If there is any popular response I will write
a short introduction to Group Theory for Cubists.
= - = - = - = - = - = - = - = - = - = - = - = - =
Problem: the cube is known to have "small mountains",
states that are locally maximally distant from the
home state but not globally so. The Pons Asinorum
is an example: it has been proved (a refreshing relief
from unmitigated conjecture) to be 12 qtw from home
and all its neighbors are closer.
The cube's Cayley graph in its six generators is vertex-
transitive (all the states look the same) like all Cayley
Graphs. In addition, because all the generators are
conjugate in the big group (quarter-twists plus whole-
cube motions) the Cayley graph is edge-transitive
(all transitions between adjacent states look the same).
Can anyone find a small example of an edge-transitive
graph that has local maxima that are not global?