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 Post subject: puzzle solvability
PostPosted: Tue Jul 22, 2014 7:19 am 
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Hello,

some say there are dozen people that can solve pretty much every twisty puzzle there is, physical or computer simulated. I think with plenty of time it is possible by at least trial and error(s).

I believe everybody sometimes experiences solving situations where there is seemingly no way to continue in solving without algorithms - bandaged puzzles, geared puzzles, jumblers and puzzles that can be folded come to my mind immediately.

I wonder if every twisty puzzle there is can be solved intuitively, i.e. not by trial and error? That includes also by algorithms but only in case one can perfectly understand them (performing an algorithm just because it - as a whole - cycles 3 pieces is not considered as intuitive solving). If there are puzzles which can not be solved intuitively, what makes them unsolvable intuitively? Are there some boundaries which divide the puzzles into intuitively solvable and unsolvable folders?

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 Post subject: Re: puzzle solvability
PostPosted: Tue Jul 22, 2014 10:30 am 
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Context.

bcube wrote:
I think with plenty of time it is possible by at least trial and error(s).
I would not call this solving. The algorithm you use for a solve must perform better than "turn randomly until puzzle is solved". Also, the algorithm used for solving must be repeatable. For example, if you can't solve the last layer of a 3x3x3 a viable algorithm is not to solve the first two layers over and over until the last layer happen to be solved. The same goes for sub components of the last layer like relying on luck for just the edges or corners.

bcube wrote:
I believe everybody sometimes experiences solving situations where there is seemingly no way to continue in solving without algorithms - bandaged puzzles, geared puzzles, jumblers and puzzles that can be folded come to my mind immediately.
These are completely opposite of the situations I had in mind. Bandaging and jumbling especially take away your ability to execute memorized sequences. If a puzzle is sufficiently jumbled or bandaged most sequences will run into a block partway through the sequence which forces you to adapt to the situation.

bcube wrote:
I wonder if every twisty puzzle there is can be solved intuitively, i.e. not by trial and error? That includes also by algorithms but only in case one can perfectly understand them (performing an algorithm just because it - as a whole - cycles 3 pieces is not considered as intuitive solving).
The definition of "intuitive" is too imprecise. Everyone has a different gut feeling for what is or isn't intuitive.

I think when you say "algorithm" you actually mean a "memorized move sequence". A sequence to do something like 3-cycle edges or twist a pair of corners is not an algorithm. The community has been misusing the world "algorithm" for years.

To me an algorithm for solving a puzzle is a recipe / strategy for how to handle each situation on the puzzle to take it from fully scrambled to solved. Part of the algorithm for solving it is the piece solving order, part of it is how to handle special cases like permutation parities and excess twist, and part of it is the memorized move sequences to perform useful tasks.

If you ask me, if you can develop the overall algorithm from scratch, develop all of the useful move sequences (usually commutators + conjugates) and a strategy for handling all of the special cases you're solving the puzzle "intuitively".

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 Post subject: Re: puzzle solvability
PostPosted: Tue Jul 22, 2014 1:22 pm 
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Brandon Enright wrote:
If you ask me, if you can develop the overall algorithm from scratch, develop all of the useful move sequences (usually commutators + conjugates) and a strategy for handling all of the special cases you're solving the puzzle "intuitively".

To me an "intuitive solve" and "development" are incompatible. Intuitive means you are fully aware of every move you are making and understand what each turn is doing. I devised my own method from scratch in 1980. (EDIT: Apart from the moves I describe in my next post) I did it by trying stuff, writing down what it did and then memorising the moves and when to use them. I still have no idea why the moves do what they do or the path of the parts I am moving. Apart from one side I would never say I could solve a cube intuitively. Relying on memory for any move immediately rules out intuition, though that is only my interpretation. Obviously my interpretation means no solve can be 100% intuitive though I am happy to relax my rules when I feel like it.

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Last edited by Tony Fisher on Wed Jul 23, 2014 12:05 pm, edited 1 time in total.

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 Post subject: Re: puzzle solvability
PostPosted: Tue Jul 22, 2014 2:07 pm 
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Tony Fisher wrote:
Intuitive means you are fully aware of every move you are making and understand what each turn is doing.

You've perfectly captured the problem with using the word intuitive for solving.

Every Dictionary Ever wrote:
in·tu·i·tive / inˈt(y)o͞oitiv
adjective

using or based on what one feels to be true even without conscious reasoning; instinctive

Your "fully aware of every move you are making and understand what each turn is doing" implies conscious thought and reasoning about the moves.


Perhaps it isn't clear but when I develop a commutator to perform some useful sequence, I am fully aware of what every move does and why. I don't memorize the move sequences. Instead I memorize the constraints the sequence needs to satisfy and the moves that satisfy those constraints. I don't think in terms of moves but in terms of piece interactions and the moves come naturally from that.

