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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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