On 10/23/95 at 20:42:30 Mikko Haapanen said:

I have a question (yes, again). This subject may be discussed here before,
but i don't understand set theory or other high math, so i ask:

If i had a 3x3x3 cube and i pull out a corner piece. I turn it and push
back. Now the cube cannot be solved. I think the cube is now 'on the other
orbit'. If i pull now an edge piece and flip it, the cube is again on some
other orbit.

>Only one of those orbits are legal. How many different illegal orbits there are
?

In the terms you are using, there are 12 orbits. Of these, 1 is
"legal" (contains Start), and 11 are "illegal" (do not contain
Start).

There is a factor of 3 from twisting the corners. Pull out a corner
piece. There are 3 ways to put it back in. You can put it back in
the way it came out, you can twist it right, or you can twist it left.

There is a factor of 2 from flipping the edges. Pull out an edge
piece. There are 2 ways to put it back in, flipped or unflipped.

There is a factor of 2 from parity. The edges can be said to be in
even parity or in odd parity, and the corners can be said to be in
even parity or odd parity. Normally, the corners and edges are in
the same parity. A quarter turn changes the parity both for the
edges and for the corners. But pull out 2 edges pieces (or 2 corner
pieces). Put them back where they came from, and their parity
remains the same. Exchange them, and their parity changes.

We therefore have 12=3x2x2.

However (and draw a deep breath), for every expert there is an equal
and opposite expert. This use of the term "orbit" agrees with some
experts. However, other experts would say that the corners form an
orbit, that the edges form an orbit, and that the face centers form
an orbit.

I don't know which use of the term orbit is correct (perhaps both are
in the proper context). But in any case, if you take a cube apart,
there are 12 disjoint sets of positions that you choose from when you
put the cube back together.

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