## Re: Is there a way to figure the number of cases for a step?

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• Hey guys, I was wondering if it would be possible to create a cube explorer like program that will solve cubes with undefined points in the configuration, yet
Message 1 of 17 , Aug 3, 2006
Hey guys, I was wondering if it would be possible to create a cube
explorer like program that will solve cubes with undefined points in
the configuration, yet it can still identify isomorphic combinations!

That would be awsome.

Also, If you create a static reference, the cases are reduced
significantly, I just compiled 132 roux 2nd Block cases. Yes, roux
is supposed to be intuitive. But, there are just hard cases,
especially to do using the URM subset. All the cases I generated
did this, =D very nice.

Anyway, Figure out what is isomorphic (any cube state with the same
moves whether inversed, reflected, or applied from a different angle)
Since we don't have a good hybrid of ACube and CubeExplorer, just go
over the tables you've generated and recognize the cases for
yourself.

Btw, the benifit to 2-gen F2B in roux is that you can identify the
CMLL permutation case while permuting the last C/E pair.

Also I'm curious what is your method for? What do the steps involve?

I've worked with the Acube a lot lately, generating everything from
my new BLD algs for a new method, getting faster btw at execution.
Not too far from my goal. Anyway, BLD algs, Roux F2B algs, a set of
24 algs that would permute the corners and orient edges on 4th c/e
pair insertion.

The roux algs are by far the biggest group, my BLD algs are the
deepest, and the C/E pair (Permute/Orient) algs.

Like I said have a reference point, such as in F2L algs, you align
the Corner above the C/E slot to recognize the case.

--- In speedsolvingrubikscube@yahoogroups.com, d_funny007
>
> No, in some cases (I think yours would apply), you should look
> for "diagonal mirroring". Although the simple mirroring plus U
> rotations *might* be enough/analgous/equivalent, but I have put
> little thought into this as I am currently on vacation!
>
> As a long time member of this fourm, I'd like to say that it is
very
> good to see another hardcore math/cs person like Bruce here! I've
> been keeping up with his posts on this other fourm he uses too.
Very
> techincal stuff that I once wanted to see here, but after further
> thought, it just wouldn't fit here. I was always the one rushing
to
> answer math questions, but I wasn't particularly patient in the
past :
> (.
>
> I can try a verification of his computation when I get the chance.
It
> is most challenging :).
>
>
> -Doug
>
>
> --- In speedsolvingrubikscube@yahoogroups.com, "athefre"
> <athefre@> wrote:
> >
> > Thanks, 111 is better than 140, but not much.
> >
> > If you could reduce the number using mirrors and inverses, how
much
> > would it be? If you don't mind. I've been working hard for a
> month
> > trying to perfect everything so I can get to work on finding the
> > algorithms for the idea I choose.
> >
> > Inverse = backwards
> > Mirror = LUL'ULU2L' is the mirror of R'U'RU'R'U2R
> >
> > Correct?
> >
> > --- In speedsolvingrubikscube@yahoogroups.com, "Bruce Norskog"
> > <brnorsk@> wrote:
> > >
> > > Hi,
> > >
> > > Yes, you're right. I considered rotations of the E layer, but
not
> > more
> > > complicated adjustment moves like R2 E R2. If you allow that,
> then
> > the
> > > middle multipliers in my table all become 1, and you can just
> > multiply
> > > the first and third number. With that, my 140 cases (excluding
the
> > > do-nothing case) get reduced to 111 cases. (I think I did the
> > > arithmetic correctly.) Again, I haven't looked at using
mirrors
> and
> > > inverses to reduce the number of algorithms further.
> > >
> > > Sorry, it looks like my table's formatting wasn't preserved,
at
> > least
> > > if viewed from the Yahoo web site. You would think the Preview
> > button
> > > would actually show you what your post was going to look like,
> > > wouldn't you? In Preview, it looked fine, but the actual post
> > appears
> > > to have all "redundant" space characters stripped out.
> > >
> > > - Bruce
> > > --- In speedsolvingrubikscube@yahoogroups.com, "athefre"
> <athefre@>
> > > wrote:
> > > >
> > > > Thanks. All of what you said sounds right. But there is
one
> > thing
> > > > I'm not sure if you considered that I may have looked over
in
> > your
> > > > post.
> > > >
> > > > What about the "empty spaces" available in E for the cases
> where
> > 2 E
> > > > edges need to be placed? Like, if you have an empty space
at
> FR
> > and
> > > > BR or you can have the spaces at FR and BL (although you
could
> do
> > > > R2ER2 before the algorithm).
> > > >
> > > > If it really is 140 cases then that is WAY too many for me
to
> > make
> > > > and learn. I'm definitly going with my other option.
> > > >
> > > > --- In speedsolvingrubikscube@yahoogroups.com, "Bruce
Norskog"
> > > > <brnorsk@> wrote:
> > > > >
> > > > > Hi,
> > > > >
> > > > > From what I understand, you have 4 corner cubies in the U
> layer
> > to
> > > > be
> > > > > put into correct relative order (orientation doesn't
matter).
> > You
> > > > have
> > > > > 10 edges that can be permuted around without changing
> > orientation.
> > > > Of
> > > > > those 10 edges, 4 are E-layer edges which can be considered
> > > > > indistinguishable from each other. These E-layer edges are
all
> > > > > required to end up in the E layer. The other set of 6
edges
> can
> > also
> > > > > be considered to be indistinguishable from each other. The
U
> > layer
> > > > can
> > > > > be rotated before (and after, if you want the corners
> correctly
> > > > placed
> > > > > relative to the center) the algorithm. Likewise, the E
layer
> > can be
> > > > > rotated before and after the algorithm. (Rotating after to
> get
> > the
> > > > > E-layer centers back into correct position, if needed.)
> > > > >
> > > > > So to count the different cases you can have, consider the
> > different
> > > > > cases of where the E-layer edges can be, and count the
cases
> > for
> > > > each
> > > > > of the possible corner permutation situations (no swap,
swap 2
> > > > > adjacent, swap to diagonally opposite). First break down
the
> > edge
> > > > > cases by how many might be in each layer. For each
possible
> > number
> > > > of
> > > > > E-layer edges in each of the layers, determine the number
of
> > cases
> > > > > possible for each of the corner permutation situations.
> > > > >
> > > > > Then build a table of all the possibilities:
> > > > >
> > > > > (best viewed using fixed-width font)
> > > > >
> > > > > U-E-D no swap adj. swap diag. swap
> > > > > ----- ------- --------- ----------
> > > > > 4 0 0 1*1*1 = 1 1*1*1 = 1 1*1*1 = 1
> > > > > 3 1 0 1*1*1 = 1 4*1*1 = 4 2*1*1 = 2
> > > > > 3 0 1 1*1*2 = 2 4*1*2 = 8 2*1*2 = 4
> > > > > 2 2 0 2*2*1 = 4 6*2*1 = 12 4*2*1 = 8
> > > > > 2 1 1 2*1*2 = 4 6*1*2 = 12 4*1*2 = 8
> > > > > 2 0 2 2*1*1 = 2 6*1*1 = 6 4*1*1 = 4
> > > > > 1 3 0 1*1*1 = 1 4*1*1 = 4 2*1*1 = 2
> > > > > 1 2 1 1*2*2 = 4 4*2*2 = 16 2*2*2 = 8
> > > > > 1 1 2 1*1*1 = 1 4*1*1 = 4 2*1*1 = 2
> > > > > 0 4 0 1*1*1 = (1) 1*1*1 = 1 1*1*1 = 1
> > > > > 0 3 1 1*1*2 = 2 1*1*2 = 2 1*1*2 = 2
> > > > > 0 2 2 1*2*1 = 2 1*2*1 = 2 1*2*1 = 2
> > > > > --- --- ---
> > > > > 25 72 44
> > > > >
> > > > > So I get 25+72+44 = 141 cases. The 1 in parentheses in the
> table
> > > > > indicates the case where no algorithm needs to be
performed.
> So
> > if
> > > > you
> > > > > don't count that case, then I get 140.
> > > > >
> > > > > I have not considered the diagonal symmetry in the above,
but
> > then,
> > > > I
> > > > > understand you were not asking for that to be taken into
> > > > consideration.
> > > > >
> > > > > I just thought I would add my own comments about the edge
> > > > orientation
> > > > > issue.
> > > > >
