The entire cube is 2-gen!

[cuBerBruce]

I recalled a post on the Domain of the Cube forum where someone had an example of using GAP to determine an expression for a puzzle state in terms of a set of generators. So I thought I would use GAP to try generating U,F,R,L and D with respect to generators UF and RBL to see what would be produced. The spoiler below contains the result. I edited the GAP output to a somewhat more readable format. I hope I didn't mess anything up in the editing. I haven't verified the expressions.

Let a = UF and b = RBL to interpret the result.

Spoiler

U = ba'b'abab'a'bab'a'5ba4b'aba'b'a'4ba5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a2b
a'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'abab'aba'b'a'bab'a'bab'a'ba2b'a5ba2
b'aba2b'a5bab'a'ba10b'aba'b'a'ba'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3b
a2b'a'4ba'b'abab'a'2ba2b'abab'a2ba'b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'a
b'a'b'2a'2b'a'2b'2a'b'aba'b'a'10bab'a'8bab'aba'3b'a'2ba'6b'a2bab'a'3ba'b'
a2ba'3b'a2ba3b'a'2ba'b'a'6ba'b'aba'2b'abab'2abab'2a'2b'a'2b'a'2b'2a'bab'
a'bab'a'bab'a'ba2b'a5ba'b'a'7ba5b'a2bab'a'ba'4b'a4bab'a4bab'a4ba'5b'
a'7ba5b'a7ba'3b'a'5ba3b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3b'aba2b'a3ba'b'aba
b'a'ba2b'aba'b'a'ba'b'a2ba'b'a'3ba2b'a'4ba'b'a'ba'b'a'ba'2b'a'3ba'2b'
a'ba3b'a'2ba'b'2a'ba'b'2a'ba'b4a'b'a'b'4a'ba'b'abab'a'bab'a'5ba4b'aba'
b'a'4ba5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a2ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'
b'abab'aba'b'a'bab'a'bab'a'ba2b'a5ba2b'aba2b'a5bab'a'ba10b'aba'b'
a'ba'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3ba2b'a'4ba'b'abab'a'2ba2b'abab'
a2ba'b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'ab'a'b'2a'2b'a'2b'2a'b'aba'b'
a'10bab'a'8bab'aba'3b'a'2ba'6b'a2bab'a'3ba'b'a2ba'3b'a2ba3b'a'2ba'b'a'6ba'
b'aba'2b'abab'2abab'2a'2b'a'2b'a'2b'2a'bab'a'bab'a'bab'a'ba2b'a5ba'b'
a'7ba5b'a2bab'a'ba'4b'a4bab'a4bab'a4ba'5b'a'7ba5b'a7ba'3b'a'5ba3b'aba'5b'a5b
a'5b'a4bab'aba'2b'aba3b'aba2b'a3ba'b'abab'a'ba2b'aba'b'a'ba'b'a2ba'
b'a'3ba2b'a'4ba'b'a'ba'b'a'ba'2b'a'3ba'2b'a'ba3b'a'2ba'b'2a'ba'b'2a'b
a'b4a'b'a'b'4a'ba'b'abab'a'bab'a'5ba4b'aba'b'a'4ba5b'a'2bab'a'bab'a'5b
a'b'a'8ba'5b'a5ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'a'2bab'a'5ba'2b'aba'
b'aba'b'aba'b'a'3ba5b'a'ba'b'a2ba4b'a5ba6b'a'2bab'a'bab'aba5b'a4ba'b'
a7ba'5b'a9ba'5b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3b'aba2b'a'ba'b'a'2ba6b'
a2ba3b'a'ba'b'a3ba'b'aba2b'a'bab'aba4b'a'5ba'5b'a9ba'b'a'2ba'b'a'b
a'2b'a2b'a5b3ab'ab'ab'2a'b'a'b'a'b'3aba'b'a'bab'a'bab'a'ba2b'a5ba2
b'aba2b'a2ba'b'a'2ba'3b'a5ba3b'a'3bab'a'4ba'5b'a6bab'a'bab'a'bab'a'ba2
b'a5ba'2b'a'3ba5b'a'bab'a'4ba'b'a2ba2b'a'2bab'a'3ba'b'a2ba'b'a'4ba'b'aba
b'a'b'a'2b3ab'a'b'3a'b'a'b'a'b'a'2b'a'3b'abab'a'bab'a'5ba4b'aba'b'a'4b
a5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a2ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'aba
b'aba'b'a'bab'a'bab'a'ba2b'a5ba2b'aba2b'a5bab'a'ba10b'aba'b'a'ba'
b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3ba2b'a'4ba'b'abab'a'2ba2b'abab'a2ba'
b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'ab'a'b'2a'2b'a'2b'2a'b'aba'b'a'10bab'
a'8bab'aba'3b'a'2ba'6b'a2bab'a'3ba'b'a2ba'3b'a2ba3b'a'2ba'b'a'6ba'b'aba'2
b'abab'2abab'2a'2b'a'2b'a'2b'2a'bab'a'bab'a'bab'a'ba2b'a5ba'b'a'7ba5b'
a2bab'a'ba'4b'a4bab'a4bab'a4ba'5b'a'7ba5b'a7ba'3b'a'5ba3b'aba'5b'a5ba'5b'
a4bab'aba'2b'aba3b'aba2b'a3ba'b'abab'a'ba2b'aba'b'a'ba'b'a2ba'b'a'3b
a2b'a'4ba'b'a'ba'b'a'ba'2b'a'3ba'2b'a'ba3b'a'2ba'b'2a'ba'b'2a'ba'b4a'b'
a'b'4a'5ba5b'a8bab'a5ba'b'aba'b'a2ba'5b'a4bab'a'ba'4b'a5ba'b'aba'b'
a'bab'a2bab'a'ba3b'a'5ba2b'aba'b'a'2ba5b'a'4ba4b'a5bab'a4ba5b'a4bab'
a6ba5b'a4ba'b'a7ba'5b'a9ba'5b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3b'aba2b'a'b
a'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'aba2b'a'bab'aba4b'a'5ba'5b'a9ba'b'a'2b
a'b'a'ba'2b'a2b'a5b3ab'ab'ab'2a'b'a'b'a'b'2a'b'a2bab'a'ba3b'a'5ba2
b'aba'b'a'2ba5b'a'4ba4b'a5bab'a4ba5b'a4bab'a4ba3b'a'5ba2b'aba'b'a'6ba5
b'a2bab'aba'b'a'bab'a'bab'a'ba2b'a5ba2b'aba2b'a2ba'b'a'2ba'3b'a5ba3b'
a'3bab'a'4ba'5b'a9ba'b'a'10bab'a'7ba'b'a'3ba5b'a'2bab'a2ba'b'aba2b'a'ba
b'aba4b'a'4ba'b'a'6ba'b'aba2b'a'ba'b'a'ba'2b'aba'b'a'

