Date: 16 SEP 1980 0746-EDT
From: RP at MIT-MC (Richard Pavelle)
Subject: number of reachable states
To: KATZ at USC-ISIF
CC: RP at MIT-MC, CUBE-LOVERS at MIT-MC
Date: 15 Sep 1980 1842-PDT
From: Alan R. Katz
I have seen the number 4.3 * 10^19 for the number of reachable states
for the cube, can anyone tell me how you calculate it? This may have
been answered before in this list, but I couldn't find it.
The number is (12! * 2^12 * 8! * 3^8)/12. This comes from the following.
There are 8 corners and there are 3 positions- hence 8!*3^8. There are
12 edges with 2 positions hence 12!*2^12. Finally, the /12 comes from
parity considerations. Only 1/4 of the positions in the flippling of
two edges are possible while 1/3 of the toppling of two edges are possible.
Also, someone mentioned that one can make a checkerboard pattern from
the Pons Asinorum by trebly rotating the centers by a simple
transformation. Can anyone tell me this transformation? (again I may
have missed reading it)
The moving of centers is easy- 4 moves of the center slice while rotating
the cube 90 degrees in your hand between moves. With the transformation
in hand you can move the centers easily to possible positions.