Unscrambling the cube

Dear AlphaNumeric:

You are utterly incorrect about Noodler's puzzle. I believe this topic has become too challenging for your mind and thus fustration has set in. I will advise you to rein in any feeling of envy or jealousy that you may be experiencing at this moment. I hope that we can continue to delight ourselves with this marvellous thread.

LC, Ph.D., Los Alamos National Laboratory.
Los Alamos, New Mexico.

 

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So there are three formulations for n. The parameter n is the number of positions over all the A,B corresponding to conjugations. This is the FTM or face turn metric, which in the pocket version (N=2) has 9 initial positions for n = 1. This is doubled at N=3 because of the extra layer.

Note the formulas have not been stress-tested properly, and the sequence \( s_i \) 'starts' at i = 2.
Backwards induction implies \( s_{-1}, s_{-2}, ..., \). We need to 'stop' at \( s_0 \).
This means deciding that negative s don't exist - you can't turn s 'inside out'.

Actually the only way to do this is virtually - in an optical mirror or a computer model. The 2-cube is a kind of mirror for the full group, because the model includes 'forbidden' reflections in 2-d, but virtually.

p.s. may I remind any bored members of the audience that analytical comments, such as any towards constructive, or deconstructive arguments about the formulations is welcome, and, there is an exit over to my right...

I will also inform detractors that these formulations are not my intellectual property, I am also evaluating them. However they were put together by people who have published papers on such trivia as the shortest path problem, wrt the Rubik group.

 

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I mentioned Enigma - this was certainly enigmatic for its time, without Turing's insights into computable numbers and programming a machine to write a code, things might have been quite different for history.

But that aside, Rubik's original puzzle quickly became enigmatic, it was 'modded' almost immediately and the pocket rocket appeared. There is a k-complete table for this reduced ("edge-less") version, by which I mean the lengths of all algorithms have been computed. The formulations used to do this are not the above, which are specific to N=3 and d=3. I expect the set of formulas are also reduced, so that the group's symmetry is preserved between N=2 and N=3.

(the convention here is that normal text "N" is the number of sections, in the edge basis, the italicised form is recursive, it acts to enumerate all possible positions given n).

Note the symmetry here: there are twice as many slices when N goes from 2 to 3, and the number of sections per face increases geometrically from 4 to 9. The 'deck' goes from two to three layers, and the middle layer has no elements belonging to the N=2 group.

In the 2-cube there are no elements of rotation, but reflection instead, so the colors permute over reflections. A way to model this is to color only the middle slice layer of an original, and keep it oriented so the elements stay in-plane. This is in fact what the computer model of the 2-cube does, pivoting rows or columns of elements one at a time.

Each of the formulas above, given a value for n, will return the same value which is the number of positions n can 'step to' via recursion. If n is 1, that means only one member of the FTM is selected (X), if n is 2, two members are selected (XY), and fully permuted, and so on.

Here, colors can be minimized: the initial color-symmetry has 1 position in the full topograph of G = all faces with equal but separate color. Separate is the operative term, since a pair of opposite faces can be the same color (interference is impossible, opposing pairs commute), this demonstrates that color is also a separating (selective) operator, colors are fixed values, like integers.

The formulas return 'black + white', not color; you need to color the numbers yourself or get a computer to do it, or maybe buy a few billion plastic ones and start really early every day, for a few billion years, see how it goes. Then move up to the 4-dimensional version.

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It looks like a cube (or like an open box if you leave one or more sides uncolored), but it's really a device for encoding stuff. The contents are still under analysis.

 

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You see, it occured to me (and this has I see occured to others) that the "digital" aspects of such devices are obvious, and so is the connection to the domain of capacitance and inductance - the volume of information or the entropy in any 'word' s_k has a 'group velocity' which is the position it occupies in the s(n) topograph. At MIT they called the positions cubicles and the individual elements cubies.

The cubicles are equivalent to a capacitive component (the capacitance is color), together as a symmetric object which is imaginary since the sides of real plastic elements are what forms them, When the color is "none" the capacitance is zero, but not the capacity.