F = ba'b'a'ba3b'a'bab'a'bab'a2ba4b'a'3bab'a'bab'a'ba2b'a'5ba'2b'a'3ba5b'
a'2bab'a'7ba5b'a4ba'b'a3ba'3b'a'5ba3b'a2bab'a'2bab'aba'b'a'bab'a'2ba3b'a'3b
a2b'a'ba5b'a'2bab'a'5ba'b'a'4ba'5b'a'4ba'b'a'5ba'4b'a4ba'5b'a2bab'a'ba'2
b'a5ba'3b'aba'b'a'2bab2ababab2a'ba'ba'b'3a'5ba'2ba2b'abab'a2bab'a'9ba5b'
a5ba'4b'a'ba'b'aba'2b'a'bab'a'3bab'aba'3b'a'2ba'6b'a2bab'aba'2b'a'ba'3
b'a'ba2b'a'ba'b'a'4ba5b'a'5ba5b'a'ba5b'a'9ba5b'a'7bab'a'4ba'5b'a'6ba'b'
a'4ba'5b'a'4ba'b'a'5ba'4b'a4ba'5b'a2bab'a'ba'2b'a5ba'3b'aba'b'a'2ba'b'aba
b'a'bab'a'5ba4b'aba'b'a'4ba5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a5b4abab'4a
b'ab2ab'ab2ab'a2ba'3b'aba2b'a3ba2b'abab'abab'a4ba'2b'a3bab'a'2bab'aba
b'a'ba'2b'aba'b'a'bab'a'3ba'2b'a'ba'3b'a'ba2b'a'ba'b'a'4ba5b'a'5ba5b'
a'ba'3b'a5ba3b'a'7ba'5b'a7ba5b'a'4ba'b'a'4ba'b'a'4ba4b'aba'b'a'2ba'5b'a7ba
b'a'5ba'2b'aba'b'aba'b'aba'b'ab2a2ba2ba2b2a'b'a'b2a'b'a'ba2b'a'bab'
a6bab'a2ba'3b'a'2ba3b'a'2bab'a3ba'b'a'2ba6b'a2ba3b'a'ba'b'a8ba'b'a10bab'
a'bab2a2ba2b2aba'ba'b'2a'b'aba'bab'a2ba'2b'a'bab'a'bab'a'2ba'b'a'ba'2b'
a2ba'b'a'bab'a4ba'2b'a3bab'a'3bab'aba'3b'a'2ba'6b'a2bab'abab'a'ba'10b'aba'
b'a'5ba'2b'a'ba'2b'a'5ba'2b'aba'b'aba'b'abab'a'ba'b'a'bab'a'ba3b'a'5b
a2b'aba'b'a'6ba5b'a'2ba5b'a8bab'a5ba'b'aba'b'a2ba'5b'a4bab'a'ba'4b'a5b
a'b'aba'b'a'ba3ba2bababab3aba'b'3a2baba'b'a'bab'a4bab'a'2bab'a3ba'b'
a2ba'2b'a'2bab'a4ba'b'aba'5b'a3ba2b'a'5ba'2b'aba'b'aba'b'aba'b'a'6ba5b'
a4ba'b'a3ba'3b'a'5ba3b'a2bab'a'2ba'2b'a'ba'2b'a'5ba'2b'aba'b'aba'b'abab'a'
b3ababab2a'ba'ba'b'3a'5ba'2ba2b'abab'a2bab'a'9ba5b'a5ba'4b'a'ba'b'aba'2b'
a'bab'a'3bab'aba'3b'a'2ba'6b'a2bab'aba'2b'a'ba'3b'a'ba2b'a'ba'b'a'4ba5b'
a'5ba5b'a'ba5b'a'9ba5b'a'7bab'a'4ba'5b'a'ba'b'aba'b'a2ba'6b'a'5ba'4b'a'2ba
b'aba'5b'a3bab'a'bab'a'bab'a'ba2b'a5ba'b'a2bab'a'ba3b'a'5ba2b'aba'b'
a'6ba5b'a'5ba5b'a8bab'a5ba'b'aba'b'a2ba'5b'a4bab'a'ba'4b'a5ba'b'aba'b'
a'bab'ab4abab'4ab'ab2ab'ab2ab'a2ba'3b'aba2b'a3ba2b'abab'abab'a4ba'2b'
a3bab'a'2bab'abab'a'ba'2b'aba'b'a'bab'a'3ba'2b'a'ba'3b'a'ba2b'a'ba'b'
a'4ba5b'a'5ba5b'a'ba'3b'a5ba3b'a'7ba'5b'a7ba5b'a'4ba'b'a'4ba'b'a'4ba4b'ab
a'b'a'2ba'5b'a7bab'a'5ba'2b'aba'b'aba'b'aba'b'ab2a2ba2ba2b2a'b'a'b2a'
b'a'ba2b'a'bab'a6bab'a2ba'3b'a'2ba3b'a'2bab'a3ba'b'a'2ba6b'a2ba3b'a'ba'
b'a8ba'b'a10bab'a'bab2a2ba2b2aba'ba'b'2a'b'aba'bab'a2ba'2b'a'bab'a'bab'
a'2ba'b'a'ba'2b'a2ba'b'a'bab'a4ba'2b'a3bab'a'3bab'aba'3b'a'2ba'6b'a2ba
b'abab'a'ba'10b'aba'b'a'5ba'2b'a'ba'2b'a'5ba'2b'aba'b'aba'b'abab'a'ba'b'
a'bab'a'ba3b'a'5ba2b'aba'b'a'6ba5b'a'2ba5b'a8bab'a5ba'b'aba'b'a2ba'5b'
a4bab'a'ba'4b'a5ba'b'aba'b'a'bab'ab4abab'4ab'ab2ab'ab2ab'a2ba'3b'aba2b'
a3ba2b'abab'abab'a4ba'2b'a3bab'a'2bab'abab'a'ba'2b'aba'b'a'bab'a'3ba'2b'
a'ba'3b'a'ba2b'a'ba'b'a'4ba5b'a'5ba5b'a'ba'3b'a5ba3b'a'7ba'5b'a7ba5b'a'4b
a'b'a'4ba'b'a'4ba4b'aba'b'a'2ba'5b'a7bab'a'5ba'2b'aba'b'aba'b'aba'b'ab2
a2ba2ba2b2a'b'a'b2a'b'a'ba2b'a'bab'a6bab'a2ba'3b'a'2ba3b'a'2bab'a3ba'
b'a'2ba6b'a2ba3b'a'ba'b'a8ba'b'a10bab'a'bab2a2ba2b2aba'ba'b'2a'b'aba'ba
b'a2ba'2b'a'bab'a'bab'a'2ba'b'a'ba'2b'a2ba'b'a'bab'a4ba'2b'a3bab'a'3ba
b'aba'3b'a'2ba'6b'a2bab'abab'a'ba'10b'aba'b'a'5ba'2b'a'ba'2b'a'5ba'2b'ab
a'b'aba'b'abab'a'ba'b'a'bab'a'ba3b'a'5ba2b'aba'b'a'6ba5b'a'2ba5b'a8ba
b'a5ba'b'aba'b'a2ba'5b'a4bab'a'ba'4b'a5ba'b'aba'b'a'bab'