> > > > > I agree with Doug in that the key in what you said was
that F
> > and B
> > > > > moves flip four edges.
> > > > >
> > > > > From that I assume you mean, that to be oriented:
> > > > > - an edge cubie that belongs in the M or S slice, and is
> > currently
> > > > > located in one of those slices, must have its U or D
facelet
> > > > aligned
> > > > > with the U or D center
> > > > > - an edge cubie that belongs in the M or S slice, and is
> > located in
> > > > > the E slice, must have it U or D facelet aligned with the
F
> or
> > B
> > > > center.
> > > > > - an edge cubie that belongs in the E slice, and is
located
> in
> > the
> > > > E
> > > > > slice, must have its F or B facelet aligned with the F or
B
> > center
> > > > (or
> > > > > equivalently, its R or L face aligned with the R or L
center)
> > > > > - an edge cubie that belongs in the E slice, and is
located
> in
> > the
> > > > M
> > > > > or S slice, must have its F or B face aligned with the U
or D
> > > > center.
> > > > >
> > > > > When an edge is in the inner slice that it belongs to, its
> > usually
> > > > > assumed that the edge would be oriented if each of its
> facelets
> > is
> > > > > aligned with the same color center, or the center that is
> > opposite
> > > > > that center. (Someone could define edge orientation in a
way
> > such
> > > > that
> > > > > the above would not be the case, but I would say this is
> rare.)
> > But
> > > > > when an edge is moved to a different inner slice than the
one
> it
> > > > > belongs in, it is not generally as clear what it means to
be
> > > > oriented.
> > > > >
> > > > > Doug mentioned a way of defining edge orientation such
that
> > moving L
> > > > > or R a quarter-turn flips four edges. There is yet another
> way
> > of
> > > > > defining edge orientation that I have used in computer
> analyses
> > of
> > > > the
> > > > > cube. You can define edge orientation such that moving any
of
> > the
> > > > > layers U, D, L, R, F, or B a quarter-turn flips all four
> edges
> > > > moved.
> > > > > This is the most symmetrical way of defining edge
> orientation.
> > But
> > > > > define edge orientation in the way that makes the most
sense
> > for
> > > > your
> > > > > situation. With your way, you can keep all edges oriented
> > simply by
> > > > > avoiding F, F', B, and B' moves (F2 and B2 okay, of
course).
> > > > >
> > > > > - Bruce
> > > > > --- In speedsolvingrubikscube@yahoogroups.com, "athefre"
> > <athefre@>
> > > > > wrote:
> > > > > >
> > > > > > Yeah, it was supposed to say "DFL".
> > > > > >
> > > > > > I don't really understand or know anything about
inverses
> and
> > > > mirrors
> > > > > > and symmetry and all of that crazy stuff but hopefully
this
> > helps:
> > > > > >
> > > > > > -Add in the inverses the stuff like that but tell me how
> many
> > > > > > distinct cases there are with those included and without.
> > > > > >
> > > > > > -Don't count U adjustments. I don't mind having to
> U
> > > > before
> > > > > > doing an algorithm.
> > > > > >
> > > > > > So far I'm thinking it's around 102. If so, no way.
I'm
> > going
> > > > with
> > > > > > my other option. This is what I've been counting:
> > > > > >
> > > > > > Already permuted: 17 cases
> > > > > > Diagonal swap: 18 cases (1 for E edges already in E)
> > > > > > Adjacent swap: 69 cases (same as above)
> > > > > >
> > > > > > Is there a site that describes these kinds of things?
> > > > > >
> > > > > >
> > > > > > --- In speedsolvingrubikscube@yahoogroups.com,
d_funny007
> > > > > > <no_reply@> wrote:
> > > > > > >
> > > > > > > That does help. Actually I use a different EO
> definition...
> > I
> > > > treat
> > > > > > > L and R as flipping 4 edges.
> > > > > > >
> > > > > > > Also, could you double check this: "The algorithm must
not
> > > > > > > mess up UFL, DL, DBL, DB, DBR, or DFR." It doesn't
feel
> > right.
> > > > Are
> > > > > > > you sure you don't mean 'DFL' there? Also what would
you
> > count
> > > > as a
> > > > > > > distinct case? I could group diagonally-symmetric
cases
> as
> > one.
> > > > I
> > > > > > > could even group cases that use inverse algorithms
> > together. If
> > > > U
> > > > > > > layer is not free for the first turn, than you could
get
> > what I
> > > > > > like
> > > > > > > to think of as a single case counted 4 times.
> > > > > > >
> > > > > > > This question sounds familiar, like I've already heard
> > > > something
> > > > > > > similar before, but it is definately a hard one and
may
> > take
> > > > some
> > > > > > > time.
> > > > > > >
> > > > > > >
> > > > > > > -Doug
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > > --- In
speedsolvingrubikscube@yahoogroups.com, "athefre"
> > > > > > > <athefre@> wrote:
> > > > > > > >
> > > > > > > > I'm not too sure what you mean, but I'm using yellow
on
> > top,
> > > > blue
> > > > > > > on
> > > > > > > > the right, orange in the front. All of the yellow
and
> > white
> > > > > > edges
> > > > > > > > face the white or yellow center (it doesn't matter)
and
> > all
> > > > of
> > > > > > the
> > > > > > > > blue and green edges are facing the blue or green
> > centers.
> > > > It's
> > > > > > > like
> > > > > > > > Petrus, the edges are oriented that way, and if you
do
> F
> > or B
> > > > it
> > > > > > > > messes up 4 edges.
> > > > > > > >
> > > > > > > > Does that help.
> > > > > > > >
> > > > > > > > --- In
speedsolvingrubikscube@yahoogroups.com, "Stefan
> > > > Pochmann"
> > > > > > > > <pochmann@> wrote:
> > > > > > > > >
> > > > > > > > > --- In
> > speedsolvingrubikscube@yahoogroups.com, "athefre"
> > > > > > > <athefre@>
> > > > > > > > > wrote:
> > > > > > > > > >
> > > > > > > > > > All edges on the cube are already oriented
before
> > going
> > > > to
> > > > > > this
> > > > > > > > > > step.
> > > > > > > > >
> > > > > > > > > There's no general definition for orientation so
you
> > need
> > > > to
> > > > > > > > provide
> > > > > > > > > one.
> > > > > > > > >
> > > > > > > >
> > > > > > >
> > > > > >
> > > > >
> > > >
> > >
> >
>
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