## Patterns upon patterns

What is the basis for reason? And mathematics?

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Philosophy Explorer
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### Re: Patterns upon patterns

Arising_uk wrote:
Fri Jul 14, 2017 1:56 am
Philosophy Explorer wrote: Not really because you're adding 22 with the first set of numbers, 45 with the second set and 76 with the last set.

PhilX
Not really, I'm adding one to the first and then producing the rest by adding one to the next. It fits the pattern you gave, i.e. it produces the pattern you gave and it reaches the correct result.
Do you think my puzzle is as trivial as that? Look at it from the viewpoint of a mathematician and not just as a philosopher. You don't have to add any numbers to the group to reach the end result (but it does require, to the best of my knowledge, a new concept that's alien to the internet. By going through this exercise, I'm making you think).

PhilX

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### Re: Patterns upon patterns

vegetariantaxidermy wrote:
Fri Jul 14, 2017 12:51 am
Not everything is on the internet. I mean, please. A rabidly nationalistic, barely literate yank moron, who has never before strung together more than one sentence on here, two at the most, suddenly writes an ENTIRE paragraph in English rather than bastardised-by-Americans English?? Give me a break. It's not hard to tell when someone hasn't written something him/herself.
"...please." Are you plagiarizing Bob Evenson? PhilX

Arising_uk
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### Re: Patterns upon patterns

Philosophy Explorer wrote:Do you think my puzzle is as trivial as that? ...
No idea but it doesn't sound like philosophy of mathematics nor does it appear to do with Logic as given your question my answer is logically correct I'd have thought?
Look at it from the viewpoint of a mathematician and not just as a philosopher. ...
Then post it in the Lounge or on a Mathematical puzzle site.
By going through this exercise, I'm making you think).
Not sure it's thinking but add the first and third numbers and the second and fourth and then multiply them. No idea what concept this is supposed to represent?

Philosophy Explorer
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### Re: Patterns upon patterns

Arising_uk wrote:
Fri Jul 14, 2017 8:35 am
Philosophy Explorer wrote:Do you think my puzzle is as trivial as that? ...
No idea but it doesn't sound like philosophy of mathematics nor does it appear to do with Logic as given your question my answer is logically correct I'd have thought?
Look at it from the viewpoint of a mathematician and not just as a philosopher. ...
Then post it in the Lounge or on a Mathematical puzzle site.
By going through this exercise, I'm making you think).
Not sure it's thinking but add the first and third numbers and the second and fourth and then multiply them. No idea what concept this is supposed to represent?
Nice thinking Arising. Have you also tried multiplying the first and second numbers together, multiplying the second and third numbers together, multiplying the third and fourth numbers together, and multiplying the fourth and first numbers together, and add up the intermediate results to get the final result? (IOW, what you've did and what I've did are algebraically the same).

Now look at the results and rearrange them to come up with a series.

PhilX

vegetariantaxidermy
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### Re: Patterns upon patterns

Philosophy Explorer wrote:
Fri Jul 14, 2017 10:58 am
Have you also tried multiplying the first and second numbers together, multiplying the second and third numbers together, multiplying the third and fourth numbers together, and multiplying the fourth and first numbers together, and add up the intermediate results to get the final result? (IOW, what you've did and what I've did are algebraically the same).

Now look at the results and rearrange them to come up with a series.

PhilX
Blimey. Isn't the point of a puzzle or riddle supposed to be a simple solution that takes you by surprise?

Philosophy Explorer
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### Re: Patterns upon patterns

vegetariantaxidermy wrote:
Fri Jul 14, 2017 8:49 pm
Philosophy Explorer wrote:
Fri Jul 14, 2017 10:58 am
Have you also tried multiplying the first and second numbers together, multiplying the second and third numbers together, multiplying the third and fourth numbers together, and multiplying the fourth and first numbers together, and add up the intermediate results to get the final result? (IOW, what you've did and what I've did are algebraically the same).

Now look at the results and rearrange them to come up with a series.

PhilX
Blimey. Isn't the point of a puzzle or riddle supposed to be a simple solution that takes you by surprise?
It is and Arising with a little prodding got it.

PhilX

vegetariantaxidermy
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### Re: Patterns upon patterns

You didn't say they all had the same pattern.