It really bugs me when people suggest I don't solve intuitively when I pre-search for useful commutators and then go on to use a definition of intuitive solving that clearly violates the English-language definition of the word "intuitive".

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 Post subject: Re: puzzle solvability
PostPosted: Wed Jul 23, 2014 1:36 am 
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Brandon Enright wrote:
These are completely opposite of the situations I had in mind.


Ok, allow me to explain it. I think you mean something like: "Ok, I have scrambled BiCube. Lets solve it just like 3x3x3. Oh wait, I can not do this move. Well, lets use algorithms known for 3x3x3. Oh wait, I can not do this move. Ok, I must solve it really intuitively rather than algorithimc way." Well, I am thinking completely opposite of the situation. I have scrambled BiCube and I think like this: "Lets make this move. Oh, I can not. Lets somehow avoid it then in a clever way so that it is possible to do it aftewards. But wait, I can not do it due to this bandaging over here." After short time I feel I am no longer master of my algorithm / solution (because of so many backstreets which lead to inability to trace pieces). I somehow try to not scramble the cube more than it is now and get to the "acceptable position" back again. The effect of that is ussually not the same as intended before starting an algorithm (instead of cycling required pieces I cycle something else). So, even if I can solve BiCube, I don´t consider it as intuitive solution.

Brandon Enright wrote:
The definition of "intuitive" is too imprecise.

Yeah. I like very much what Tony said about it.
Tony Fisher wrote:
Intuitive means you are fully aware of every move you are making and understand what each turn is doing.
I feel it the same way. When I try to solve 3x3x3 and I begin with the cross, I consider as intuitive to solve edge-corner pair and place it correctly. My solution will be neither shortets nor fastest but I can trace those pieces and solve them intuitively (despite of used algorithm).

You see, since I am no longer master of my own algorithm in case of BiCube, I don´t consider it as intuitive solve. Same with Gear Cube Extreme for me. I can solve it, but it was rather trial and error and see what happened afterwards. Recently I solved AI 4x4x4 cube. I was trying to cycle edges. After 72 moves I was like: "Oh man, from edges perspective, it does nothing! What a waste." Then I noticed:"Oh wait, centers cycled. Maybe I can use this to solve them". So it was not on purpose, but I managed to solve the puzzle.

It is like focusing hard on beta radiation (edges) and after some time say "well, this is going nowhere". Then realize "wait a second, even when my accelerating tube was shielded, a film in my pocket blackened. I must have been discovering new kind of radiation (centers)". The discovery was not planned - it was rather side effect / coincidence. If it is side effect, I do not consider it as intuitive solution.

Brandon Enright wrote:
I think when you say "algorithm" you actually mean a "memorized move sequence".


Yes. But not only memorized sequence, also a sequence found somewhere on the internet (i.e. not developed by myself). Since I used to be a speedcuber, I use an algorithm in that sence (memorized move sequence).

Brandon Enright wrote:
A sequence to do something like 3-cycle edges or twist a pair of corners is not an algorithm. The community has been misusing the world "algorithm" for years.


Thank you for the heads up. When 3-cycle is not an algorithm, what is it then? From my perspective, it is intuitive algorithm (when performed intuitively), otherwise just an algorithm. How do you call move sequence nowadays (if not an algorithm)?

Brandon Enright wrote:
To me an algorithm for solving a puzzle is a recipe / strategy for how to handle each situation on the puzzle to take it from fully scrambled to solved. Part of the algorithm for solving it is the piece solving order, part of it is how to handle special cases like permutation parities and excess twist, and part of it is the memorized move sequences to perform useful tasks.

If you ask me, if you can develop the overall algorithm from scratch, develop all of the useful move sequences (usually commutators + conjugates) and a strategy for handling all of the special cases you're solving the puzzle "intuitively".


As I was trying to say before, I developed my own algorithms for both BiCube and Gear Cube Extreme (among others) but I still consider it as coincidence (it was not on purpose - I rather did something and saw what happened than, as Tony said, were fully aware of every move I was making and understand what each turn is doing. ), therefore I don not consider it as intuitive solution.

Brandon Enright wrote:
Perhaps it isn't clear but when I develop a commutator to perform some useful sequence, I am fully aware of what every move does and why. I don't memorize the move sequences. Instead I memorize the constraints the sequence needs to satisfy and the moves that satisfy those constraints. I don't think in terms of moves but in terms of piece interactions and the moves come naturally from that.


I consider that as intuitive solving.

