Component Colorings II: Diamonds and Triamonds

Here’s a nice coincidence: the numbers of tri-diamonds and di-triamonds are both 9, which is the right amount to tile a regular hexagon of side length 3. And both sets can! Behold the di-triamonds:

3-coloring the triamonds here isn’t hard. The tiling seems to want to have a bunch of points where four triamonds meet, which disrupts chains of forced colors. The challenge is adding more challenges on top of 3-coloring. I suspect that there is no strict 3-coloring of the triamonds. One possibility is a sort of meta-coloring of the di-triamonds where no two di-triamonds with the same color pair may be adjacent. The above diagram doesn’t qualify because there are blue-red di-triamonds touching each other. Problem #60: Find such a meta-coloring.

The diamonds in the tri-diamonds are even easier to 3-color. Enough so that 3-coloring them so each tri-diamond has all three colors (the equivalent of the poorly thought out problem #58 with the tri-dominoes) was no challenge at all. Perhaps there is something to be done with symmetry. Notice that, ignoring color and the tri-diamond outlines, the diamonds in the figure below have an an axis of reflection symmetry. I wonder if, for some tiling, some form of symmetry on the diamonds is possible where colors are included.

The meta-coloring idea above suggests a way to salvage Problem #58. Instead of a three coloring of the dominoes in a tri-domino tiling, we could look for a 4-coloring of the dominoes where every tri-domino contains 3 of the 4 colors, and there is simultaneously a meta-4-coloring of the tri-dominoes where no two adjacent tri-dominoes are missing the same color.