R = ba'b'abab'a'bab'a'5ba4b'aba'b'a'4ba5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a2b
a'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'abab'aba'b'a'bab'a'bab'a'ba2b'a5ba2
b'aba2b'a5bab'a'ba10b'aba'b'a'ba'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3b
a2b'a'4ba'b'abab'a'2ba2b'abab'a2ba'b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'a
b'a'b'2a'2b'a'2b'2a'b'aba'b'a'10bab'a'8bab'aba'3b'a'2ba'6b'a2bab'a'3ba'b'
a2ba'3b'a2ba3b'a'2ba'b'a'6ba'b'aba'2b'abab'2abab'2a'2b'a'2b'a'2b'2a'bab'
a'bab'a'bab'a'ba2b'a5ba'b'a'7ba5b'a2bab'a'ba'4b'a4bab'a4bab'a4ba'5b'
a'7ba5b'a7ba'3b'a'5ba3b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3b'aba2b'a3ba'b'aba
b'a'ba2b'aba'b'a'ba'b'a2ba'b'a'3ba2b'a'4ba'b'a'ba'b'a'ba'2b'a'3ba'2b'
a'ba3b'a'2ba'b'2a'ba'b'2a'ba'b4a'b'a'b'4a'ba'b'abab'a'bab'a'5ba4b'aba'
b'a'4ba5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a2ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'
b'abab'aba'b'a'bab'a'bab'a'ba2b'a5ba2b'aba2b'a5bab'a'ba10b'aba'b'
a'ba'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3ba2b'a'4ba'b'abab'a'2ba2b'abab'
a2ba'b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'ab'a'b'2a'2b'a'2b'2a'b'aba'b'
a'10bab'a'8bab'aba'3b'a'2ba'6b'a2bab'a'3ba'b'a2ba'3b'a2ba3b'a'2ba'b'a'6ba'
b'aba'2b'abab'2abab'2a'2b'a'2b'a'2b'2a'bab'a'bab'a'bab'a'ba2b'a5ba'b'
a'7ba5b'a2bab'a'ba'4b'a4bab'a4bab'a4ba'5b'a'7ba5b'a7ba'3b'a'5ba3b'aba'5b'a5b
a'5b'a4bab'aba'2b'aba3b'aba2b'a3ba'b'abab'a'ba2b'aba'b'a'ba'b'a2ba'
b'a'3ba2b'a'4ba'b'a'ba'b'a'ba'2b'a'3ba'2b'a'ba3b'a'2ba'b'2a'ba'b'2a'b
a'b4a'b'a'b'4a'ba'b'abab'a'bab'a'5ba4b'aba'b'a'4ba5b'a'2bab'a'bab'a'5b
a'b'a'8ba'5b'a5ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'a'2bab'a'5ba'2b'aba'
b'aba'b'aba'b'a'3ba5b'a'ba'b'a2ba4b'a5ba6b'a'2bab'a'bab'aba5b'a4ba'b'
a7ba'5b'a9ba'5b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3b'aba2b'a'ba'b'a'2ba6b'
a2ba3b'a'ba'b'a3ba'b'aba2b'a'bab'aba4b'a'5ba'5b'a9ba'b'a'2ba'b'a'b
a'2b'a2b'a5b3ab'ab'ab'2a'b'a'b'a'b'3aba'b'a'bab'a'bab'a'ba2b'a5ba2
b'aba2b'a2ba'b'a'2ba'3b'a5ba3b'a'3bab'a'4ba'5b'a6bab'a'bab'a'bab'a'ba2
b'a5ba'2b'a'3ba5b'a'bab'a'4ba'b'a2ba2b'a'2bab'a'3ba'b'a2ba'b'a'4ba'b'aba
b'a'b'a'2b3ab'a'b'3a'b'a'b'a'b'a'2b'a'3b'abab'a'bab'a'5ba4b'aba'b'a'4b
a5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a2ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'aba
b'aba'b'a'bab'a'bab'a'ba2b'a5ba2b'aba2b'a5bab'a'ba10b'aba'b'a'ba'
b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3ba2b'a'4ba'b'abab'a'2ba2b'abab'a2ba'
b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'ab'a'b'2a'2b'a'2b'2a'b'aba'b'a'10bab'
a'8bab'aba'3b'a'2ba'6b'a2bab'a'3ba'b'a2ba'3b'a2ba3b'a'2ba'b'a'6ba'b'aba'2
b'abab'2abab'2a'2b'a'2b'a'2b'2a'bab'a'bab'a'bab'a'ba2b'a5ba'b'a'7ba5b'
a2bab'a'ba'4b'a4bab'a4bab'a4ba'5b'a'7ba5b'a7ba'3b'a'5ba3b'aba'5b'a5ba'5b'
a4bab'aba'2b'aba3b'aba2b'a3ba'b'abab'a'ba2b'aba'b'a'ba'b'a2ba'b'a'3b
a2b'a'4ba'b'a'ba'b'a'ba'2b'a'3ba'2b'a'ba3b'a'2ba'b'2a'ba'b'2a'ba'b4a'b'
a'b'4abab'a'ba3b'a'5ba2b'aba'b'a'2ba5b'a'4ba4b'a5bab'a4ba5b'a4bab'a6ba5b'
a4ba'b'a7ba'5b'a9ba'5b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3b'aba2b'a'ba'b'a'2b
a6b'a2ba3b'a'ba'b'a3ba'b'aba2b'a'bab'aba4b'a'5ba'5b'a9ba'b'a'2ba'b'
a'ba'2b'a2b'a5b3ab'ab'ab'2a'b'a'b'a'b'2a'b'a'4ba5b'a'5ba5b'abab'a'ba3
b'a'5ba2b'aba'b'a'ba'5b'a2ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'aba'b'aba'
b'aba2b'a'5ba3b'a'bab'a'bab'a'7ba5b'a4ba'b'a'2ba'3b'a5ba3b'a'3bab'a'4ba'5
b'a9ba'b'a'2ba'b'a'3ba5b'a'2bab'a2bab'aba'b'abab'a'ba2b'aba'b'a'ba'b'
a2ba'b'aba2b'a'bab'aba4b'a'3ba'b'a3ba'4b'a'3bab'a3ba'b'a'ba'b'a2ba'
b'aba'b'a'ba'b'a'4b'a'bab2ab'ab'a'b'2a'2b'a'2b'3a'ba'b'2abab'2a'2b'a'2b'a'2
b'a'2b'aba'5b'a4ba'b'a2ba'5b'a3bab'a2ba'b'a'3ba2b'a'4ba'2b'a3ba3b'a'3bab'
a3ba'2b'a'b'a'bab'a'b3ab'a'b'

L = b'ab'a'ba'5b2ab'aba'3b'a4ba'3b'a'2bab'a3ba'b'a2ba'b'a3ba'4b'abab'a'ba'2
b'aba'b'a'bab'a'4bab'a'5ba'b'a'4ba'5b'a'3ba2b'a'10ba'2b'a'2ba2ba2ba2ba2b2a'
b'a'b2ab'ab3a2ba2b2aba'ba'b'2a'b'aba4bab'abab'a'bab'a'2bab'abab'a'3ba'
b'a3ba4b'a'3bab'a3ba'4b'a'ba'b'aba'2b'a'bab'a'2bab'abab'a'ba'2b'aba'b'
a'bab'a'ba'b'a'2ba'b'a2ba'5b'a3bab'a2bab'a'9ba5b'a4ba'b'a3ba'3b'a'5ba3b'
a2bab'a'4ba'5b'a7ba'b'aba'b'aba'3b'a5ba'2b'a'bab'a'ba2b2ababab2a'ba'ba'
b'3a'5ba'2ba2b'abab'a2bab'a'9ba5b'a5ba'4b'a'ba'b'aba'2b'a'bab'a'3bab'aba'3
b'a'2ba'6b'a2bab'aba'2b'a'ba'3b'a'ba2b'a'ba'b'a'4ba5b'a'5ba5b'a'ba5b'a'9b
a5b'a'7bab'a'4ba'5b'a'ba'b'aba'b'abab'a'ba'b'aba'b'a'4ba6b'aba'b'a'7ba'2
b'aba'b'aba'b'aba'b'a2ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'a'6ba5b'a8bab'
a5ba'b'aba'b'a2ba'5b'a4bab'a'ba'4b'a5ba'b'aba'b'a'bab'ab4abab'4ab'ab2a
b'ab2ab'a2ba'3b'aba2b'a3ba2b'abab'abab'a4ba'2b'a3bab'a'2bab'abab'a'b
a'2b'aba'b'a'bab'a'3ba'2b'a'ba'3b'a'ba2b'a'ba'b'a'4ba5b'a'5ba5b'a'ba'3
b'a5ba3b'a'7ba'5b'a7ba5b'a'4ba'b'a'4ba'b'a'4ba4b'aba'b'a'2ba'5b'a7bab'a'5b
a'2b'aba'b'aba'b'aba'b'ab2a2ba2ba2b2a'b'a'b2a'b'a'ba2b'a'bab'a6bab'a2b
a'3b'a'2ba3b'a'2bab'a3ba'b'a'2ba6b'a2ba3b'a'ba'b'a8ba'b'a10bab'a'bab2a2b
a2b2aba'ba'b'2a'b'aba'bab'a2ba'2b'a'bab'a'bab'a'2ba'b'a'ba'2b'a2ba'b'
a'bab'a4ba'2b'a3bab'a'3bab'aba'3b'a'2ba'6b'a2bab'abab'a'ba'10b'aba'b'a'5ba'2
b'a'ba'2b'a'5ba'2b'aba'b'aba'b'abab'a'ba'b'a'bab'a'ba3b'a'5ba2b'aba'b'
a'6ba5b'a'2ba5b'a8bab'a5ba'b'aba'b'a2ba'5b'a4bab'a'ba'4b'a5ba'b'aba'b'
a'ba6b'a'6bab'a'bab'a'5ba'b'a'8ba'2b'a5ba'3b'a'ba'b'a'ba'5b'a7ba5b'a'4ba'5
b'a4

