mrCage
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- Jun 17, 2006
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Any nxnx cube larger than size 3 can be solved entirely without 3x3x3 algs. I would say the 15-puzzle or pyraminx is the 'mother' of all puzzle solving
Per
Per
Is it true?
- Thread starter TrollingHard
- Start date
Christopher Mowla
Premium Member
Certainly, anyone who has learned the 4X4X4 or 5X5X5 and then applied that knowledge to solve larger size cubes is certain that the 4X4X4 is technically the minimum in introducing methods to solve an entire new category of puzzles (although knowing how to solve the 5X5X5 is probably a better representation of larger cube sizes since it has a central slice): ones with inner-layer orbits. This obviously applies to higher order minxs as well. The reason this is true (although obvious to those who solve large puzzles) is due to the fact that the same algorithms used on the puzzles with one inner-layer orbit can be directly translated to puzzles with multiple inner-layer orbits.
Obviously, r2 U2 r2 (Uu)2 r2 u2 cannot be successfully translated to the 5X5X5 (or any other odd cube for that matter) without discoloring the front and back centers a little, but having an X permutation of wing edges between two composite edges of odd cubes can be ignored using select edge pairing techniques.
This is the obvious/common knowledge. I will now present the non-obvious.
There are algorithms which cannot be directly translated to larger cube sizes than, say the 4X4X4 and 5X5X5, or even not applicable to larger cube sizes at all. Though not ideal for speedsolving purposes, these algorithms do exist.
Since we are discussing the minimum representation for NXNXN cubes, the first step is to realize that in this new "universe", even and odd size cubes must be treated differently because the central slice on odd cubes DOES matter now. The 6X6X6 is the minimal representation for even cubes and the 7X7X7 is the minimal representation for odd cubes. The reason is simple: those two cube sizes are the smallest cube sizes which contain more than one inner-layer orbit, namely two.
Since the 6X6X6 and 7X7X7 cubes have two inner-layer orbits, there are three possible cases an algorithm must pass to be officially transferable to them. Although obvious, I am going to mention these three cases anyway, so that I can assign them case numbers for future reference.
Using the 7X7X7 as a model,
Let 0's represent the wing edges in the orbits for which the algorithm is translated to and let X's represent the slices which the algorithms are not be translated to.
Case 1:
X00X00X
Case 2:
X0XXX0X
Case 3:
XX0X0XX
Unlike any big cube algorithms you have ever seen before, there are algorithms that only work for case 1 (treating a 6X6X6 as a 4X4X4 and a 7X7X7 as a 5X5X5, or more generally, turning half of the inner-layer slices on a big cube as if they were one).
There is another category of algorithms which work for all three cases, but only on even cube sizes (4X4X4 and greater). This category contains some algorithms which are rather complicated.
And there is a third category of algorithms which only work on the 4X4X4: they cannot be translated to larger cube sizes (besides matching case 1 on even cube sizes) without scrambling them.
Despite these different cases, it turns out that if an algorithm survives all three cases (for both the 6X6X6 and 7X7X7 cubes), it can be applied to all big cube sizes.
In summary, for a person who has strong determination and patience, the 4X4X4 and the center commutator is sufficient to eventually learn how to solve NXNXN cubes. At the same time, there exists algorithms which do not support the idea of direct tranlation. So truly technically speaking, the 6X6X6 and the 7X7X7 are the minimal sizes to represent all big even and odd cubes, respectively.
So, TrollingHard, you're instinct for the 6X6X6 as the minimum cube size representation was correct (for even cubes, that is), although you didn't know exactly in what way it was.
Obviously, r2 U2 r2 (Uu)2 r2 u2 cannot be successfully translated to the 5X5X5 (or any other odd cube for that matter) without discoloring the front and back centers a little, but having an X permutation of wing edges between two composite edges of odd cubes can be ignored using select edge pairing techniques.
This is the obvious/common knowledge. I will now present the non-obvious.
There are algorithms which cannot be directly translated to larger cube sizes than, say the 4X4X4 and 5X5X5, or even not applicable to larger cube sizes at all. Though not ideal for speedsolving purposes, these algorithms do exist.
Since we are discussing the minimum representation for NXNXN cubes, the first step is to realize that in this new "universe", even and odd size cubes must be treated differently because the central slice on odd cubes DOES matter now. The 6X6X6 is the minimal representation for even cubes and the 7X7X7 is the minimal representation for odd cubes. The reason is simple: those two cube sizes are the smallest cube sizes which contain more than one inner-layer orbit, namely two.
Since the 6X6X6 and 7X7X7 cubes have two inner-layer orbits, there are three possible cases an algorithm must pass to be officially transferable to them. Although obvious, I am going to mention these three cases anyway, so that I can assign them case numbers for future reference.
Using the 7X7X7 as a model,
Let 0's represent the wing edges in the orbits for which the algorithm is translated to and let X's represent the slices which the algorithms are not be translated to.
Case 1:
X00X00X
Case 2:
X0XXX0X
Case 3:
XX0X0XX
Unlike any big cube algorithms you have ever seen before, there are algorithms that only work for case 1 (treating a 6X6X6 as a 4X4X4 and a 7X7X7 as a 5X5X5, or more generally, turning half of the inner-layer slices on a big cube as if they were one).
There is another category of algorithms which work for all three cases, but only on even cube sizes (4X4X4 and greater). This category contains some algorithms which are rather complicated.
And there is a third category of algorithms which only work on the 4X4X4: they cannot be translated to larger cube sizes (besides matching case 1 on even cube sizes) without scrambling them.
Despite these different cases, it turns out that if an algorithm survives all three cases (for both the 6X6X6 and 7X7X7 cubes), it can be applied to all big cube sizes.
In summary, for a person who has strong determination and patience, the 4X4X4 and the center commutator is sufficient to eventually learn how to solve NXNXN cubes. At the same time, there exists algorithms which do not support the idea of direct tranlation. So truly technically speaking, the 6X6X6 and the 7X7X7 are the minimal sizes to represent all big even and odd cubes, respectively.
So, TrollingHard, you're instinct for the 6X6X6 as the minimum cube size representation was correct (for even cubes, that is), although you didn't know exactly in what way it was.
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