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