D = ba'b'abab'a'bab'a'5ba4b'aba'b'a'4ba5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a2b
a'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'abab'aba'b'a'bab'a'bab'a'ba2b'a5ba2
b'aba2b'a5bab'a'ba10b'aba'b'a'ba'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3b
a2b'a'4ba'b'abab'a'2ba2b'abab'a2ba'b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'a
b'a'b'2a'2b'a'2b'2a'b'aba'b'a'10bab'a'8bab'aba'3b'a'2ba'6b'a2bab'a'3ba'b'
a2ba'3b'a2ba3b'a'2ba'b'a'6ba'b'aba'2b'abab'2abab'2a'2b'a'2b'a'2b'2a'bab'
a'bab'a'bab'a'ba2b'a5ba'b'a'7ba5b'a2bab'a'ba'4b'a4bab'a4bab'a4ba'5b'
a'7ba5b'a7ba'3b'a'5ba3b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3b'aba2b'a3ba'b'aba
b'a'ba2b'aba'b'a'ba'b'a2ba'b'a'3ba2b'a'4ba'b'a'ba'b'a'ba'2b'a'3ba'2b'
a'ba3b'a'2ba'b'2a'ba'b'2a'ba'b4a'b'a'b'4a'ba'b'abab'a'bab'a'5ba4b'aba'
b'a'4ba5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a2ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'
b'abab'aba'b'a'bab'a'bab'a'ba2b'a5ba2b'aba2b'a5bab'a'ba10b'aba'b'
a'ba'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3ba2b'a'4ba'b'abab'a'2ba2b'abab'
a2ba'b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'ab'a'b'2a'2b'a'2b'2a'b'aba'b'
a'10bab'a'8bab'aba'3b'a'2ba'6b'a2bab'a'3ba'b'a2ba'3b'a2ba3b'a'2ba'b'a'6ba'
b'aba'2b'abab'2abab'2a'2b'a'2b'a'2b'2a'bab'a'bab'a'bab'a'ba2b'a5ba'b'
a'7ba5b'a2bab'a'ba'4b'a4bab'a4bab'a4ba'5b'a'7ba5b'a7ba'3b'a'5ba3b'aba'5b'a5b
a'5b'a4bab'aba'2b'aba3b'aba2b'a3ba'b'abab'a'ba2b'aba'b'a'ba'b'a2ba'
b'a'3ba2b'a'4ba'b'a'ba'b'a'ba'2b'a'3ba'2b'a'ba3b'a'2ba'b'2a'ba'b'2a'b
a'b4a'b'a'b'4a'ba'b'abab'a'bab'a'5ba4b'aba'b'a'4ba5b'a'2bab'a'bab'a'5b
a'b'a'13ba5b'abab'a'ba3b'a'5ba2b'aba'b'a'ba'5b'a4bab'a'5ba'2b'aba'b'ab
a'b'aba'b'a'3ba5b'a'ba'b'a2ba4b'a5bab'a3ba2b'aba'2b'abab'a'ba'2b'aba
b'a'2ba'b'a'7ba5b'a'2ba5b'aba'b'a'2ba'3b'a5ba3b'a'3bab'a'4ba'5b'a6bab'a'b
a2b'a5ba'b'a'bab'a'4ba5b'a'2ba2b'a5ba'2b'aba'b'aba'b'a'ba'2b'abab'a'7ba
b'a'ba'b'a'ba'3b'a'4ba5b'a8bab'a5ba'b'aba'b'a2ba'5b'a4bab'a'ba'4b'a5b
a'b'aba'b'a'bab'a2bab'a'ba3b'a'5ba2b'aba'b'a'2ba5b'a'4ba4b'a5bab'a4ba5
b'a4bab'a6ba5b'a4ba'b'a7ba'5b'a9ba'5b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3
b'aba2b'a'ba'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'aba2b'a'bab'aba4b'a'5ba'5b'
a9ba'b'a'2ba'b'a'ba'2b'a2b'a5b3ab'ab'ab'2a'b'a'b'a'b'2a'b'a'4ba5b'a'5b
a5b'abab'a'ba3b'a'5ba2b'aba'b'a'ba'5b'a4bab'a'5ba'2b'aba'b'aba'b'aba'b'
a'3ba5b'a'ba'b'a2ba4b'a5ba6b'a'2bab'a'bab'aba5b'a4ba'b'a7ba'5b'a9ba'5
b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3b'aba2b'a'ba'b'a'2ba6b'a2ba3b'a'ba'
b'a3ba'b'aba2b'a'bab'aba4b'a'5ba'5b'a9ba'b'a'2ba'b'a'ba'2b'a2b'a5b3ab'a
b'ab'2a'b'a'b'a'b'2a'b'a'bab'aba5b'a4ba'b'a7ba'5b'a9ba'5b'a'4ba5b'a4ba
b'a'ba10b'aba'b'a'ba'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3ba2b'a'4ba'
b'abab'a'2ba2b'abab'a2ba'b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'ab'a'b'2a'2
b'a'2b'2a'b'a'2bab'a'bab'a'ba2b'a5ba'b'a'ba2b'a4ba'2b'a3bab'a'4ba'5b'
a7bab'a'2ba'b'a6bab'a'2ba'b'2a'ba'b2a'b'ab'a'b'2a'b3a'b'5a'2b

 

[rokicki]

Wow

Very nice! I wonder if Lucas would care to actually try one of those
out by hand, and verify it?

Now all we need is a generic optimal solver in *any* metric (that takes the generators to use) and we can see what these "solutions" might look like
in a shortest form.

They'd be pretty long, I suspect, even in optimal form.

 

[qqwref]

Someone should make a simulator that lets the player solve the cube using these two generators (and their inverses). It would be entertaining to see if anyone can do it.

 

[Lucas Garron]

Very nice! I wonder if Lucas would care to actually try one of those
out by hand, and verify it?

