Hi. I'm really pressed for time, but I'll drop a couple of
comments.
Alan pretty well said it--there are half-twisters and there
are quarter-twisters and the included message is one of the former.
I strongly favor the latter, since then all the moves are
equivalent, (M-conjugate, to you archive-readers). But Singmaster's
book, though in the other camp, is too good to ignore.
To extend the argument I gave on 9 January to the case
where quarter-twists and half-twists are counted equally (we call
such a move a `htw' whether it is quarter or half) let PH[n] be the
number of (3x3x3-cube) positions at exactly n htw from SOLVED. Then
PH[0] = 1 PH[1] <= 6*3*PH[0] PH[2] <= 6*2*PH[1] + 9*3*PH[0] PH[n] <= 6*2*PH[n-1] + 9*2*PH[n-2] for n > 2.
Solving yields the following upper bounds:
htw new total htw new total 0 1 1 10 2.447*10^11 2.646*10^11 1 18 19 11 3.267*10^12 3.531*10^12 2 243 262 12 4.360*10^13 4.713*10^13 3 3240 3502 13 5.820*10^14 6.292*10^14 4 43254 46756 14 7.769*10^15 8.398*10^15 5 577368 624124 15 1.037*10^17 1.121*10^17 6 7706988 8331112 16 1.385*10^18 1.497*10^18 7 102876480 111207592 17 1.848*10^19 1.998*10^19 8 1373243544 1484451136 18 2.467*10^20 2.667*10^20 9 18330699168 19815150304
At least 18 htw are required to reach all the 4.325*10^19
positions of the cube. This is the same argument that was used in
Singmaster's fifth edition, p. 34, and is the best I know. Lest ye
be tempted to pull the trick I did in the January message, remember
that half-twists are even permutations, so there is no assurance
that half the positions are an odd distance from SOLVED. This is
illustrated in the 2x2x2 case, where more than half of the
positions are at a particular odd distance.
And yes, all of Thistlethwaite's analysis seems to use the
half-twist metric. I am quite surprised, however, to hear the rumor
of 41 htw. As of Singmaster's fifth edition, the figure was 52 htw
``... but he hopes to get it down to 50 with a bit more computing
and he believes it may be reducible to 45 with a lot of
searching.'' If anyone has harder information on the situation, I
would like to hear it.
Well, back to real work. I saw a Rubikized tetrahedron in a
shop window earlier this evening; I'm not sure whether I'm relieved
or infuriated that the store was closed for the day.