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Subject:

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.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU