Finish the Sequence! (+1) (+1) (+2) (+3)

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coppro
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Finish the Sequence! (+1) (+1) (+2) (+3) (0)

Can you figure it out?

09-12-2007 at 04:17 AM

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CuriousShyRabbit
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (0)

Maybe I'm thinking way too simple, but is the next one

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09-12-2007 at 04:22 AM

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Rabscuttle
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (+1)

(0)

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09-12-2007 at 04:37 AM

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Oneiromancer
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (0)

I think we even did some sequences in the old Puzzle Tag thread. For example, what are the next 3 entries in these series:

50 33 25 20 16 __ __ __

and

Monday Thursday Thursday Sunday Tuesday _____ _____ _____

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09-12-2007 at 05:05 AM

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CuriousShyRabbit
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (+2)

I did enjoy looking through the old Puzzle Tag thread when it was closer to the top.

Too bad it's dead...

And now I do understand that this thread is about mod points and not puzzles, but I can't resist...

50, 33, 25, 20, 16,

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Monday, Thursday, Thursday, Sunday, Tuesday,

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09-12-2007 at 05:56 AM

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Hikari
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (0)

I don't have the points to do it via modpoints, but, is it...

(+1) (+1) (+2) (+3) (+5) (+8) (+13) (+21) (+34) (+55) (+89) ....?

It looks like the start to the Fibonacci Sequence to me.

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09-12-2007 at 06:28 AM

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bdwing
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (0)

Am I the only one seeing (+1)(+1)(+2)(+3)(0)? (Except for Rabscuttle's post which was (+1)(+1)(+2)(+3)(+1).)

That makes it just a bit harder.

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09-12-2007 at 01:20 PM

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Syntax
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (+2)

I'm with Hikari that it *looks like* the Fibonacci series.
However, the problem with short series like this there can be a lot of possible answers.

For example, here's an explanation of why the next number in 1,1,2,3 is (could be) 6...

Take the integer series:

 1  2  3  4  5

Square each number (called series I):

 1  4  9 16 25

Starting at 0 and adding consecutive integers starting with 3, we also create another series (called series II):

 0  3  7 13 19

Now take away series II from I:

 1  1  2  3  6

QED

Question is, how many possible numbers can be explained mathematically for a given truncated series, and whether this is infite. If not, what is the correlation between the length of the original series alongside its statistical features and the number of possible mathematically reasoned answers.

Yes, numerical series are a passion of mine

09-12-2007 at 03:09 PM

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Kevin_P86
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (+1)

Syntax wrote:
Starting at 0 and adding consecutive integers starting with 3, we also create another series (called series II):
 0  3  7 13 19

Umm... you seem to have added 3, 4, 6 and 6

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09-12-2007 at 04:02 PM

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Syntax
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (0)

lol! Good catch. Well, my theory wouldn't work otherwise so I'll leave it as is

[EDIT]

Though the obvious explanation is that you must also remove x where x is the nth position in the series divided by 4 rounded upwards...

[Last edited by Syntax at 09-12-2007 04:52 PM]

09-12-2007 at 04:49 PM

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Jacob
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (+1)

OK, then. What is the next number in this sequence?

2,4,4,2,3,4,6,__

(I agree, for any finite list of numbers you can find a justification for any continuation you want. So it's more a question of Guess what the problemsetter's thinking.)

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09-13-2007 at 12:33 AM

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CuriousShyRabbit
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (+1)

Jacob wrote:
OK, then. What is the next number in this sequence?

2,4,4,2,3,4,6,__

The next 3 numbers are:

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09-13-2007 at 01:42 AM

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The Stew Boy
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (0)

Ok, what's the next number in the series 1,-1,2,6,21,43?

[Last edited by The Stew Boy at 09-13-2007 07:38 AM : Typo]

09-13-2007 at 07:38 AM

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Casebier
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (0)

Did you guys realize that if you take the number 41 (a prime) and cumulatively add the even numbers from 2 to 40, every result is also a prime number?

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09-15-2007 at 03:57 PM

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Znirk
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (0)

Casebier wrote:
Did you guys realize that if you take the number 41 (a prime) and cumulatively add the even numbers from 2 to 40, every result is also a prime number?

Would you believe I've taken the number 41 and cumulatively added the even numbers from 2 to 40 many, many times, and I never noticed this?

09-15-2007 at 11:59 PM

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Someone Else
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (+5) (+8) (0)

Ok, a mod point goes to whoever can put the next number in this sequence.

09-17-2007 at 10:42 AM

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Anson
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (0)

Too easy. 13.

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09-17-2007 at 11:02 AM

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Tuttle
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (+5) (+8) (0)

You mean if I spend 62 mod points to put a (+13) next to your heading, you'll give me a mod point back? Bargain.