Philosophy Explorer
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### Re: Patterns upon patterns

vegetariantaxidermy wrote:
Fri Jul 14, 2017 8:56 pm
You didn't say they all had the same pattern.
Which they are you referring to?

PhilX

wtf
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### Re: Patterns upon patterns

Philosophy Explorer wrote:
Fri Jul 14, 2017 10:58 am
Nice thinking Arising. Have you also tried multiplying the first and second numbers together, multiplying the second and third numbers together, multiplying the third and fourth numbers together, and multiplying the fourth and first numbers together, and add up the intermediate results to get the final result? (IOW, what you've did and what I've did are algebraically the same).
Oh you are seeking a mathematical function that maps 4-tuples of natural numbers to naturals? In that case I propose the function f such that

f(1, 2, 3, 4) = 24,

f(2, 3, 4, 5) = 48,

f(3, 4, 5, 6) = 80,

and f(n, m, r, s ) = 0 for any other values.

That's the trouble with vaguely-defined puzzles. They don't test your mathematical ability to solve the problem, they only test your psychological ability to figure out what the questioner is thinking.

Here is another classic example. What is the next number in this sequence: 1, 2, 4, 8, 16?

If you are clever enough to read my mind, then you know that the obvious answer is of course 31. It's the number of distinct regions formed by cutting a circle with n chords.

http://mathworld.wolfram.com/CircleDivi ... hords.html

Londoner
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### Re: Patterns upon patterns

I know nothing of maths but I understood there is never any logical compulsion towards any particular answer. The rule might be that '...until you get to any number above 30, thereafter you multiply by 17', or anything else. This comes in Wittgenstein to do with 'rule following' and 'meaning', as further discussed by Kripke.

That said, I can't help feeling that 'cutting a circle with n chords' trick question was simultaneously annoying and delightful.

(I always used to do very well in IQ tests that had this sort of 'complete the sequence' question, but in my case this ability didn't seem to signal any general mathematical ability.)

Philosophy Explorer
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### Re: Patterns upon patterns

wtf wrote:
Sat Jul 15, 2017 1:50 am
Philosophy Explorer wrote:
Fri Jul 14, 2017 10:58 am
Nice thinking Arising. Have you also tried multiplying the first and second numbers together, multiplying the second and third numbers together, multiplying the third and fourth numbers together, and multiplying the fourth and first numbers together, and add up the intermediate results to get the final result? (IOW, what you've did and what I've did are algebraically the same).
Oh you are seeking a mathematical function that maps 4-tuples of natural numbers to naturals? In that case I propose the function f such that

f(1, 2, 3, 4) = 24,

f(2, 3, 4, 5) = 48,

f(3, 4, 5, 6) = 80,

and f(n, m, r, s ) = 0 for any other values.

That's the trouble with vaguely-defined puzzles. They don't test your mathematical ability to solve the problem, they only test your psychological ability to figure out what the questioner is thinking.

Here is another classic example. What is the next number in this sequence: 1, 2, 4, 8, 16?

If you are clever enough to read my mind, then you know that the obvious answer is of course 31. It's the number of distinct regions formed by cutting a circle with n chords.

http://mathworld.wolfram.com/CircleDivi ... hords.html
Not so wtf.

Look at the title, "Patterns upon patterns." It suggests there may be (and there are) more numbers involved. Let's add 0 1 2 3 to 8 to my list. Now look at the right-hand side (abbreviated RHS). On the RHS, we have the numbers 8, 24, 48, 80... (the three dots mean that particular list of numbers goes on indefinitely). Do you see a particular pattern with those numbers?

I do. In fact that pattern is one of the famous series in math. But before you can see it, factor out the 8. So now we have:

8•1 = 8
8•3 = 24
8•6 = 48
8•10 = 80...

Do you recognize the series 1, 3, 6, 10...? This is the famous triangular series and the process I used which led to those numbers I call cyclic multiplication.

This is merely scratching the surface. I've found out more with cyclic multiplication (e.g. other series) and it could lead towards other areas I haven't considered (it already has).

PhilX

wtf
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### Re: Patterns upon patterns

Philosophy Explorer wrote:
Sat Jul 15, 2017 12:18 pm
Not so wtf.
Not so that my function f satisfies the conditions of the problem? But it does.

Not so that these kinds of puzzles are psychology tests and not math tests? But they are.

