No pieces of the same color touching (side or corner).
No pieces of the same shape touching (side or corner).
This variation, and others (such as colors or shapes touching,
while the other does not and filling other shapes), were posed
in 2000 by Kate Jones
of Kadon Enterprises.
The 8x10 has 248 unique solutions. The 9x9 has 231.
90 of the 9x9's have 180-degree rotational symmetry.
Selections of them are presented below.
8 x 10
A remarkable solution with two congruent halves.
Every piece has piece of the same shape in the same location
in the opposite half.
A second half-and-half solution: each half has two of each shape.
The other two solutions are nearly divided in half along the
The two solutions could have their red and yellow
pieces interchanged OR just the red and yellow squares,
since each adjoins only blue and green, OR just the other four
pairs of pieces, since they form a chain which adjoins
only blue and green also.
The next two have just the red and yellow T's and Z's swapped.
The last two have just the red and blue bars swapped.
Fourth and fifth rows:
Show nine solutions with many pieces in common and just
a minority swapped in various ways.
9 x 9
Those with black frames come from the 90 solutions which
have symmetry: with a 180-degree rotation
and swapping of colors, the solution remains the same. The ones
with gray frames (and an asterisk) do not have this property.
A ring of pieces and a center. Alternating strips of red/green and
blue/yellow, aligned roughly from upper right to lower left.
A ring of blue and yellow, with red/green stripes above, below, and in the center.
Two "S"s, intersecting in the center. Blue/yellow as an S, red/green
resembling a "u n".
Two solutions which divide in half.
They differ by having the red and green bars swapped.
Alternating upper-right/lower-left strips, resembling a staircase.
All the square pieces in the interior.
The square pieces in the four corners.
The shape is symmetrical but the coloring is not. If the green and
blue squares were swapped, the coloring would be symmetrical, too.
Three 3x9 near-rectangles, with two actual 3x8 rectangles.
The last two have the same shape. By swapping some same-shape
pieces we get a solution grouped by color. Surely there are more
ways to explore these arrangements, but their investigation awaits