# Just Two (From the Pillowbook III)

A while ago we learned how to tile curvy 3×3 squares with pillows. Most of the possible tilings need at least three different kinds of pillows. The only way to tile a curvy 3×3 square was using the Blue and Yellow. This changes when we look at tilings of larger curvy rectangles. For instance, below is a tiling of a 5×7 rectangle with Red and Blue: I have overlaid the curvy rectangle with a 3×4 ragged rectangle, tiled by L-trominoes. Each L-tromino is replaced with a cluster of 3 Reds, and the T-junctions of the L-tromino tiling are filled with Blues. As we have seen that every ragged rectangle whose area is divisible by 3 can be tiled with L-trominos. This gives plenty of examples. In fact, every
tiling of a curvy rectangle with just Blue and Red comes from an L-tromino tiling of a ragged rectangle.

To see this, one can look at the possible ways a red pillow can be surrounded by blue and red pillows, and one almost finds that each red pillow belongs to a unique L-tromino. There is one exception that causes a little bit of headache that leads to circular clusters as in the example below (dark red and pink). But one can show anyway that such clusters can be tiled (in multiple ways) with L-trominoes.

Another challenge is to find a curvy square that can be tiled with just Blue and Red. That this is impossible follows from the deficit formula: We need to have the area r+b to be a square and r-2b =2 for r Reds and b Blues. But this implies that -1 is a square modulo 3, which is false.

Tiled squares are possible with other two color combinations. The example of a tiling of curvy 5×5 square tiled by Yellow and Green is deceivingly simple. The next case of the 7×7 square below is more complicated. Can you find a pattern? The only really simple case is tilings by Yellow and Blue. All curvy rectangles can be tiled, and in only just one way. A mean little exercise is to ask somebody to tile any curvy rectangle with Green and Red. There is no solution, because the deficit formula tells us that r-g=2, but r+g needs to be odd, because curvy rectangles have odd area.

There are a few more color combinations to consider. For instance using either orange or purple pillows together with a second color is impossible. By the deficit formula, this would require to be either a single yellow pillow or precisely two red pillows. For purple this means that there would be a line of purple pillows through the rectangle. But such lines always end at one concave and one convex segment, which can’t be. For orange this would require al least two orange corner pillows, which also doesn’t work.