Not so that the first few values of the chord sequence are 1, 2, 4, 8, 16, 31? But they are.

Not so that this example is often used to demonstrate the problem with underdetermined puzzles? But it is. The link I gave mentions that.

Can you please be specific as to "Not so" refers to?

You have something in mind but we are not mind readers. You have additional constraints and conditions in your head that you are not telling us about; and then feeling superior because people can't read your mind.

You posed a puzzle but at the end of your last response you alluded to your mysterious research that, once again, you haven't taken the trouble to tell us about. We're all dim if we can't read your mind.

This is what I take from your last response.

Do you understand that the the conditions of the problem exactly as you have written and explained them, are underdetermined? That means there are many many different solutions, infinitely many. Guessing what you have in mind is a problem in psychology, not math.

Philosophy Explorer
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### Re: Patterns upon patterns

wtf wrote:
Sat Jul 15, 2017 10:40 pm
Philosophy Explorer wrote:
Sat Jul 15, 2017 12:18 pm
Not so wtf.
Not so that my function f satisfies the conditions of the problem? But it does.

Not so that these kinds of puzzles are psychology tests and not math tests? But they are.

Not so that the first few values of the chord sequence are 1, 2, 4, 8, 16, 31? But they are.

Not so that this example is often used to demonstrate the problem with underdetermined puzzles? But it is. The link I gave mentions that.

Can you please be specific as to "Not so" refers to?

You have something in mind but we are not mind readers. You have additional constraints and conditions in your head that you are not telling us about; and then feeling superior because people can't read your mind.

You posed a puzzle but at the end of your last response you alluded to your mysterious research that, once again, you haven't taken the trouble to tell us about. We're all dim if we can't read your mind.

This is what I take from your last response.

Do you understand that the the conditions of the problem exactly as you have written and explained them, are underdetermined? That means there are many many different solutions, infinitely many. Guessing what you have in mind is a problem in psychology, not math.
Not so that I wasn't seeking the function you brought up
(Arising has figured it out).

PhilX

Arising_uk
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### Re: Patterns upon patterns

Philosophy Explorer wrote:...
(Arising has figured it out).

PhilX
I didn't really figure anything out as I still have no idea what concept you say you are demonstrated, I gave you two solutions to your puzzle and you didn't like the first but did the second, go figure.

Averroes
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### Re: Patterns upon patterns

PhilX wrote:I do. In fact that pattern is one of the famous series in math. But before you can see it, factor out the 8. So now we have:

8•1 = 8
8•3 = 24
8•6 = 48
8•10 = 80...

Do you recognize the series 1, 3, 6, 10...? This is the famous triangular series and the process I used which led to those numbers I call cyclic multiplication.

This is merely scratching the surface. I've found out more with cyclic multiplication (e.g. other series) and it could lead towards other areas I haven't considered (it already has).
Here is what I think lies underneath the pattern you found.

Lets us take one of the sequence that you have provided, as an example:
1,2,3,4 => 24

Now, if you take the first member of this sequence (i.e. 1) and add it to the third member of the sequence (i.e. 3), you will get 4.
Lets us now take the second member of the sequence (i.e. 2) and add it to the fourth (i.e. 4), we get 6.
If we now multiply those two numbers, we get 24!

If we do this to the other sequences you have provided the answer is as you have provided in the OP.
0,1,2,3 => [(0+2)*(1+3)]= [2*4]= 8.
2,3,4,5 => [(2+4)*(3+5)]= [6*8]= 48.
3,4,5,6 => [(3+5)*(4+6)]= [8*10 = 80.

Now, how does all this relates to the triangular numbers? Well, here is the relation:

In effect, what we have been doing by this addition and multiplication scheme can be given a general representation as follows:

n, (n+1), (n+2), (n+3).

What we have been doing is then this: [n+ (n+2)] * [(n+1)+(n+3)]

The latter formula can be expanded into the quadratic expression 4*(n^2 + 3n + 2) and then factorized as 4*(n+1)(n+2)!

You will recall that (n+1)(n+2)/2 is the formula for evaluating the triangular numbers! (See Wikipedia for a refresher)

So, our formula, can be expressed as 4*2 [(n+1)(n+2)/2] = 8*[(n+1)(n+2)/2], for n ∈ +Z □

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