Now all we need is a generic optimal solver in *any* metric (that takes the generators to use) and we can see what these "solutions" might look like
in a shortest form.

1) Have you heard of ksolve?

2) Did you read the part of my post where I estimate alg length? I don't think the optimal solution would be that bad.

By hand, this would take a while:

Spoiler

U turn:
R B L F' U' L' B' R' U F R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F U F L' B' R' U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' F' U' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F U F R B L U F L' B' R' F' U' R B L F' U' F' U' L' B' R' U F U F U F U F U F R B L F' U' F' U' F' U' L' B' R' U F R B L F' U' L' B' R' U F R B L U F L' B' R' U F R B L F' U' L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F U F L' B' R' U F U F U F U F U F R B L U F U F L' B' R' U F R B L U F U F L' B' R' U F U F U F U F U F R B L U F L' B' R' F' U' R B L U F U F U F U F U F U F U F U F U F U F L' B' R' U F R B L F' U' L' B' R' F' U' R B L F' U' L' B' R' F' U' F' U' R B L U F U F U F U F U F U F L' B' R' U F U F R B L U F U F U F L' B' R' F' U' R B L F' U' L' B' R' U F U F U F R B L F' U' L' B' R' F' U' F' U' F' U' R B L U F U F L' B' R' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' U F R B L U F L' B' R' F' U' F' U' R B L U F U F L' B' R' U F R B L U F L' B' R' U F U F R B L F' U' L' B' R' U F R B L F' U' L' B' R' U F R B L U F U F L' B' R' F' U' F' U' R B L F' U' L' B' R' U F L' B' R' F' U' R B L U F R B L R B L U F L' B' R' U F L' B' R' F' U' L' B' R' L' B' R' F' U' F' U' L' B' R' F' U' F' U' L' B' R' L' B' R' F' U' L' B' R' U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F L' B' R' U F R B L F' U' F' U' F' U' L' B' R' F' U' F' U' R B L F' U' F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F R B L U F L' B' R' F' U' F' U' F' U' R B L F' U' L' B' R' U F U F R B L F' U' F' U' F' U' L' B' R' U F U F R B L U F U F U F L' B' R' F' U' F' U' R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' U F R B L F' U' F' U' L' B' R' U F R B L U F L' B' R' L' B' R' U F R B L U F L' B' R' L' B' R' F' U' F' U' L' B' R' F' U' F' U' L' B' R' F' U' F' U' L' B' R' L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F U F L' B' R' U F U F U F U F U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' U F U F R B L U F L' B' R' F' U' R B L F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F R B L U F L' B' R' U F U F U F U F R B L U F L' B' R' U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' U F U F U F U F U F U F U F R B L F' U' F' U' F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F L' B' R' U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F R B L U F L' B' R' U F R B L F' U' F' U' L' B' R' U F R B L U F U F U F L' B' R' U F R B L U F U F L' B' R' U F U F U F R B L F' U' L' B' R' U F R B L U F L' B' R' F' U' R B L U F U F L' B' R' U F R B L F' U' L' B' R' F' U' R B L F' U' L' B' R' U F U F R B L F' U' L' B' R' F' U' F' U' F' U' R B L U F U F L' B' R' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' F' U' R B L F' U' L' B' R' F' U' R B L F' U' F' U' L' B' R' F' U' F' U' F' U' R B L F' U' F' U' L' B' R' F' U' R B L U F U F U F L' B' R' F' U' F' U' R B L F' U' L' B' R' L' B' R' F' U' R B L F' U' L' B' R' L' B' R' F' U' R B L F' U' R B L R B L R B L R B L F' U' L' B' R' F' U' L' B' R' L' B' R' L' B' R' L' B' R' F' U' R B L F' U' L' B' R' U F R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F U F L' B' R' U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' F' U' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F U F R B L U F L' B' R' F' U' R B L F' U' F' U' L' B' R' U F U F U F U F U F R B L F' U' F' U' F' U' L' B' R' U F R B L F' U' L' B' R' U F R B L U F L' B' R' U F R B L F' U' L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F U F L' B' R' U F U F U F U F U F R B L U F U F L' B' R' U F R B L U F U F L' B' R' U F U F U F U F U F R B L U F L' B' R' F' U' R B L U F U F U F U F U F U F U F U F U F U F L' B' R' U F R B L F' U' L' B' R' F' U' R B L F' U' L' B' R' F' U' F' U' R B L U F U F U F U F U F U F L' B' R' U F U F R B L U F U F U F L' B' R' F' U' R B L F' U' L' B' R' U F U F U F R B L F' U' L' B' R' F' U' F' U' F' U' R B L U F U F L' B' R' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' U F R B L U F L' B' R' F' U' F' U' R B L U F U F L' B' R' U F R B L U F L' B' R' U F U F R B L F' U' L' B' R' U F R B L F' U' L' B' R' U F R B L U F U F L' B' R' F' U' F' U' R B L F' U' L' B' R' U F L' B' R' F' U' R B L U F R B L R B L U F L' B' R' U F L' B' R' F' U' L' B' R' L' B' R' F' U' F' U' L' B' R' F' U' F' U' L' B' R' L' B' R' F' U' L' B' R' U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F L' B' R' U F R B L F' U' F' U' F' U' L' B' R' F' U' F' U' R B L F' U' F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F R B L U F L' B' R' F' U' F' U' F' U' R B L F' U' L' B' R' U F U F R B L F' U' F' U' F' U' L' B' R' U F U F R B L U F U F U F L' B' R' F' U' F' U' R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' U F R B L F' U' F' U' L' B' R' U F R B L U F L' B' R' L' B' R' U F R B L U F L' B' R' L' B' R' F' U' F' U' L' B' R' F' U' F' U' L' B' R' F' U' F' U' L' B' R' L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F U F L' B' R' U F U F U F U F U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' U F U F R B L U F L' B' R' F' U' R B L F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F R B L U F L' B' R' U F U F U F U F R B L U F L' B' R' U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' U F U F U F U F U F U F U F R B L F' U' F' U' F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F L' B' R' U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F R B L U F L' B' R' U F R B L F' U' F' U' L' B' R' U F R B L U F U F U F L' B' R' U F R B L U F U F L' B' R' U F U F U F R B L F' U' L' B' R' U F R B L U F L' B' R' F' U' R B L U F U F L' B' R' U F R B L F' U' L' B' R' F' U' R B L F' U' L' B' R' U F U F R B L F' U' L' B' R' F' U' F' U' F' U' R B L U F U F L' B' R' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' F' U' R B L F' U' L' B' R' F' U' R B L F' U' F' U' L' B' R' F' U' F' U' F' U' R B L F' U' F' U' L' B' R' F' U' R B L U F U F U F L' B' R' F' U' F' U' R B L F' U' L' B' R' L' B' R' F' U' R B L F' U' L' B' R' L' B' R' F' U' R B L F' U' R B L R B L R B L R B L F' U' L' B' R' F' U' L' B' R' L' B' R' L' B' R' L' B' R' F' U' R B L F' U' L' B' R' U F R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F U F L' B' R' U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' F' U' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F U F R B L U F L' B' R' F' U' R B L F' U' F' U' L' B' R' U F U F U F U F U F R B L F' U' F' U' F' U' L' B' R' U F R B L F' U' L' B' R' F' U' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L F' U' F' U' L' B' R' U F R B L F' U' L' B' R' U F R B L F' U' L' B' R' U F R B L F' U' L' B' R' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' F' U' R B L F' U' L' B' R' U F U F R B L U F U F U F U F L' B' R' U F U F U F U F U F R B L U F U F U F U F U F U F L' B' R' F' U' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' U F R B L U F U F U F U F U F L' B' R' U F U F U F U F R B L F' U' L' B' R' U F U F U F U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F R B L U F L' B' R' U F R B L F' U' F' U' L' B' R' U F R B L U F U F U F L' B' R' U