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# Just Shut Up and Think

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Use the full power of your rational mind to answer, as best as possible, the following open-ended problems:

1) Predict the next five numbers in each sequence and justify your reasoning:

a) 0,1,2,3,4,5,6,7,8,9,10,11,....

b) 0,1,3,7,15,31,63,127,...

c) 0,1,-1,3,-5,11,-21,...

d) 0,0,1,2,1,-2,-3,2,9,6,-11,...

e) 0,0,1/3,1/3,2/15,7/90,73/630,...

2) Do the same as above except come up with a different answer and justify your reasoning

4) Why?

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1)

a) These are just the non-negative integers in order.  The next five are 12, 13, 14, 15, 16.

b) These numbers are one less than consecutive powers of 2.  The next five are 255, 511, 1023, 2047, 4095.

c)  After initializing with the first two numbers, each number can be obtained by subtracting the previous number from twice the number before that.  The next five are 43, -85, 171, -341, 683.  Another rule that gives the exact same answer forever is, after initializing with the first number,  each number is 1 minus twice the previous number.

d)  I don't see an elegant answer here so far, although I presume there is one.  I notice that the ninth number is 9 and the eleventh number is -11, but I don't know if this is helpful or just a coincidence.

e)  I'm not sure about an elegant answer for this one.  If we treat these numbers as reduced forms of 0/1, 0/2, 2/6, 8/24, 16/120, 56/720, 584/5040, ..., then the denominators are successive factorials, but I don't see offhand where the numerators are coming from.  If we look at ratios of consecutive nonzero numerators, we get 4, 2, 3.5, 10 3/7, which looks a little strange.  Are these the first several coefficients of the Taylor series of a reasonably simple function?

2)

a)  To get a number, add up the previous 11 numbers and divide by 5.  The next five are 13.2, 15.64, 18.368, 21.4416, 24.92992.  If we extend the sequence backwards so that it follows this rule, the number before the 0 is 5, and the number before that is 4.

For the rest I will give a strategy that is conceptually lazy but computationally not so lazy, and for now I will be too lazy to carry out the computations.

b) Fit a polynomial of degree at most 7.

c) Fit a polynomial of degree at most 6.

d) Fit a polynomial of degree at most 10.

e) Fit a polynomial of degree at most 6.

To answer 3) and 4), we must choose a criterion for deciding which is "better".  The first two approaches that come to my mind are

i)  Use elegance, which would give us 1) as being better than 2).

ii)  Demand some context before answering.

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Well, I have a standard answer to all "open ended problems" (because that way I don't have to read them first), but, presumably, it's just as good as any of the other answers, right? That's what "open ended" means? So:

1) the answer is potato, to all of them.

2) the "different" answer is still potato...but a different one.

3 and 4) the first potato, because I don't like different potatoes. I like all my potatoes the same.

Edited by Nicky
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@Doug Morris

I just looked up "genius" in the dictionary, and it said "see Doug Morris"

Your second answers are not conceptually lazy at all. You are now on your way to finding a general solution. But why fit polynomials in particular? Is that the only class of functions you can try to fit to the given data? Suppose that 1, -1, 1, -1, 1, -1, 1, -1, ... had been one of the sequences. Any polynomial fit here would result in the terms tending towards (+/-) inf at some point, yet that doesn't seem "elegant". That being said, is any class of functions equally as good a space to search through as any other? Why might one or the other be better?

Also, how would you define "elegance"?

@Nicky

"Open-ended" means that I will never give you the correct answer although there is one. And just because a problem is "open-ended" does not mean that some answers are not better than others. That being said, at least you tried, but I'm not at all convinced that you gave the problem 100% effort.

Edited by SpookyKitty
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Spooky Kitty,

Thank you for the compliment.

For your alternating sequence my 1) answer would be (-1) to the power of n + 1 or a sinusoid.  I could still use a polynomial for 2).

For c) and d) we could experiment with a sinusoid times an exponential plus something simple and see what that gets us.

Polynomials are a conceptually simple general solution, but we could try all sorts of functions.  One option would be to say that the given finite sequence just repeats.

Providing a context might help us decide which is better.

Elegance, like beauty, is at least to some extent in the eye of the beholder, and I'm not sure how to give a good definition of either off the top of my head.  But elegance is partly about simplicity.

Edited by Doug Morris
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20 hours ago, Nicky said:

1) the answer is potato, to all of them.

2) the "different" answer is still potato...but a different one.

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8 hours ago, Doug Morris said:

Polynomials are a conceptually simple general solution, but we could try all sorts of functions.  One option would be to say that the given finite sequence just repeats.

Why not try every function?

Can you attempt a definition of elegance?

Edited by SpookyKitty
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It would literally take forever to try every function.

I have a variety of things to spend my time on.  Attempting a definition of elegance is not at the top of the list now.  But it is a worthy activity and maybe I'll come back to it later.  I would certainly welcome a good attempt from anyone else.

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Elegance can mean all kinds of things, but the common meanings tend to range between two polar opposites: practical simplicity and pretentious sophistication.