You can change the last, by simply coloring part (any part at all) of the device, which immediately orients it in 3-space, now, you can identify the marked element and any others that were marked as well (you store an initial pattern). There are n places that this can be in of which only some will have information entropy, when the initial segment is "expanded" to full capacitance/volume level.

Likewise, using pairs of colors the capacitance is increased digitally. There is a spectrality \( s(0),\; s(1),\; ..., \;s(k),\; ..., \;s(n) \), and so some kind of theory: that the spectrum of fixed color has more spectrality in the sliced-up cube. Also the cube group is only a subgroup of the full group of Platonic solids, so far only a handful have higher-dimensional models.

This is part of the 'spectral theory' of the full group. It requires a vector space which is normal(ly computable) H, an algebra (of coloring) A, and a set of derivative and integral forms, which three form a triple or tuple. Einstein's puzzle and the listing are an example of a Euclidean problem that involves coloring and orientation. I guess he thought spectrally.

 

So, the space of sliced cube-puzzles:

This is the same as the space of passive components in a circuit. The black plastic is passive because i) a cube is not self-assembling, and ii) black is conventionally the absence of color.

You can assemble a disassembled cube blindfolded which proves that any dynamics the colors lend the object are "forgettable". Blind assembly yields either: a black cube with no colors, a colored cube which is extended to the full permutation group, or a randomly colored cube (with up to N^3 colors). In short you can't tell without looking for color or its absence, anything about color.

So with a black structure, you have the entirely passive form, which is structural, and you have "equations of motion" that tell you the object is closed under rotations.

Colors are not passive, even when restricted to a narrow "visible" range, and having a low intensity. Color is a complex function of frequency and the microstructure of a surface, which can be specular, or "shiny", etc, so that a perceived color depends on various things. The sky is a different shade of blue depending how close the part of the sky is to the horizon, etc.

Adding color alters the structure, because now there's a way to track motions - the colors "record" these for you, by creating coordinates. If you color only one facet you can see it move around, but it helps to have another color-mark as a reference, that stays stuck. As soon as a pair of colored elements exists you have a permutation group.

There is a general swap function in the structure: you swap black for color, you swap elements in the structure by swapping their positions and/or orientations. Characterizing this swap function should say a lot more about the general switching character of the structure, and its computational connections.

 

Painfully I read this thread... Ouch! yet still entirely unsure as to the point of it..

There is much freely available code to solve cubes.

But this got my attention.

then the formula????

What??? There are many ways to determine if a number is machine computable , but what has this to do with the cube???

The Enigma is not a computing device.

What is a recursive number?? Never heard of it personally, are you referring to Irrational numbers??

When you post a formula please have the curtsy to at least define the variables specifically towards the problem at hand.

 

Blindman, noodler is the new account of someone who got banned for just posting nonsense on topics he knew nothing about. The new account just seems to have taken this up a notch. He doesn't even bother with being coherent any more, just posting snippets of his thoughts after reading unrelated Wikipedia pages. Every time he's confronted he just gets aggressive and refuses to back up any of the claims he makes of understanding this.

I wonder what the point is, he's not going to make anyone say "OMG, you're a genius!" with his incoherent nonsense and he's certainly not actually addressing the original point of the thread, to solve a Rubik's cube.

 

The title of THIS thread has little to do with Alphanumeric's subject (the one he hasn't managed to figure out yet).
For which I apologise profusely: I should have just listed a solution for his benefit.
After all he's the only person here worth upsetting.


For the rest: when you read the title, did you immediately conclude that you would find a cube-solving method? Are you aware that there are thousands of these all over the Web? Why would I repeat this, and what's wrong with your jump to an obviously false conclusion?

Please only presume that this thread has no benefit to you, or anyone else, and we can all get some sleep, ok?

Unless you happen to be good at designing algorithms and determining recurrences, I'm not really interested.


Now, has anyone looked at the Wiki entry for the Pocket cube, and have they worked out what the recurrence is?

