I just got some very nice examples of circle (group-theory) puzzles -
similar to the Lorente Grill puzzles shown in Hofstadters' Sci.Am.
article in July, 1982 (p. 26). They are made by Douglas Engel, under
the name of General Symmetrics, Inc, 2935 W. Chenango, Englewood, Co,
80110. He is selling some as experimental items, and one version he
hopes to have marketed "soon". They all consist of two intersection
circles, each of which rotate; I have 3 kinds: the "21 piece", similar
to 2/3 of Lorente's Trebol puzzle; the "35 piece", similar to the
one in the upper right corner of the article "Another Grill puzzle by
Lorente", and a "19 piece", similar to the 21 piece, except the circles
intersect in the center. Each of these are quite different from each other;
the 35 piece has 3 different pieces which rotate in 4 different cycles
(compare to the Cubes' 3 kinds of pieces - edge, corner, and center).
In the 35 piece, a straightforward "in in out out" (R'LRL') type move
will move the centers in a 3-cycle, the "square" pieces in a pair of
2-cycles, and the "triangles" in 2 separate 5-cycles.
(Doug is selling these at $10.00 for the 19 or 21 piece, $20 for the
Doug Engel has also written a paper called "Some Problems Suggested
By Circle Puzzles", mostly asking questions about various kinds of
symmetry in this type, whether there are an infinite number of them,
etc. I wonder what the best way to talk about them and analyze them
from a group theory perspective is. I'll try to put in more information
and theories as I work on them; I'd like to hear any thoughts or
suggestions. Right now, I want to go home for dinner.