So you can tell a lot about a person from their definition of elegance.

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On 6/9/2018 at 11:30 AM, Nicky said:

Well, I have a standard answer to all "open ended problems" (because that way I don't have to read them first), but, presumably, it's just as good as any of the other answers, right? That's what "open ended" means?

An open ended problem has many possible answers, and usually has multiple acceptable answers, but that doesn't mean that any answer is just as good as any of the other answers.  There can be good answers, bad answers, and silly answers, and within each group there are likely to be variations of degree and quality.

For example, consider the open ended question "What do you intend your situation to be in five years?"  Which answer has what quality would depend a lot on the person giving it.

One kind of answer would have the form "I intend to make progress XYZ in my career and/or my relationships and/or my fitness and health and/or the breadth of my interests."  This is basically a good approach to an answer.  Answers of this form would vary in quality, but some might be equally good, and what answer was how good would depend a lot on the person doing the answering.

For some people, it would be a good answer to say "I will have beaten this addiction that is plaguing me, I will have been clean and sober for at least four years, I will have repaired, restored, or replaced much of what I have lost due to this addiction, and I will have made a good start on building further."

"I will have extensively explored the pleasures of cocaine and heroin."

"I will have gotten extensive and gratifying revenge on that ^&*%^#\$ who rejected me."

"Potato."

"i will be worthy of a unicorn."

"I will have literally physically visited Gondor, Rohan, Rivendell, and the hobbits' Shire.  I will be staying in one of those places, consolidating the learning and growth I have achieved in these travels."

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9 hours ago, Doug Morris said:

An open ended problem has many possible answers, and usually has multiple acceptable answers﻿, but that doesn't mean that any answer is just as good as any of the other answers.﻿  ﻿There can be good answers, bad answers, and silly answers, and within each group there are likely to be variations of degree and quality. ﻿

I bet you can't prove that my answer isn't of the highest degree of acceptability, goodness, quality, or any other word you wish to use.

9 hours ago, Doug Morris said:

﻿ For example, consider the open ended question "What do you intend your situation to be in five years?" ﻿﻿ ﻿Which answer has what quality would depend a lot on the person giving it.

So the question "Is the pen in your hand blue?" is also an open ended question, because it also depends on who's answering it?

Quote

One kind of answer would have the form "I intend to make progress XYZ in my career and/or my relationships and/or my fitness and health and/or the breadth of my interests."  This is basically a good approach to an answer.

Sure. That's because you asked a meaningful question. The quality of a person's answer to your question can be evaluated objectively...because there is an objective standard for evaluating people's goals and decisions. That standard is Ethics. Specifically, rational selfishness. That's what makes the answer "I want to be the next Mother Theresa", for instance, an objectively bad answer.

On the other hand, I have not been made aware of any standard you might use to evaluate my answer above, and objectively call it worse than yours. So it's not worse than yours. You can't even pick on "potato" not being a number, because we're using a digital medium, so it is.

Edited by Nicky
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14 hours ago, Nicky said:

Elegance can mean all kinds of things, but the common meanings tend to range between two polar opposites: practical simplicity and pretentious sophistication.

@Nicky and @Doug Morris

Is it possible to measure the "simplicity" of a provided answer?

Also, just drop this side argument about what an open-ended question is and isn't. It's just fucking stupid and not even remotely productive.

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17 hours ago, Doug Morris said:

It would literally take forever to try every function.

Suppose you had an infinite amount of time on your hands. How exactly would you go about searching through every possible function? How would you know when you've found the best one possible? What do you imagine distinguishes it from all the rest?

Edited by SpookyKitty
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46 minutes ago, SpookyKitty said:

Also, just drop this side argument about what an open-ended question is and isn't. It's just fucking stupid and not even remotely productive. ﻿﻿﻿﻿

Productive? Only thing you're producing in this thread is nonsense, crazy lady. How am I gonna make you any less productive with a side argument?

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19 minutes ago, Nicky said:

Productive? Only thing you're producing in this thread is nonsense, crazy lady. How am I gonna make you any less productive with a side argument? ﻿

What are you even hoping to achieve by insulting me? Read the thread title and apply yourself to the given problem.

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10 hours ago, SpookyKitty said:

Suppose you had an infinite amount of time on your hands. How exactly would you go about searching through every possible function? How would you know when you've found the best one possible? What do you imagine distinguishes it from all the rest?

If I was willing to undertake a search of that kind, my initial inclination would be, rather than trying to search through every possible function, to search through every possible algorithm.  If I take Church's thesis as a working assumption, the possible algorithms are recursively enumerable, so this could be a mechanical process.  I would have to devise a system of time allocation that would allow for lack of knowledge about which computations terminate.

Before trying to answer your other two questions, we should first ask, can we be certain there is a best one possible?  Could it be that no matter how good one of them is, there's another one that's better?

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To avoid recursion in c), express each number as a sum of consecutive powers of -2, starting with an empty sum.

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Or to also avoid a summation, express it as -2 to a power, minus 1, divided by -3.

