Regarding the polyforms with centers of gravity in the center of a
tile but no rotational symmetry: It seems to me that these are simply
the combination of two symmetric, extended (possibly disconnected)
polyforms in such a way as to make the combination not have symmetry.
The T made of squares consists of 3 squares in a line at the bottom,
and 3 squares in a vertical line above this; these groups have their
centers of gravity in the centers of two squares located two units
apart, and they are of the same size, so the center of gravity of the
whole thing is on the square between those two.
The polyiamond consists of a large triangle, with its center on the
central triangle, and another form consisting of two disconnected
triangles, whose center of gravity is in the same place. The polyhex
works in a very similar manner.
Many, many such shapes can be composed in this way. For instance,
start with two groups of squares connected like so:
#
#
#
###
Each block of 3 has its center of gravity on the central square, and
the center of gravity on the square diagonally between them.
This square itself is another polyform with center of gravity on this
square, so it can be added to unify the shape.
#
#
##
###
Regarding using Tipover to play out rolling-block puzzles: It is a
shame the Tipover board is only 6x6, as rolling block puzzles tend to
have larger grids. Or maybe we'll have to cut them apart and fasten
the grid parts together in the manner that Bluff combines deluxe
Scrabble boards to make the expandable board for his game Word Bluff.
4 of them would make a nice large grid.
Joseph DeVincentis
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I found a polyomino shape that works with no symmetry at all. It probably
isn't the smallest (16 squares), but I wanted to see if it could be done
at all:
X
X X XXX
X XXX
XXX XX
X
I hope that is readable.
-Jeremy Galvagni
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Dear Ed,
I'm sure you will hear this several times, but I noticed that the polyiamonds
are equivalent to a special case of the polyhexes. Just cut off 1/9 of a triangle
from each corner, and the resulting hexagon will have the same center of gravity.
Sincerely,
Bryce Herdt