   Date: Fri, 10 Jan 92 18:32:35 -0500 (EST)   From: Dan Hoey <hoey@aic.nrl.navy.mil >   Subject: Re: Rubik's Cube, the minimax number of moves

I wonder how large the necessary tables for Thistlethwaite's method
for the cube are? I seem to recall reading that there were a few
hundred entries....

Well, this is the information from Singmaster's _Notes_on_Rubik's_
_Magic_Cube_ (1980). Thistlethwaite's method is to work from group to
subgroup as follows:

```G0: <F,B,L,R,T,D>
G1: <F^2,B^2,L,R,T,D>
G2: <F^2,B^2,L^2,R^2,T,D>
G3: <F^2,B^2,L^2,R^2,T^2,D^2>
G4: <I>
```

The following table shows the number of cosets (the index of each
subgroup in the next larger group). Then I include the number of HT
moves proven, anticipated, and best possible, from the 1980 N_o_R_M_C.
Finally, I include the number of HT claimed in the 1987 R_C_C. It is
interesting to note that the improvement did not occur where
Thistlethwaite anticipated it.

```Step | Number of Cosets  |     Number of HT, 1980      | #HT, 1987
|                   |  Proven  Anticipated  Best  |   Proven
|                   |                             |
1  | G0:G1 =     2,048 |     7        7         7    |      7
2  | G1:G2 = 1,082,565 |    13       12        10    |     13
3  | G2:G3 =   663,552 |    15       14 ?      13 ?  |     15
4  | G3:G4 =    29,400 |    17       17        15 ?  |     15
-----+-------------------+-----------------------------+-----------
Total HT  |    52       50 ?      45 ?  |     50
Total QT  |   101       97 ?      87 ?  |     97
```

I had thought the tables contained one entry for each coset, and so
there would be over a million entries for step 2. However, I was
surprised just now to notice in N_o_R_M_C that tables were only needed
in step 4, and then only 172 entries, so there must be some
abbreviation or algorithmic approach. Of course, when Knuth's
students improved step 4, they may have changed it to use a huge
lookup table; I don't know. Still, this is much better than the
situation I expected in my note two days ago.

In listing QT I assume that in we can require steps 1, 2, and 3 to
each end with a quarter-turn. So the number of QT is at most three
less than twice the number of HT.

And, more important, have they been published, or does anyone have
them in an electronic format?

The bibliography in N_o_R_M_C mentions Thistlethwaite's algorithms as
being in typescript, but I don't know if they were available by
request, and I don't know anyone who has them. I don't know anything
about the improvements by Knuth's students, and there's nothing in
the R_C_C bibliography that looks like a Stanford tech report.

Dan     