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.

Raggedex 01

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:

ragged3-01.jpg

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:

Ragged4

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:

Raggedframe

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.

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