F R B L U F U F L' B' R' F' U' R B L F' U' L' B' R' F' U' F' U' R B L U F U F U F U F U F U F L' B' R' U F U F R B L U F U F U F L' B' R' F' U' R B L F' U' L' B' R' U F U F U F R B L F' U' L' B' R' U F R B L U F U F L' B' R' F' U' R B L U F L' B' R' U F R B L U F U F U F U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F U F U F U F U F R B L F' U' L' B' R' F' U' F' U' R B L F' U' L' B' R' F' U' R B L F' U' F' U' L' B' R' U F U F L' B' R' U F U F U F U F U F R B L R B L R B L U F L' B' R' U F L' B' R' U F L' B' R' L' B' R' F' U' L' B' R' F' U' L' B' R' F' U' L' B' R' L' B' R' L' B' R' U F R B L F' U' L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F U F L' B' R' U F U F U F U F U F R B L U F U F L' B' R' U F R B L U F U F L' B' R' U F U F R B L F' U' L' B' R' F' U' F' U' R B L F' U' F' U' F' U' L' B' R' U F U F U F U F U F R B L U F U F U F L' B' R' F' U' F' U' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F U F R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F U F L' B' R' U F U F U F U F U F R B L F' U' F' U' L' B' R' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' U F U F R B L U F U F L' B' R' F' U' F' U' R B L U F L' B' R' F' U' F' U' F' U' R B L F' U' L' B' R' U F U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' U F R B L U F L' B' R' F' U' L' B' R' F' U' F' U' R B L R B L R B L U F L' B' R' F' U' L' B' R' L' B' R' L' B' R' F' U' L' B' R' F' U' L' B' R' F' U' L' B' R' F' U' F' U' L' B' R' F' U' F' U' F' U' L' B' R' U F R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F U F L' B' R' U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' F' U' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F U F R B L U F L' B' R' F' U' R B L F' U' F' U' L' B' R' U F U F U F U F U F R B L F' U' F' U' F' U' L' B' R' U F R B L F' U' L' B' R' U F R B L U F L' B' R' U F R B L F' U' L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F U F L' B' R' U F U F U F U F U F R B L U F U F L' B' R' U F R B L U F U F L' B' R' U F U F U F U F U F R B L U F L' B' R' F' U' R B L U F U F U F U F U F U F U F U F U F U F L' B' R' U F R B L F' U' L' B' R' F' U' R B L F' U' L' B' R' F' U' F' U' R B L U F U F U F U F U F U F L' B' R' U F U F R B L U F U F U F L' B' R' F' U' R B L F' U' L' B' R' U F U F U F R B L F' U' L' B' R' F' U' F' U' F' U' R B L U F U F L' B' R' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' U F R B L U F L' B' R' F' U' F' U' R B L U F U F L' B' R' U F R B L U F L' B' R' U F U F R B L F' U' L' B' R' U F R B L F' U' L' B' R' U F R B L U F U F L' B' R' F' U' F' U' R B L F' U' L' B' R' U F L' B' R' F' U' R B L U F R B L R B L U F L' B' R' U F L' B' R' F' U' L' B' R' L' B' R' F' U' F' U' L' B' R' F' U' F' U' L' B' R' L' B' R' F' U' L' B' R' U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F L' B' R' U F R B L F' U' F' U' F' U' L' B' R' F' U' F' U' R B L F' U' F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F R B L U F L' B' R' F' U' F' U' F' U' R B L F' U' L' B' R' U F U F R B L F' U' F' U' F' U' L' B' R' U F U F R B L U F U F U F L' B' R' F' U' F' U' R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' U F R B L F' U' F' U' L' B' R' U F R B L U F L' B' R' L' B' R' U F R B L U F L' B' R' L' B' R' F' U' F' U' L' B' R' F' U' F' U' L' B' R' F' U' F' U' L' B' R' L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F U F L' B' R' U F U F U F U F U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' U F U F R B L U F L' B' R' F' U' R B L F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F R B L U F L' B' R' U F U F U F U F R B L U F L' B' R' U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' U F U F U F U F U F U F U F R B L F' U' F' U' F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F L' B' R' U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F R B L U F L' B' R' U F R B L F' U' F' U' L' B' R' U F R B L U F U F U F L' B' R' U F R B L U F U F L' B' R' U F U F U F R B L F' U' L' B' R' U F R B L U F L' B' R' F' U' R B L U F U F L' B' R' U F R B L F' U' L' B' R' F' U' R B L F' U' L' B' R' U F U F R B L F' U' L' B' R' F' U' F' U' F' U' R B L U F U F L' B' R' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' F' U' R B L F' U' L' B' R' F' U' R B L F' U' F' U' L' B' R' F' U' F' U' F' U' R B L F' U' F' U' L' B' R' F' U' R B L U F U F U F L' B' R' F' U' F' U' R B L F' U' L' B' R' L' B' R' F' U' R B L F' U' L' B' R' L' B' R' F' U' R B L F' U' R B L R B L R B L R B L F' U' L' B' R' F' U' L' B' R' L' B' R' L' B' R' L' B' R' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' U F U F U F U F U F U F U F U F R B L U F L' B' R' U F U F U F U F U F R B L F' U' L' B' R' U F R B L F' U' L' B' R' U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F R B L U F L' B' R' F' U' R B L F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F R B L F' U' L' B' R' U F R B L F' U' L' B' R' F' U' R B L U F L' B' R' U F U F R B L U F L' B' R' F' U' R B L U F U F U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L U F U F L' B' R' U F R B L F' U' L' B' R' F' U' F' U' R B L U F U F U F U F U F L' B' R' F' U' F' U' F' U' F' U' R B L U F U F U F U F L' B' R' U F U F U F U F U F R B L U F L' B' R' U F U F U F U F R B L U F U F U F U F U F L' B' R' U F U F U F U F R B L U F L' B' R' U F U F U F U F U F U F R B L U F U F U F U F U F L' B' R' U F U F U F U F R B L F' U' L' B' R' U F U F U F U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F R B L U F L' B' R' U F R B L F' U' F' U' L' B' R' U F R B L U F U F U F L' B' R' U F R B L U F U F L' B' R' F' U' R B L F' U' L' B' R' F' U' F' U' R B L U F U F U F U F U F U F L' B' R' U F U F R B L U F U F U F L' B' R' F' U' R B L F' U' L' B' R' U F U F U F R B L F' U' L' B' R' U F R B L U F U F L' B' R' F' U' R B L U F L' B' R' U F R B L U F U F U F U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F U F U F U F U F R B L F' U' L' B' R' F' U' F' U' R B L F' U' L' B' R' F' U' R B L F' U' F' U' L' B' R' U F U F L' B' R' U F U F U F U F U F R B L R B L R B L U F L' B' R' U F L' B' R' U F L' B' R' L' B' R' F' U' L' B' R' F' U' L' B' R' F' U' L' B' R' L' B' R' F' U' L' B' R' U F U F R B L U F L' B' R' F' U' R B L U F U F U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L U F U F L' B' R' U F R B L F' U' L' B' R' F' U' F' U' R B L U F U F U F U F U F L' B' R' F' U' F' U' F' U' F' U' R B L U F U F U F U F L' B' R' U F U F U F U F U F R B L U F L' B' R' U F U F U F U F R B L U F U F U F U F U F L' B' R' U F U F U F U F R B L U F L' B' R' U F U F U F U F R B L U F U F U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L U F U F L' B' R' U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' U F U F R B L U F L' B' R' U F R B L F' U' L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F U F L' B' R' U F U F U F U F U F R B L U F U F L' B' R' U F R B L U F U F L' B' R' U F U F R B L F' U' L' B' R' F' U' F' U' R B L F' U' F' U' F' U' L' B' R' U F U F U F U F U F R B L U F U F U F L' B' R' F' U' F' U' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F U F U F U F U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' F' U' F' U' R B L U F L' B' R' U F U F R B L F' U' L' B' R' U F R B L U F U F L' B' R' F' U' R B L U F L' B' R' U F R B L U F U F U F U F L' B' R' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' U F R B L U F U F L' B' R' F' U' R B L F' U' L' B' R' F' U' R B L F' U' F' U' L' B' R' U F R B L F' U' L' B' R' F' U'