If you've never seen this, it looks like this:

Code:

n 	f 	q
0 	1 	1
1 	9 	6
2 	54 	27
3 	321 	120
4 	1847 	534
5 	9992 	2256
6 	50136 	8969
7 	227536 	33058
8 	870072 	114149
9 	1887748 	360508
10 	623800 	930588
11 	2644 	1350852
12 	0 	782536
13 	0 	90280
14 	0 	276

For some unfathomable reason, both sequences add to the same total? What could this mean?

I've made the observation that there are 12 terms in the f-sequence, this is the same number (I also observed) as the number of edges in a cube.

M_12 is a subgroup of the 3x3x3, so bIg deal, who cares? Not me for starters, nor does the 3x3x3 I own.

You see, the ignorant look at one of these and see a plastic cube-shaped object with colored stickers on it. The object itself remains what it is, despite what the ignorant and neurotic have to say about it.

I really don't care if you don't understand what they are, I only care if I do, ok? Your problems are all yours...

 

Holy shit, I just realised, if the Enigma devices aren't computers, how did the Germans use them to encode messages?

How do you rewrite a message in code, without using a computer? If the Enigma machines used by the Germans in WW2 don't qualify as computers, then what IS a computer, or what is a machine that rewrites numbers? Did Alan Turing have the wrong idea about this, and in that case, how come the Brits decoded the German intelligence? What did they use?

This revelation, that encoding isn't computing, could be really important... can't think who to, but it must be pretty important - or it could also be completely wrong. Wonder which it is?

 

Well being a computer programmer for the past 28 years i think I know what im on about.

Can you at last explain how "determining recurrences" relates to anything in this thread. The only time I have used recurrence is in generating random numbers and some nice pictures.

Come now noodler you need to be consistent.

Thanks AlphaNumeric for the heads up..

 

They are rather simple mechanical devices. It lacks two important components, an instruction set (program), and conditional branching.

So if i change the keys on a type writer does that make it a computer, it can encode but it far from a Turing complete machine which from memory was only first created in 1941 by the Germans.

 

Sure, it's nice to think you know what you're talking about, lots of people do.

It was really your statement about Enigma, that had me thinking you have no idea what a computer or computation actually is. A lot of people don't, including I would guess, Alphanumeric, who no doubt is happy believing they know how to write a program, so that's got that covered...

If you can't see what recurrence has to do with Rubik's cube, all I can think of suggesting is that you have another look at algorithm design and inductive logic.
The cube is recursive, this is kind of vividly obvious. When you put one together from a kitset, that's recursive too.

Can you explain how you've been programming for 28 years, and have a question about how recursion is related to the Rubik's cube??

Maybe Alphanumeric knows. He';s got a brain the size of a planet after all.

 

"unfathomable". LOL Your surprised that the total number of combinations for quarter turns and full turns are the same?

I can use sarcasm as well noodler.

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How so? Are you talking about computing or mathematics. The two are very different.

 

.At the moment, I'm not surprised that you've focussed on a sarcastic statement and tried to imply that I actually meant it. This is a common tactic from people who are at a loss.

 

Ok show me the most simple recursive algorithm you can come up with. Any language.

 

How is the cube puzzle, and the rest of the group of puzzles recursive?
What is the difference between computing and mathematics, do you think you know and can you say what it is?

Perhaps use contradiction. Assume the cube is not recursive and show this is contradictory. Or assume it is recursive and show the same thing. One or the other must be false, right?

Example of recursion:

\( a_n\; =\; a_{n-1}\; +\; a_{n-2} \)

 

Sorry i mixed up computing recursive with mathematics recurrence theorems. Guess ive been reading your posts to long...

But please do explain how a cube solution is recursive and not sequential. As all solutions I have seen are sequential.

 

Ok, I can do that.

A program (based on an algorithm) iterates through an input dataset and computes results stepwise, there is no actual recursion. An algorithm is coded in some language so that it gets written as a recursive formula, but in fact a program, being a sequence of steps, is never recursive. Loops are sequential, but return to the same conditional test - the test is made sequentially. So any loop could be rewritten as a sequence, because loops are iterated.

There's a sound logical reason that programmers don't write programs in sequential steps, but use loops and recursive formulas.

If you want me to demonstrate the cube's algorithms include loops and conditional branching, can do.

 

Please im trying to understand but what does it mean with the n below the alpha???