Rounded Hexominoes
There are 580 Rounded Hexominoes. Unfortunately the number of pieces,
which have an odd number of white squares under checkerboard coloring, is odd. Therefore a construction
with two axis of symmetry isn't possible.
What can we do?
If we take a 83x42 rectangle and leave a hole shaped like an additional even piece,
we have 83x42/2=1743 white places minus the even number of white places of the hole. So the number of white places is
odd in consistence with the pieces.
We can add a monomino to get a 59x59 square with 1741 black and 1740 white places. The position of the monomino
must be a white place to leave an odd number of white places for the other pieces.
Center or corner positions aren't possible.
To get two axis of symmetry we can break the grid structure. First we construct a 61x61 square
with a rectangular hole of size 13x19. The 3474 places can be filled with 579 pieces leaving one odd piece.
If this piece is symmetric, it can be placed out of the grid to get a symmetric construction as a whole.
Among the 1105 one-sided Rounded Hexominoes are an odd number of odd pieces, too.
As with the two-sided pieces I constructed a 93x93 - 45x45 ring for 1104 pieces and placed a symmetric
odd piece in the center but out of the grid. Since only symmetry between the squares is regarded,
we have two axis of symmetry. Due to the large number of
pieces I decided to solve one part of the pattern using no symmetric pieces. Then this solution was reflected
and the remaining area was filled with the symmetric and the remaining couples of asymmetric pieces.
A subset of the Rounded Hexominoes are the 51 Rounded Tridominoes. With this set I got three 6x17 rectangles.
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