So, now when I explained (at least I hope so) better what is meant by "intuitive", lets get back to the original question, shall we? I wonder if every twisty puzzle there is can be solved intuitively, i.e. not by trial and error? That includes also by algorithms but only in case one can perfectly understand them (performing an algorithm just because it - as a whole - cycles 3 pieces is not considered as intuitive solving). If there are puzzles which can not be solved intuitively, what makes them unsolvable intuitively? Are there some boundaries which divide the puzzles into intuitively solvable and unsolvable folders?

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 Post subject: Re: puzzle solvability
PostPosted: Wed Jul 23, 2014 5:13 am 
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I will never forget a light bulb moment regarding intuitive solving back in 1980. I rarely read about solving so I don't know the technical terms. It suddenly occurred to me that on a Rubik's Cube, if I took a corner out of the top layer and put it back with a different orientation I could then turn the top layer (90, 180 or 270 deg), reverse the original move and I would have done nothing other than change the orientation of two top layer corners. The same logic could then be applied to many puzzles in many different ways and no actual moves would need to be remembered.
When I see kids today blindly learning solutions from books & internet it's a shame that they miss these kind of incredibly satisfying moments and understanding of their puzzles.

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Last edited by Tony Fisher on Wed Jul 23, 2014 12:01 pm, edited 1 time in total.

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 Post subject: Re: puzzle solvability
PostPosted: Wed Jul 23, 2014 8:55 am 
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Brandon Enright wrote:
A sequence to do something like 3-cycle edges or twist a pair of corners is not an algorithm.

It definitely is an algorithm. Just not one for solving the puzzle but for a simpler task.

I dislike our usage of the term, though, as I suspect many cubers don't know the general meaning and many non-cubers who are told we use "algorithms" don't know what it means at all, so it's just confusing them rather than helping.

Tony Fisher wrote:
I rarely read about solving so I don't know the technical terms.

What you describe (doing A, then B, then undoing A, then undoing B) is called a "commutator" (you might now want to reread the previous messages using that term :)).


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 Post subject: Re: puzzle solvability
PostPosted: Wed Jul 23, 2014 2:44 pm 
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I think the use of intuition in this thread is completely misguided. Doing a single turn or a number of turns intuitively does NOT indicate that you know exactly why you are doing those turns. Intuition means doing something subconsciously, not the opposite, Maybe that was indeed the intended meaning, but from waht was said it appears to be the opposite :wink:

Per

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 Post subject: Re: puzzle solvability
PostPosted: Wed Jul 23, 2014 4:36 pm 
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perfredlund wrote:
I think the use of intuition in this thread is completely misguided. Doing a single turn or a number of turns intuitively does NOT indicate that you know exactly why you are doing those turns. Intuition means doing something subconsciously, not the opposite, Maybe that was indeed the intended meaning, but from waht was said it appears to be the opposite :wink:

Per
Exactly! :) There is one post, though, that has pointed this out, already. Please, have a look at the last post from Brandon (from yesterday)

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 Post subject: Re: puzzle solvability
PostPosted: Wed Jul 23, 2014 5:52 pm 
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perfredlund wrote:
I think the use of intuition in this thread is completely misguided. Doing a single turn or a number of turns intuitively does NOT indicate that you know exactly why you are doing those turns. Intuition means doing something subconsciously, not the opposite, Maybe that was indeed the intended meaning, but from waht was said it appears to be the opposite :wink:

Per

Good point. Yes indeed the opposite is meant. Now you have destroyed further use of the word you owe us an alternative!

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 Post subject: Re: puzzle solvability
PostPosted: Wed Jul 23, 2014 7:27 pm 
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Brandon Enright wrote:
Tony Fisher wrote:
Intuitive means you are fully aware of every move you are making and understand what each turn is doing.

You've perfectly captured the problem with using the word intuitive for solving.

Every Dictionary Ever wrote:
in·tu·i·tive / inˈt(y)o͞oitiv
adjective

using or based on what one feels to be true even without conscious reasoning; instinctive

Your "fully aware of every move you are making and understand what each turn is doing" implies conscious thought and reasoning about the moves.