[Last edited by Tuttle at 09-17-2007 11:13 AM : fixed subject]

09-17-2007 at 11:12 AM

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Briareos
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (+5) (+8) (+13) (+1)

Someone Else wrote:
Ok, a mod point goes to whoever can put the next number in this sequence.

Here you go.

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09-17-2007 at 11:29 AM

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Someone Else
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (0)

Ok, I have to give it to you.

09-17-2007 at 07:17 PM

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Hikari
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (+5) (+8) (0)

Someone Else wrote:
Ok, a mod point goes to whoever can put the next number in this sequence.

Hikari wrote:
I don't have the points to do it via modpoints, but, is it...

(+1) (+1) (+2) (+3) (+5) (+8) (+13) (+21) (+34) (+55) (+89) ....?

It looks like the start to the Fibonacci Sequence to me.

I like how you asked for the next number in a sequence that was already posted earlier on the page.

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09-17-2007 at 09:42 PM

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Briareos
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (+5) (+8) (0)

Hikari wrote:
I like how you asked for the next number in a sequence that was already posted earlier on the page.

Duh... who doesn't know the Fibonacci sequence when he sees it?

But anyhow, I was the first to actually put the next number in the sequence, so there...

np: Luke Vibert - Comphex (Chicago, Detroit, Redruth)

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09-17-2007 at 11:07 PM

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zex20913
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (0)

Syntax wrote:

Question is, how many possible numbers can be explained mathematically for a given truncated series, and whether this is infinite {sic}. If not, what is the correlation between the length of the original series alongside its statistical features and the number of possible mathematically reasoned answers.

Yes, numerical series are a passion of mine

There are indeed an infinite number of possibilities for any truncated sequence.

Say you're given n terms in the sequence, and you have to determine the next number.

Treat each term, in order, as y-values for a graph. The x-values can be either 0...n-1 or 1...n, depending on what you want.

You can find a polynomial that will describe all of these points. (I'm pretty sure the polynomial is also of degree n or less, but not 100% certain).

Next, you can choose any y-value for your next x, and find a polynomial that satisfies all of those. Since you can choose any y-value to be the next number, and find a proper polynomial, the amount of terms is infinite.

Question from me: Is the cardinality of the solution set to any sequence aleph-null or aleph-one?

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09-19-2007 at 01:06 AM

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The Stew Boy
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (0)

The Stew Boy wrote:
Ok, what's the next number in the series 1,-1,2,6,21,43?

No-one's gotten this yet.

It is an actual sequence, a recursive one.

09-19-2007 at 08:19 AM

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Sillyman
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (0)

The cardinality would be aleph-one, unless you are restraining this to integers, as the solution set would be the set of all complex numbers. At least, I think that would be the case, as you only said aleph-one and aleph-null, and rational numbers are uncountably infinite.

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[Last edited by Sillyman at 09-19-2007 09:37 PM]

09-19-2007 at 09:35 PM

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zex20913
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (0)

Rational numbers are actually countable. If you set them all up as an infinite by infinite matrix with Aij=i/j, it covers all rational numbers. Then, start at the upper left, and count diagonally up and to the right. When you hit the top, go down one from the beginning of your diagonal "ray". This will count all rational numbers.

There is also a known proof that all algebraic numbers are countable (numbers that can be shown to be roots of a polynomial). I don't know this proof, though. So my question really stems from if you can find a polynomial that will cover a transcendental number.

Also, the cardinality of the complex numbers is the maximum of (the cardinality of the real part or the cardinality of the imaginary part), so all we need to ask is to show it for one of them.

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09-19-2007 at 10:10 PM

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Casebier
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (0)

Well, if we're talking about Infinities here...

There's a proof that shows there is no Intelligent Life in the universe (including Earth).
Granted that there are an Infinite number of worlds in the universe, but only a Finite number of worlds capable of sustaining Intelligent Life:

Lim( x->∞ ) [y/x] = 0

Where y = Any finite number

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[Last edited by Casebier at 09-20-2007 06:28 PM]

09-20-2007 at 06:27 PM

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silver
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Re: Finish the Sequence! (+1) (+1) (+2) (+3) (0)

But I don't grant your premise, because if there were an infinite number of worlds, there would be an infinite number of worlds capable of sustaining life. (Actually, I don't believe your premise because I believe there are a finite number of worlds, but I'm on better logical grounds going after the second clause).

Even though we "merely" mean aleph-null (countable infinity):

Let's say that for every X dead worlds there is a living world. No matter how large X is, you can still count to X+1 infinitely many times. Like if X were 1: how big is the set of even integers? Still infinite. No matter how big you make X, it's still infinite.

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09-20-2007 at 08:08 PM

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