# Grain (Polyforms I)

Go and purchase four types of wood, in different colors, with distinctive grain. Cut it into four different sizes, 2×1, 1×2, 3×1, and 1×3 inches long and tall, of the same thickness, and so that pieces of the same wood type all have the same dimensions and orientation of the grain. Here is my humble illustration of what will look much more beautiful in reality:

These are grained dominoes and polyominoes, a special case of more general grained polyominoes. In puzzles with polyominoes, you are typically tasked with tiling a certain shape with certain types of polyominoes. The presence of grain allows for variations of the rules. For instance, guided by esthetically considerations, we might demand that the grain needs to be horizontal, and that no two tiles of equal type touch along an edge or part of an edge.

Here, for instance, only the first tiling of the 5×3 rectangle follows the rules: The second has two tiles of the same kind meeting at a part of their edge, while the third does not preserve the grain.

Now for the puzzles: Above is one of the four different ways to tile the 8×7 rectangle, not counting symmetries. At this point, solving such puzzles involves mainly trial and error. The divide-and-conquer strategy that works for the ungrained case, namely using small, already tiled, rectangles to tile larger rectangles, does not work here, because lining up tiled rectangles will usually violate the rules.

There is, however, some interesting structure emerging, that one can see better in a coloring that distinguishes more clearly between horizontal and vertical tiles. In the solution of the 11×9 rectangle (one of two), one can see bands of horizontal (blue) tiles and of vertical (red) tiles that extend from the top to the bottom edge.

Finally, below is the only solution for tiling the 13×13 square, ignoring symmetries of course: