From:

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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 <-