I sent the results of a quarter-turn analysis of these puzzles to
Cube-Lovers in several messages during August, 1984. I modified a
program written by Karl Dahlke to get these results. I counted both
positions and local maxima at every distance up to the diameter of 14
quarter-turns. In case you don't have the archives handy, here are
the results:

  Quarter        2^3 Puzzle           Corners of 3^3 Puzzle
   Turns   Positions  Local Maxima   Positions  Local Maxima
____________________________________________________________
     0            1        0                1           0
     1            6        0               12           0
_____2___________27________0______________114___________0___
     3          120        0              924           0
     4          534        0             6539           0
_____5_________2256________0____________39528___________0___
     6         8969        0           199926         114
     7        33058       16           806136         600
_____8_______114149_______53__________2761740_______17916___
     9       360508      260          8656152       10200
    10       930588     1460         22334112       35040
____11______1350852____34088_________32420448______818112___
    12       782536   402260         18780864     9654240
    13        90280    88636          2166720     2127264
____14__________276______276_____________6624________6624___

The first column agrees with Dik Winter's findings. As Michael Reid
surmised, the diameters of the two groups are the same.

My hazy recollection is that the 2^3 program ran for a few minutes on
a Vax 750, while the corners program took a couple of hours.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil