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I have found a Hamiltonian circuit for the entire 2x2x2 cube group (3674160 elements).

I note that my solution was developed independently from the solution for the <U,R> subgroup posted by cubemir.

For compactness, I use variables to define sub-sequences of moves.

The following conventions are used.

Upper case letters are used for the basic quarter-turn moves of the 2x2x2 puzzle.

· U represents turning the top face a quarter-turn clockwise.

· V represents turning the top face a quarter-turn counterclockwise (conventionally denoted U').

· R represents turning the right face a quarter-turn clockwise.

· S represents turning the right face a quarter-turn counterclockwise (conventionally denoted R')

· F represents turning the front face a quarter-turn clockwise (used in Appendix B).

· G represents turning the front face a quarter-turn counterclockwise clockwise (conventionally denoted F').

Variables denoted with lower case letters represent sequences of multiple moves.

Many of them are defined in terms of other lower case variables.

So recursive expansion is requred to get a plain sequence of individual moves.

If a letter is followed by an apostrophe, then the inverse of the indicated sequence is to be used.

This means that the order of the moves and direction of the moves must be reversed.

A line with a letter and an equal sign starts the definition for a maneuver.

The definition may extend over several lines until there is a line where another definition is started.

The variable z is used to represent the entire Hamiltonian circuit.

Spoiler

I note that my solution was developed independently from the solution for the <U,R> subgroup posted by cubemir.

For compactness, I use variables to define sub-sequences of moves.

The following conventions are used.

Upper case letters are used for the basic quarter-turn moves of the 2x2x2 puzzle.

· U represents turning the top face a quarter-turn clockwise.

· V represents turning the top face a quarter-turn counterclockwise (conventionally denoted U').

· R represents turning the right face a quarter-turn clockwise.

· S represents turning the right face a quarter-turn counterclockwise (conventionally denoted R')

· F represents turning the front face a quarter-turn clockwise (used in Appendix B).

· G represents turning the front face a quarter-turn counterclockwise clockwise (conventionally denoted F').

Variables denoted with lower case letters represent sequences of multiple moves.

Many of them are defined in terms of other lower case variables.

So recursive expansion is requred to get a plain sequence of individual moves.

If a letter is followed by an apostrophe, then the inverse of the indicated sequence is to be used.

This means that the order of the moves and direction of the moves must be reversed.

A line with a letter and an equal sign starts the definition for a maneuver.

The definition may extend over several lines until there is a line where another definition is started.

The variable z is used to represent the entire Hamiltonian circuit.

Code:

```
b=URURUR
a=bURUR
i=bbUR
c=VRiiVR
n=VRa
d=nn
e=ncUR
f=URcaVR
g=ncabVR
h=nbcaVR
j=ccaVR
k=bbVR
l=ncc
m=nUR
o=VRUR
r=adcURefknbhaodcURfccncabdodcURjabVRdccfcgdccfccVRicaURdnVRdljURURejaocdc
cfcglccadVRddURciVRcjURcecmcURcadVRdgaboURnbeccad
s=nbgaVRdblinVReckdbcanbVRmcURmcidoUR
t=URURnUUaURdVRihkhcecncabdVRdcURfURURhaVRdlfabnbVRdeckcfVRigkURdURcfccaUR
dVRdejhhejaoURdcURURefaVRegkglURcadVRddURciVRchejaoliVRVRigmcidVRnbeUReU
RcadVRdcabVRhURefaVRdhcjURgmefURmciddUUaURdVRicknbefaodcURjabnodlinboURe
URnbcadVRdgciVRcgncgcaURdVRddURURckURjURceURncabdVRccdcURfURURlURVRicaUR
dnVRdmccVRijaVRhcVRijaVRnbefaodURURglfaVReckdcURfabVRmhidnUUaURdeefknbha
odcURfccncabdodcURjabVRdccfcgdccfccVRicaURdnVRdljURURejaocdccfcglccadVRd
dURciVRcjURcecmcURcadVRdgaboURnbeccadVRnbgaVRdblinVReckdbcanbVRmcURmcidd
UUaURdVRihkhcecncabdVRdcURfURURhaVRdlfabnbVRdeckcfVRigkURdURcfccaURdVRde
jhhejaoURdcURURefaVRegkglURcadVRddURciVRchejaoliVRVRigmcidVRnbeUReURcadV
RdcabVRhURefaVRdhcjURgmefURmciddUUaURdVRicknbefaodcURjabnodlinboUReURnbc
adVRdgciVRcgncgcaURdVRddURURckURjURceURncabdVRccdcURfURURlURVRicaURdnVRd
mccVRijaVRhcVRijaVRnbefaodURURglfaVReckdcURfabVRmhidnUUaURdeefknbhaodcUR
fccncabdodcURjabVRdccfcgdccfccVRicaURdnVRdljURURejaocdccfcglccadVRddURci
VRcjURcecmcURcadVRdgaboURnbeccadVRnbgaVRdblinVReckdbcanbVRmcURmciddUUaUR
dVRihkhcecncabdVRdcURfURURhaVRdlfabnbVRdeckcfVRigkURdURcfccaURdVRdejhhej
aoURdcURURefaVRegkglURcadVRddURciVRchejaoliVRVRigmcidVRnbeUReURcadVRdcab
VRhURefaVRdhcjURgmefURmciddUUaURdVRicknbefaodcURjabnodlinboUReURnbcadVRd
gciVRcgncgcaURdVRddURURckURjURceURncabdVRccdcURfURURlURVRicaURdnVRdmc
w=cVRijaVRhcVRijaVRnbefaodURURglfaVReckdcURfabVRmhidnU
u=krVRsURtwF
v=uuuuuuuuaVRrVRsURtwUUG
p=FVw't'SVs'RRRUr'SUSVb'SFRbURVRrVSSSsURtwUG
q=GVw't'SVs'SUr'SUa'SFVVw't'SVs'SUr'SUa'FRrVRsUSSStwUaG
x=krVRsURtcVRijaVRhcVRijaVRnbefaodURURglfaVRnVRUpbURUpbURUpbockdcURfabVRVR
kaboURUpbURUpbURUpURdURURdnUFkrVRsURtVRUpaabVRVRijaVRhcVRicnaUpbdVRnbefa
odURURgnoURUpbURUpbURUpURVRcfaVReckVpaVRUpbURcURURcURUpbnbVRVRkabnbURUpb
URdURURdnUFaVRrVRsURtVRUpaabVRVRijaVRhcVRijaVRnbeURoUpaidodURURglfaVRenb
URUqbURVRknVRUpbURnabURVpURfabVRVRkURUpUpbUpUpcUpURUpUpURVRbUpbVRbURUpVR
UpbURoUpbUUUG
z=vvvvvvuuuuuux
```

Last edited: Dec 26, 2011