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On 6/11/2018 at 9:32 PM, SpookyKitty said:

Is it possible to measure the "simplicity" of a provided answer?

Simplicity refers to a count of things: the more things, the less simple. If you can analyze the structure of an "answer" in terms of concepts and propositions introduced in the answer, you can count. In the other sense of "possible" (i.e. "can you actually do it"), no, you probably can't: it requires reducing a sentence or sequence of sentences to concepts and propositions.

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Is counting all that's involved?  What, exactly, do you count?  Are all concepts and propositions equal for this purpose?  What about relationships among the concepts and propositions?

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Bearing in mind the cognitive role of propositions and concepts, an answer is relative to a knowledge context. Also bear in mind that simplicity is a comparative measure. Let’s start with a developmentally-early context, where a person doesn’t have the concept “polygon” but they do have the concepts “hexagon” and “octagon” (also no doubt “triangle” and “square” – but none of the other gons). If my answer is “These are all plastic hexagons”, we rely on at least two concepts, and a relationship between them (the proposition states that relationship, i.e. “those which are both plastic and hexagon”). Every answer asserts a relationship between the proposition and existents. We can count that as 4, and can nit-pick later over the technology of relating sentences to propositions.

In that context, I could also answer that “These are all plastic polygons”, which requires me to further introduce the concept “polygon”. At this point, I confess that I haven’t analyzed the conceptual structure of triangle, octagon etc. so I don’t know exactly what conceptual stuff has to be created to create the concept polygon. The point is that my alternative answer is, contextually, more complicated since it relies on a new concept. But we are not permabonded to the perceptual level. Once fully integrated into an epistemology, “polygon” is on a par with “hexagon”.

I find that the main impediment to people’s understanding the nature of “simplicity” is the tendency to drop cognitive context. If you have never thought about binary numbers, certain numeric and logical relations are not available to you, so an answer that relies on such concepts and propositions is more complex than one that doesn’t. If you (as proponent of an answer) incorrectly presuppose that such concepts are freely available to everyone, you have dropped context.

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DavidOdden

I would like to see your attempts to solve the above problems. Since you seem to know more about concept formation than anyone else, a conceptual breakdown of the process you used to tackle the problems would be very valuable.

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On 6/9/2018 at 9:49 PM, SpookyKitty said:

But why fit polynomials in particular? Is that the only class of functions you can try to fit to the given data? Suppose that 1, -1, 1, -1, 1, -1, 1, -1, ... had been one of the sequences. Any polynomial fit here would result in the terms tending towards (+/-) inf at some point, yet that doesn't seem "elegant".

A polynomial fit, as used here, is a method for interpolation. It cannot be used to predict numbers outside the data range. For that, you need to know beforehand what "law" the numbers are following.

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• 1 month later...

I just noticed that in d), if you take residue classes modulo 4, you get 0, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1.  That looks like a pretty strong pattern, but I'm not sure where to go from there.

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On 6/9/2018 at 1:22 AM, SpookyKitty said:

Use the full power of your rational mind to answer, as best as possible, the following open-ended problems:

1) Predict the next five numbers in each sequence and justify your reasoning:

b) 0,1,3,7,15,31,63,127,...

One reasonable response to this is to dismiss the request, and my justification for doing that would be “this isn’t a serious information question”, “you’re just playing mind games”, or something like that. The first thing that needs justifying is responding at all. That means, I have to find some benefit to myself in giving this a moment’s thought.

For me, the justification could reside an effect on the OP, or on “the rest of the world”, or some combination of the two. I know what I would want to say to the rest of the world, and it is not crucial to me whether the OP cares about / accepts my answer. A response by me would be justified, for me, just in case there is a reasonable chance that I could lay bare some fundamental epistemological and moral issues (you can see that I’m already onto that latter topic). I conclude that this is a teachable moment, which is sufficient moral justification.

My first answer is 14, 97, 32, 21. The assumed function maps from the integers {1…13} to {0,1,3,7,15,31,63,127,14,97,32,21,74}. There are uncountably many similar solutions. My second answer is 0,-1,-3,-7,-15. I assume the initial state is 8-bit binary 10000000, the operation is a version of shift-left where the low end bit is set to the opposite of the high end bit (in the input to shift). The result is interpreted as one’s complement (conventionally, +0 and -0 are not distinguished).

The request to justify my reasoning is a red herring, and a nice distractor. Both answers are extensionally correct (as are some other possibilities such as 2n-1), and “justification” doesn’t enter into the computation of correctness. However, I might want to justify chosing one solution over the other. You can only do that if you have a purpose in mind: therefore, I have to articulate a purpose (as should the OP). Now I can reveal an assumption that I entertained (did not firmly commit to, but decided was more likely true than not), namely that the OP wanted there to be some general rule which yields these number sequences. My purpose behind the first answer was to reject that assumption (which I suspect was made by the OP). Answer 1 creates an opportunity to remind the rest of the world to check their assumptions and not buy a pig in a poke. If you specifically want a rule-based answer, that needs to be part of the question (request).

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