Well, I think it's worth interpreting that definition within a more pragmatic construct. If you go be more modern understanding of how the brain works, and the conscious and unconscious contributions to awareness, then the mind, using both conscious and unconscious modalities, is fully aware, even if not completely conscious of the details. Although we are aware and conscious of everything we're doing, there is unconscious information that is accessed quicker then our conscious mind can keep up with. I call this intuitive. When I solve a side, or place a corner, I can't always explain how I'm doing it, or every detail of each combination, even though I'm aware of it. There are quick moves that are done where calculations and analysis is done unconsciously and require no memorization of algorithms. This is intuitive even though we are very much aware and conscious of what we are doing. If I swing a bat at a ball, I'm aware of what I'm doing, but the fine details are mostly unconscious calculations that I cannot predict or even completely describe. This I would call playing intuitively. Indeed, it can be said that your arm starts swinging at the ball a few milliseconds before you're conciously aware that the swinging has started. If you're thirsty, you will often times move your hands towards a glass of water split seconds before you're consciously aware of the thirst.
I believe that by this more pragmatic definition we can say that there are steps in solving puzzles that are truly intuitive as many early moves are done by allowing our unconcious "muscle memory" to quickly place pieces without much thought, while commutators (by which I mean very simple small sequence algorithms) are used to make limited changes with repetitive motions which resides more in the concious rhelm, but still with complete awareness of the effect of each move, while algorithms (By which I mean more complex then simple "commutator" versions) are more rote memorized complex series of instructions used to cause a known end outcome by a series of movements without having much conscious or unconscious awareness of the details of what's happening with the puzzle at any one moment accept for a memorized sequence of moves.

Example: Pyraminx and the 3x3x1 is all intuition, where just moving the pieces around will eventually get you solved
Dinos and skewb, all commutators (down down up up) where just one sequence of simple repetitive motions with set ups will solve it
Justin's pocket cube: All algorithms as rote instructions without knowing what the heck is going on in the middle of the sequence.


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 Post subject: Re: puzzle solvability
PostPosted: Thu Jul 24, 2014 1:28 am 
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Brandon Enright wrote:
Everyone has a different gut feeling for what is or isn't intuitive.


Once again, you have been right.

In order to get answers to my questions, I guess it is necessary to understand the questions perfectly by respondents. Thus I will try to explain confusing stuff once and for all (well, not really, but once at least).

The word "intuitive" can be understood as either
Every Dictionary Ever wrote wrote:
using or based on what one feels to be true even without conscious reasoning; instinctive


(that is a common use of the word, we can use it in every-day conversation etc. However, this in not what I meant)

or

Tony Fisher wrote:
Intuitive means you are fully aware of every move you are making and understand what each turn is doing.


(this definition is much more related to the puzzles. Even if it is not completely accurate, this is exactly what I meant)


An algorithm is a word used for move sequence (from puzzle-solving point of view). In my opinion they can be divided into two groups: intuitive ones and non-intuitive ones. Commutators and conjugates are meant to be intuitive ones (as well as algorithms in which we are fully aware of every move we are making and understand what each turn is doing), even though I can imagine situations where they are not (but lets not make it more difficult than it is now). Typically first step of solving 3x3x3 is done intuitively. If you use layer by layer method for solving 3x3x3, then you probably do first layer (or part of it) intuitively. If you begin by 2x2x2 block instead, you are probably doing this block intuitively. Same for other sized block or even if you start with corners first - you can first intuitively orient 4 corners and then solve the rest of the corners in one algorithm or more of them). Non-intuitive algorithms, for example, are used when you want to solve third layer of 3x3x3 in just one algorithm (sure, there are exceptions in which you can use only one algorithm for that and it still will be intuitive one, but for most of the cases you will be forced to use one non-intuitive algorithm if you want to solve it in just one algorithm).

I hope it is clear now for everybody. If you are going to respond in this thread, please stick with these "definitions" of "intuitive" and "algorithm", even if you can find them incorrect.

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 Post subject: Re: puzzle solvability
PostPosted: Thu Jul 24, 2014 11:22 am 
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Intuition is when something *feels* right, but you don't necessarily understand why as you haven't followed a completely logical (deductive or inductive) train of conscious thought. Intuition is about joining some dots without logic, although you may have found the dots themselves with logic. Furthermore, you can explain intuitive solves with logic retrospectively, but pure logic is not what got you there in the first place. A computer algorithm always follows a sequence of logical steps, but can never make intuitive "jumps" in the algorithm without specifically being told how and when to do so as part of the algorithm itself (although perhaps quantum computers will have that ability). Thus intuition is based on subconscious pattern recognition and associations rather than sequential steps of inductive/deductive reasoning, which are always conscious. It's right-brain vs left-brain stuff (and yes, I know this is just a figurative model), but essentially we have two different and complementary systems of thinking, as explained by Kahneman et al.

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 Post subject: Re: puzzle solvability
PostPosted: Thu Jul 24, 2014 3:52 pm 
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Tony Fisher wrote:
perfredlund wrote:
I think the use of intuition in this thread is completely misguided. Doing a single turn or a number of turns intuitively does NOT indicate that you know exactly why you are doing those turns. Intuition means doing something subconsciously, not the opposite, Maybe that was indeed the intended meaning, but from waht was said it appears to be the opposite :wink:

Per

Good point. Yes indeed the opposite is meant. Now you have destroyed further use of the word you owe us an alternative!


Would the term analytical solving fit your purpose :?:

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