Some thoughts on Local Maxima
-----------------------------

I have verified that the position I call "6 H order 2 type 2"
is a local maximum.

p175 6 H order 2 type 2  T2 B2 L2 T2 D2 L2 F2 T2              (8)

A rare example of a pattern with symmetry level 2, perhaps even
the only one of this type, and also the most symmetric of the
6 H patterns.

Nothing new here, as this was noted formerly by David Singmaster
in one of the Cubic Circulars and by Jim Saxe & Dan Hoey in the
archives.

Somewhat more interesting is the conclusion that the pattern
4 H order 2, or H's on the F,R,B,L faces (oriented like the
letter H) is also a local maximum, at least in the square's
group.

p160 4 H order 2 Type 2  B2 D2 (L2 R2 F2) ^2 T2 F2           (10)

From the archives:

We include a description of 71 local maxima, which we believe
to be all of the local maxima that can be proven using known
techniques other than exhaustive search.

Oh well, I used an exhaustive search. p160 is 10 moves long in the htw
metric, and each of the moves ( T2, D2, F2, B2, L2, R2 ) all bring
one to a position requiring nine 180 degree twists, thusly....

4 H + T2 =      L2, R2, F2   B2, T2, L2   R2, B2, F2   (9)
4 H + D2 =      L2, R2, F2   B2, D2, L2   R2, B2, F2   (9)
4 H + F2 =      B2, D2, L2   R2, F2, L2   R2, F2, T2   (9)
4 H + B2 =      F2, D2, L2   R2, F2, L2   R2, F2, T2   (9)
4 H + L2 =      R2, D2, L2   F2, B2, L2   F2, B2, T2   (9)
4 H + R2 =      L2, D2, L2   F2, B2, L2,  F2, B2, T2   (9)

----------------------------------------------------------------------

I did discover an interesting property of the "Cube in a cube" pattern
I didn't notice before.

p7a  Cube in a cube   U2 F2 R2 U3 L2 D1 (B1 R3) ^3 B1 D3 L2 U3  (15)

Let's say you are entertaining some cube guests at a cube party and
the topic is (cube) patterns. Your guests are impressed with the
efficiency of the well-memorized process. You would like to go on
to the next pattern but you don't quite remember how the inverse
goes. No problem! Rotate the whole cube so TOP becomes BACK then
BACK becomes DOWN, and finally FRONT becomes RIGHT. Simply repeat
the process p7a and your reputation as a cube expert is saved. ;->

This same idea works for the 6 X order 3 pattern as well.

And now for an unsymmetric local maximum!!

(Just kidding)

-> Mark <-