closed formula for f(a,b) though, haven't been looking for one very much.