However:

Bruce, you're awesome.

Edit: Expanded lengths of Bruce's algs: {1804, 1871, 1771, 860, 1497}
I'm willing to bet there are algs under 100 for each.

 

[rokicki]

ksolve

1) Have you heard of ksolve?

I've seen it discussed. I haven't found source anywhere, nor a binary. Has anyone seen it?

 

[Stefan]

ksolve link can be found in the wiki
http://www.speedsolving.com/wiki/index.php/PC_Software

Direct link: http://www.svekub.se/files/ksolve.zip

 

[cuBerBruce]

I played around with some other generator pairs. Apparently GAP is able to find more efficient solutions if one of the generators has low order. So in the spoiler below, I include what GAP produced for generators a = U D' and b = L F R. This time I added some spaces for additional readability and to prevent forum from inserting spaces at odd places. I also give all 6 face turns this time; with R last because for some reason it was quite a bit longer than the other 5 (at least in terms of text length).

Spoiler

a = U D'
b = L F R

U =
a' bab' abab3 a' b'2 a2 b' a' b' a' b' a' ba
b3 abababab' a' b'2 a' b' a' b' a' bab'3 abab'
a' b3 a' b' abab'3 ab'2 aba' b' a'2 b3 a'2 b'
a' bab4 a' b' a' bab' ababab' aba' b' a' b2
a' b'2 a' b' a' b10 abab' a' b'10 aba' b10 a2 b2 a
b'3 a' b'10 ab3 a' b'2 a' b' a' b10 aba' b2 ab'3 a'
b'20 ab3 a' b'2 ab' a' b2 aba' b' a' b' a' b' a'
b'10 abababab' a' b'12 ab21 a' b' a' b' a'
b'10 ababab'10 a' b' a' b' a' b10 ababab' a'

D =
a2 bab' abab3 a' b'2 a2 b' a' b' a' b' a' ba
b3 abababab' a' b'2 a' b' a' b' a' bab'3 abab'
a' b3 a' b' abab'3 ab'2 aba' b' a'2 b3 a'2 b'
a' bab4 a' b' a' bab' ababab' aba' b' a' b2
a' b'2 a' b' a' b10 abab' a' b'10 aba' b10 a2 b2 a
b'3 a' b'10 ab3 a' b'2 a' b' a' b10 aba' b2 ab'3 a'
b'20 ab3 a' b'2 ab' a' b2 aba' b' a' b' a' b' a'
b'10 abababab' a' b'12 ab21 a' b' a' b' a'
b'10 ababab'10 a' b' a' b' a' b10 ababab' a'

L =
aba' b' a' babab' a2 b2 a'2 b3 a'2 b' a' b'2 ab
a' b' a'2 b' a' bab2 aba' b'3 a'2 b'2 aba' b'
a' bab4 aba' b' a' ba' b' aba2 bab' a' b3 ab'2
a'2 b'2 a2 bab' a' ba2 b' ab'2 ab' a2 ba' b3 a' b'
a' b2 a' b' a' bab' ababab' aba' b' a' b2 a'
b'2 a' b' a' b10 abab10 ab2 ab'3 a' b'10 ab3 a' b'2
a' b' a' b10 ab'7 a' b10 ab'3 a' b20 ab'9 a' b2 ab'10
a' b'2 ab'10 a' b' a' b' a' b10 ababab'10 a' b'
a' b' a' b'10 ababab9 a'

F =
ba' b' a' bab' a'2 ba' b'2 a' ba'2 b'5 aba' b'
a' babab' a2 b2 a'2 b3 a'2 b' a' b'2 aba' b' a'2
b' a' bab2 aba' b'3 a'2 b'2 ab' a' b' a' bab6 ab
a' b' a'2 bab' a' ba'2 b' a'2 b2 aba' b' a' b'2 ab
a' b' a'2 b'10 a2 ba2 b2 ab'3 a' b'10 ab3 a' b'2 a' b9
a' b10 abab2 ab'3 a' b10 ab3 a' b'2 a' b' a' b20 ab
a' b' a' b' a' b' a' bab'10 a' b' abababab'20
a' b2 ab'10 a' b'3 a' b' a' b'10 ababab10 a' b' a'
b' a' b10 ababab'10 a' b' a' b' a' b10 ababab'10
a' b' a' b' a' b'10 ababab9 a'

B =
b'2 a'2 b4 a' b2 a' b' abab2 a' b'3 a' b' a'
b' aba2 b5 a' b5 a' bab3 ab'3 a' b'4 ab'2 ab' a'2 b
a' b2 a' ba'2 b' aba'2 b' a'2 b10 a2 ba' b' a'2 b10
a2 ba'2 b' a' b'20 abab2 ab'3 a' b'10 ab3 a' b'2 a'
b10 a' b2 ab10 a' b'2 ab9 a' b'10 aba' b2 ab10 a'
b'2 ab20 a' b' a' b' a' b10 ababab'10 a' b' a' b'
a' b'10 ababab9 a'

R =
aba' b' a' b' a' b10 ababab'10 a' b' a' b' a'
b'10 ababab9 a' b'10 aba' b' a' b' a' b10 ababa
b'11 a' b10 aba' b2 ab10 a' b'2 ab20 a' b' a' b' a'
b' a' bab10 a' b' abababab' a' b'10 aba' b2 a
b'3 a' b10 ab3 a' b'2 ab' a' b'10 abab' a' b2 ab'2
a'2 b'2 a2 bab' a' b2 ab'2 a'2 b'2 aba' b10 ab' ab3 a
b'2 abab' a' ba' b' a' b' a' ba' b' abab'2 aba
b' a' ba2 b' ab'2 ab' a2 ba' b3 a' b' a'2 b2 a2 b2
a' b'3 aba' b' a'2 b' a' bab' abab' a' b'4 a'
b' abab' a' b2 a2 b3 ab' a' b'2 a' b' aba2 bab'
a' b2 aba2 b' a' b2 abab' a' b2 aba' b' a' b'
a' b' a' bab'2 a' b' abab2 a' b' abab'2 aba'
b' a' b2 ab3 ab'2 a' b' a' b'3 ab3 a' b'3 a' bab
a' b' a' b' ab'2 ab3 a'2 b' a'2 b' a' ba' b' a'
b2 a' b2 a'2 b3 ab' a' b3 abab' a' b'2 aba' b' a'
b2 aba' b' a' b' a' b' a' bab'2 a' b' abab2 ab
a' bab' aba' b'2 a' b'3 ab'2 a2 b'2 a'2 ba' b' a'
b' abab'6 ab'5 a'

