From:

~~~ Subject:

There is an easy lower bound on the number of hypermoves needed to

solve Rubik's Revenge. If we distinguish like-colored face

centers, let us fix the BL center of the D face, permitting only

the shallow moves B1, F1, U1, U3, L1, R1, and their inverses, and

the deep moves F2, U2, R2, and their inverses. Let us compute the

number of hypermoves needed to solve just the face centers of

Rubik's Revenge. A shallow hypermove can achieve SF = 4^6 = 4096

different face center positions. A deep hypermove can achieve

DF = 7! 3^6 = 3674160 different face center positions. So in four

hypermoves, at most

1 + (SF + DF) + 2 SF DF + (SF + DF) SF DF + 2 SF DF SF DF

= 453,021,789,719,303,692,337

face center positions can be achieved. Since this is fewer than

the

23! = 25,852,016,738,884,976,640,000

face center positions of Rubik's Revenge, some face-center

positions will require at least five hypermoves.

If like-colored face centers are not distinguished, the best lower

bound I can find using this method is three hypermoves. If stomach

cubies are considered, I think both bounds increase by one, since

only deep moves can touch them.

It seems strange that this method relies only on the face center

solution. Similar arguments about edges are not as good, because

so many edge positions are achievable using shallow hypermoves.

Corners are practically irrelevant, since they can be fixed using

only shallow hypermoves.

With respect to the question of odd sequences of hypermoves, Jim

Saxe mentions that ``it is plausible that sequences of the form

SDSDS may be sufficient while sequences of the form DSDSD may

not.'' I would like to add the further plausibility that both

types may be sufficient, while neither may suffice alone.