Basis for an alternative method

[shadowslice e]

So, I've been doing some thinking and I stumbled (sort of) on the following idea when trying to incorporate orientation into the stationary point analysis of cubes (and there by solution by finding patterns in those solutions): if you orient a certain number of colours on a cube, you can solve it. If patterns could be found with orientation of each colour, then I believe this could form the basis for a very powerful FMC technique for computers, if not humans.

Let's take an example with a 2x2x2 cube. To use this method, we will consider each face as separate and con be oriented on it's own with no regard to the other pieces. To solve the cube, you only need to orient 3 colours of faces. If you can easily generate algorithms that do this according to some rule, then you should easily be able to find the optimal solution by finding the lowest point at which all of the algorithms for each face are the same in a similar way to finding the lowest common factor (that is, the shortest algorithm at which can orient all the 3 sides)

This may be somewhat difficult for a human to do which is why a way of reducing the space need to be searched would need to be found first. Perhaps the algorithms which orient a particular face could have similar properties of the matrix group they fall into (so a small amount of optimallity (if that is even a word) may need to be sacrificed in order for a human to use the patterns in a useful way.

Of course, the number will increase for each size of cube (up until 4 for 3, 5 for 4 but not beyond that- also, centres could be done through some comms or stuff like that as they don't lend themselves well to orientation. I believe some of chris' work could be helpful in this regard).


I apologise if this has been thought of before but I thought that it was worth sharing just in case it hasn't as i haven't read a post about solving a cube through only orientation before.

I would also appreciate any feedback on the idea and any suggestions as to what could be done or things i could look into to find a way to reduce the cubespace which needs to be searched (pruning tables etc).

I look forward to your critique!

SSE

 

[Lucas Garron]

I don't quite understand what you're saying, but my post from the "20-move method" thread has a few reference you might want to take a look at:

I highly recommend that you read up on Thistlethwaite (and Human Thistlethwaite), Kociemba (also see my explanation), and Jaap's computer cubing page to see if you've thought of anything that isn't already well understood.

Chances are, there is no easy way for a human to find a short combination of orientation and permutation phases without spending a lot of time searching like a computer (or using a ton of lookup tables).

If I understand correctly, you're also looking at different approaches for multi-phase solutions. I suspect it's unlikely that you can find something useful for 3x3x3, but if you know the efficacy and pitfalls of existing approaches maybe you can find something.

 

[shadowslice e]

I don't quite understand what you're saying, but my post from the "20-move method" thread has a few reference you might want to take a look at:



If I understand correctly, you're also looking at different approaches for multi-phase solutions. I suspect it's unlikely that you can find something useful for 3x3x3, but if you know the efficacy and pitfalls of existing approaches maybe you can find something.

Your explanation of the 2-phase algorithm is amazing. I wish that I could write that clearly.


Below is my attempt to reexplain the algorithm in a clearer light. It's sort of like a multiphase/single phase hybrid with some aspects of ID

Spoiler: description of the method

What I'm trying to describe in the OP is sort of like a multiphase method with ID although the phases are done somewhat simulatneously.

For example, on a 2x2x2, if you orient the red face, then the green face without deorienting the red face, then do the white face without destroying the other two (although pieces can be moved around), you will have a solved cube.

However, this would be very inefficient so I propose that algorithms of an arbitrary length be generated for each of the faces (a set to orient the red, one for the green, one for the white) which only orient a certain face without regard to the others.

At some point, there will be an algorithm which is featured in all of those groups, that is an algorithm which lies at the intersection of all the groups (there would be infinitely many of these but we would attempt to find the shortest one). This is the algorithm which will solve the cube yet we have only considered 3 faces which need to be solved (though I admit it will be more difficult for larger cubes- increasing to 5 faces for a 5x5x5 cube thought the centres could be solved like insertions and thus reducing the number of faces to consider and the number of pieces on those faces).

While the above is the core of my proposal, I recognise that for larger cubes it would not provide too much of a benefit as more faces must be oriented before it can be guaranteed that all of the cube is solved. Thus, I propose that if the orientation cases can be grouped by certian abstract features, then the intersection could be found a little more easily as each of the solutions would have similar charicteristics so the solution must satisfy all of them.

I then go on to say this may be more difficult for a human though if these patterns could be analysed and perhaps more simple tratis could be identified, a pseudosolution could reasonably be found by a human.

EDIT: Incidentally,it may be possible to show that the maximum number for any cube is 20 using this method by showing that the cubes have the most sides in different groups or that the cubes contain faces which are the most different in terms of their group properties and hence take the most number of moves to find the intersection and solve.

EDIT 2:

Spoiler: algorithms step by step

Spoiler: I suppose the brute force algorithm for a 2x2x2 would go somewhat like the following:

1) select 3 adjacent faces (in this case we will use red green and white)

2) generate all algorithms which orient all the piece on one side between two numbers. Presumably it would be reasonable to use 4-10 for the 2x2x2*
2a) generate all algorithms between 4-10 moves to orient the green face.
2b) generate all algorithms between 4-10 moves to orient the red face
2c) generate all algorithms between 4-10 moves to orient the white face.*

3) find the shortest algorthm which appears in all 3 sets.

*I suppose that you could do all the algs of length 4, compare, 5 compare etc.

Spoiler: The refined perhaps more mathematically elegant version would be

1) select 3 adjacent faces
2) identify the groups for which the faces fall into.
3) generate the shortest algorithm which transforms all the groups into to solved state.

This could be made human workable by finding more simplified or perhaps larger groups which could be used by a human in a similar way to how HTA has the steps broken down into smaller steps that a human could use.

I believe that the refined version described above could be a very powerful FMC method

 

[shadowslice e]

A further simplification for a human workable version would be to treat the edges and corners separately although this would be worse for computers as it would increase the order of the algorithm. However, if individual groups could be identified and the transformations between the groups could be found then this could be a very good step towards this being a practical method.

Another pro of this method is that it can directly get any cube to any solvable state purely by changing which pieces would need to be oriented.