Date: Tue, 16 Sep 80 09:46:00 -0400 (EDT)
From: David C. Plummer <DCP@MIT-MC >
~~~ Subject: number of reachable states
```Date: 15 Sep 1980 1842-PDT
From: Alan R. Katz <KATZ at USC-ISIF>
```

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.

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)

Reply to either me or the list.

```Alan
-------
```

Consider the corners. There are 8 of them, and they can go
anyplace. This leads to 8 factorial permutations. Each corner can
take on three orientations, so this is another factor of 3^8. But
the corners have three possible states (trarity [three way
parity]) so divide by 3. Now do the same with the edges. 12 edges
gives 12 factorial arrangements, times 2^12 oreintations. But the
edges have two parities involved, so divide by four (thus giving
rise to the 12 states of the cube, one of which has the solved
configuration as a member). So if you evaluate

```    8	   12
8!*3 *12!*2
-----------
3*4
```
```you will get 4.3 * 10^19.
```