From:

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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.