## Ragged Rectangles (From the Pillowbook II)

In a ragged rectangle, the sides zigzag diagonally as in the left figure below, which shows a ragged rectangle of dimensions 6⨉7, and within a ragged 3⨉3 square. Note that the boundary changes directions at every unit step. These shapes make interesting candidates for regions to be tiled with polyominoes. The example in this post illustrates nicely how the interplay between making examples and generalization leads to a miniature theory.

To tile a shape like this with polyominoes, it will help to know its area in terms of unit squares. This is easy: If you color the squares in a ragged a⨉b rectangle beige and brown, you will get a⨉b squares of one color, and (a-1)⨉(b-1) squares of the other color.

This right away shows that it is hopeless to tile a ragged rectangle with dominoes. The first really interesting case is to use L-trominoes. The area formula implies that we need one dimension of the rectangle to be divisible by 3, and the other to leave remainder 1 after division by 3. Thus the shortest edge that can occur has length 3, and the other them must have length 3n+1. The figure below shows how to tile any ragged rectangle of dimensions 3x(3n+1) with L-trominoes:

The next shortest edge possible has length 4, and then the other edge must have length 3n. Again, a few experiments lead to a general pattern which shows that any 4x(3n) ragged rectangle can be tiled with L-trominoes:

This covers the two basic kinds of thin and arbitrarily long rectangles. What about larger dimensions? If we already have a ragged rectangle tiled with L-trominoes, we can put a frame around it that is also tiled with L-trominoes:

These three constructions together show that a ragged rectangle can be tiled with L-trominoes if and only if its area is divisible by 3. Next time we will see how this helps us to tile curvy rectangles with pillows.

## 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: