Presentation for Rubik's cube

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secondmouse

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Mar 4, 2010, 12:02:50 PM3/4/10
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I found the following short presentation for the
miniature 2x2x2 Rubik's cube of order 3674160:

< a,b,c | a^4 = b^4 = c^4 = 1,
ababa = babab,
bcbcb = cbcbc,
abcba = bcbac,
bcacb = cacba,
cabac = abacb,
(ac)^2 (ab)^3 (cb)^2 = 1 >

See the following link for more info as to why
3 generators makes sense in this case:

http://www.jaapsch.net/puzzles/cube2.htm

By adding three more generators a^2, b^2 and c^2
and six extra relators I found another presentation
describing it in terms of the half-turn metric
(the diameter of the Cayley graph on the nine
generators including inverses is known to be 11).

Would this approach (i.e. finding short edge-cycles
of adjacent generators) be fructiferous in tackling the
much harder 3x3x3 case presented on it's usual
generators {L, R, F, B, U, D} - rather than using
semidirect or wreath products which has seemed
to be the case traditionally?

Someone must know more about this given it's
a 30-year old question of Singmaster.

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