The essence of my solution is a new paradigm. A new way to look at the cube.
The way you look at something affects how you think about it, and also what you think you can do to it. If you see it as a cube with six faces that you can turn, you would very soon be frustrated by the wild number of interactions between the things you can do. Each rotation makes changes to the four adjacent faces, and soon you lose track of what is happening.
Alternatively, you may view the cube as three layers, and attempt to solve them one by one. You would have a hard time trying to move pieces around on the second and third layer without destroying the layers that you have already built.
In the theory below, we work on subsets of the cube in manageable bits. We look only at the subset and forget about the rest of the cube, so that the complexity will be contained in the subset. More importantly, we would ask what we want to do in the subset, and why we choose the particular movements to achieve our goals.
Call him wise whose actions, words, and steps are all a clear "because" to a clear "why".
— Johann Kaspar Lavater
From Mathematics, a vector space of dimension N can be spanned by N basis vectors. It has been proven that two algorithms alone are sufficient to span the whole pattern space of the cube (and therefore can bring you to any pattern, including solving the cube). Unfortunately, these two sequences are very very long, and you need to combine them in the correct way to arrive at the desired target pattern.
On the other hand, it is obvious that the six basic face movements can solve the cube. This fact is hardly useful because the face movements themselves affect too many pieces each time. Therefore, they cannot help us to achieve meaningful goals.
Notice that corners always move to corners and edges to edges. You can never move a corner to an edge. Therefore, they are very different objects, and should be treated separately. The minimum basis vectors I have chosen are:
Since the Strategy of Eight Corners settles corners first, corner algorithms can affect edges, but edge algorithms must preserve corners.
These basis algorithms, together with their mirror images and conjugates (see below), are sufficient to solve the cube.
Notice that by looking at the cube as corners and edges, you have to handle only two scenarios: how to work with corners, and how to work with edges. For each scenario, you have only two things to do: move them around, or flip/twist them in place. That is all you need to solve the cube.
Among all algorithms, one is of special interest. An Identity Algorithm is an algorithm that moves cubelets around, but always moves each of them back to where they started from. There are infinitely many Identity Algorithms. A trivial one is
In general, for any algorithm X
Three things are always conserved by any algorithm: total number of twists (modulus 3), total number of flips (modulus 2), and total number of piece exchanges (modulus 2). This is a useful property when you are designing algorithms: you do not have to worry that the cube might end up with one single edge flipped, or two corners twisted in the same direction — these situations will not occur in any configuration that originates from START.
As a consequence, we only have algorithms flipping 2 edges. There doesn't exist any algorithm that flips a single edge. Same applies to twists (you can't twist one single corner, or two corners in the same direction) and movements (you cannot swap two corners without moving edges, or swap two edges without move corners).
Here, the four basic rules are presented. The next three sections (on corner, edge, and center algorithms respectively) can be regarded as worked examples of these rules.
e.g. X {(C1,C2,C3)-, (E1,E2),(E3)+...}
(C1,C2,C3)- means C1 is moved to C2, C2 to C3, and C3 to C1- (twisted -120°). In this form, if we say {(RFU,RBU..),..}, the three faces of the corners have one to one correspondence; that is, the R face of RFU is moved to R, F to B, and U to U.
(E1,E2) means E1 moves to E2, and E2 moves to E1.
(E3)+ means E3 is flipped.
Mathematically, the {} of ()s are viewed as a set {} of groups () with different periods. The period of the first one is 9 (C1 will go back to C1- after 3 cycles, and 3 twists bring it back to C1: therefore period = 3 * 3); the second and the third are of period 2. In other words, if you repeat the first () 9 times, the end result is no change: the three pieces are moved back to the original places on every third iteration, and after all 9 repetitions they are twisted back to the original state as well. As for the second and third (), it means if you repeat it twice, it will have no remaining effects left over.
X² eliminates all groups of period 2. In other words, by using it an even number of times, all groups of period 2 will revert back to "no change". Only groups with an odd period will remain. On the other hand, X9 eliminates the first group (since it has period 9), and only the effects of the two remaining groups are seen. Xn a useful way to isolate required effect groups and preserve other things.
It may be a little surprising that, for any algorithm X, there is always an integer n such that, Xn ≡ I, the identity algorithm, which changes nothing. For the above example, X18 ≡ I.
Similar to the above, it will cancel out most of the changes to give algorithms preserving many pieces. Therefore, it often does interesting things for you.
X⊕Y denotes applying the algorithm X and then followed by the algorithm Y.
If Y is an algorithm that flips UF and UR, and you want to flip UF and FD, what do you do? You can invent any X (without any constraint!) that moves FD to UR but leaves UF in UF. Then apply Y to flip the two edges. Finally, use X-1 to restore UR back to FD. Therefore, an algorithm to flip two specific edges allows you to flip any two edges when conjugated properly.
Commutators cancel out most of the changes. If X and Y are commutative, all changes are cancelled out ‐ their commutator will always be the identity algorithm. Algorithms are usually not commutative and something interesting is often left behind.
It was brought to my attention that Scientific American published a very good article on the Cube. Since it is not available online, you need go to a library to dig up the magazine archives.
Go to Scientific American, March 1981. The magazine cover is a Rubik’s cube. The cover story is in the Metamagical Themes column, by Martin Gardner.
In the column, besides a detailed presentation of the cube using theories not unlike what you find here, there is also a very good explanation of the conservation rules, in particular, why corner twists are always conserved. Most interestingly, the article made an unexpected observation about the cube. Since twists are always conserved, the phenomenom is similar to the conservation laws in particle physics. In particular, quarks have 1/3 charges, and like corner twists, never exist in isolation. Mesons (such as Pions) are particles made from a quark plus an anti-quark, and similarly you can have a cube with two corners twisted in different directions; baryons (such as Neutrons) are particles made from three quarks and also you can have a cube with three corners twisted in the same direction. The analogy stands because they are based on the same number theory.
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