 

[rokicki]

If a=UF and b=RBL, then:

U=a'a'a'b'b'ab'a'a'a'bbaabba'b'a'bba'a'a'baaaaaaba'bab'ab'a'a'
F=aaba'ba'b'ab'a'a'a'a'a'a'b'aaab'b'abab'b'a'a'b'b'aaaba'bbaaaa
R=b'a'babba'b'a'b'a'a'ba'b'a'a'bab'abab'a'a'b'abbbbabababab
D=a'a'baaab'b'aba'a'bab'b'abbbab'a'b'aab'b'a'baaaab'b'aabb
B=b'a'ba'a'a'a'bab'a'a'b'b'a'b'aaba'a'bba'bbaaaaba'ba'bba'b'
L=b'a'a'b'aba'a'b'b'b'a'b'a'b'a'b'a'b'b'b'aaaaaba'b'ababa'b'aba'b

These are shortest solutions. The lengths are 40, 41, 40, 40,
38, and 39 respectively.

 

[mrCage]

This is mostly old stuff. But probably new to many people anyway. Simple question/challenge. How about generating the cube supergroup,i.e. all the face center rotations. How many more generators would be needed? I don't want cube rotation as a generator

Per

 

[Tim Reynolds]

This is mostly old stuff. But probably new to many people anyway. Simple question/challenge. How about generating the cube supergroup,i.e. all the face center rotations. How many more generators would be needed? I don't want cube rotation as a generator

Per

You need at least six generators to rotate all the centers (five could only generate 4^5=2^10 center positions, but there's 4^6/2=2^11 total). So {U,D,F,B,R,L} is a minimal set of generators.

 

[rokicki]

This is mostly old stuff. But probably new to many people anyway.

Can you give any citations (beyond those already listed)? I see several
new results in this thread, that I have never seen before.

Simple question/challenge. How about generating the cube supergroup,i.e. all the face center rotations. How many more generators would be needed? I don't want cube rotation as a generator

Per

I don't know the answer to this, but I'd be surprised if the given generator pairs do not already generate the supergroup that includes center rotations.

 

[Lucas Garron]

I don't know the answer to this, but I'd be surprised if the given generator pairs do not already generate the supergroup that includes center rotations.

'cept they don't? <UF, RBL> only gives 16 configurations of center orientations, right?

 

[rokicki]

I don't know the answer to this, but I'd be surprised if the given generator pairs do not already generate the supergroup that includes center rotations.

'cept they don't? <UF, RBL> only gives 16 configurations of center orientations, right?

Very good, I totally missed the obvious. Thanks, Lucas and Tim!

 

[JBCM627]

This is mostly old stuff. But probably new to many people anyway. Simple question/challenge. How about generating the cube supergroup,i.e. all the face center rotations. How many more generators would be needed? I don't want cube rotation as a generator

Per

You need at least six generators to rotate all the centers (five could only generate 4^5=2^10 center positions, but there's 4^6/2=2^11 total). So {U,D,F,B,R,L} is a minimal set of generators.

{Rw, Uw, Fw}? But I guess those (and slice turns) would count as cube rotations?

 

[cuBerBruce]

This is mostly old stuff. But probably new to many people anyway. Simple question/challenge. How about generating the cube supergroup,i.e. all the face center rotations. How many more generators would be needed? I don't want cube rotation as a generator

Per

You need at least six generators to rotate all the centers (five could only generate 4^5=2^10 center positions, but there's 4^6/2=2^11 total). So {U,D,F,B,R,L} is a minimal set of generators.

{Rw, Uw, Fw}? But I guess those (and slice turns) would count as cube rotations?

Haha. That only generates the "keychain" supercube group. And yes, it goes outside the <U,D,L,R,F,B> supercube group. Rw, Uw, and Fw are not in the <U,D,L,R,F,B> group.

EDIT: OK, I realize the elements of the "keychain" group (whether regular cube or supercube) has half the number of elements of the full group. Since a corner cubie serves as a fixed reference (like the fixed centers of <U,D,L,R,F,B>), I believe there is still only a 1:1 mapping between keychain group elements and their corresponding "cube positions." So mapping <Rw, Uw, Fw> elements to cube positions, half of the positions of the cube are generated. So it comes very close to generating all positions, just a factor of 2 short for either the regular cube or supercube.

EDIT2: OK, from using GAP, I conclude <Rw, Uw, FwU> generates all supercube positions. <Rw, Uw, FwD> generates all supercube positions in all 24 cube orientations (the full <U, x, y> group). Of course these generators can be viewed as having implicit cube rotations.

 

[rokicki]

I've found a pair of generators of order 2 and 4 that generate the whole cube.

a=U2F1B1D2F3B3D1
b=F1B2R1L1U2F1

These are the smallest orders possible that generate the whole cube.

The count of positions at levels 1, 2, ... goes 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711 ... which I suspect many of you will recognize as the Fibonacci sequence.

 

[Stefan]

The count of positions at levels 1, 2, ... goes 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711 ... which I suspect many of you will recognize as the Fibonacci sequence.

Neat! How far does this go?

 

[rokicki]

Fib through level 32

It continues through level 32; 33 is the first level where it doesn't hold. Other pairs of generators of orders 2 and 4 may take it further; I'll experiment with some of those.

At level 0 total 1 thislev 1
At level 1 total 4 thislev 3
At level 2 total 9 thislev 5
At level 3 total 17 thislev 8
At level 4 total 30 thislev 13
At level 5 total 51 thislev 21
At level 6 total 85 thislev 34
At level 7 total 140 thislev 55
At level 8 total 229 thislev 89
At level 9 total 373 thislev 144
At level 10 total 606 thislev 233
At level 11 total 983 thislev 377
At level 12 total 1593 thislev 610
At level 13 total 2580 thislev 987
At level 14 total 4177 thislev 1597
At level 15 total 6761 thislev 2584
At level 16 total 10942 thislev 4181
At level 17 total 17707 thislev 6765
At level 18 total 28653 thislev 10946
At level 19 total 46364 thislev 17711
At level 20 total 75021 thislev 28657
At level 21 total 121389 thislev 46368
At level 22 total 196414 thislev 75025
At level 23 total 317807 thislev 121393
At level 24 total 514225 thislev 196418
At level 25 total 832036 thislev 317811
At level 26 total 1346265 thislev 514229
At level 27 total 2178305 thislev 832040
At level 28 total 3524574 thislev 1346269
At level 29 total 5702883 thislev 2178309
At level 30 total 9227461 thislev 3524578
At level 31 total 14930348 thislev 5702887
At level 32 total 24157813 thislev 9227465
At level 33 total 39088158 thislev 14930345 (oops, not a Fibonacci number)

 

[cuBerBruce]

According to my GAP calculations, a minimal generator set for generating all 2,125,922,464,947,725,402,112,000 elements of the <U, x, y> supercube group, using only (single-layer) face turns and double-layer turns in the generators, would be <Uw, LUwRw>.

As for just generating all supercube *positions* (in at least 1 cube orientation each), I believe <Uw, DRw> suffices.

Can anyone confirm these assertions?

 

[mrCage]

Thanf for intersting contributions. But all of them seem to have implicit cube rotations (wide turns). Who can find "pure" generators? Will 3 generators suffice? That's my guess anyway:rolleyes:

Per