Color a square grid (7 by 7) with three colors while satisfying some rules and ensuring some level of uniformity / symmetry. This stems from one of my side projects about assigning hydrophilic (green) and hydrophobic (blue) chains on a nanocube surface. This problem is challenging because it's freaking hard to sample.
See the pic attached below for some general ideal.
- Curvature selectivity with blue color. Blue color (or hydrophobic species) has lightning lane pass in Disney. Start blue-coloring vertex sites first, then edges, finally faces.
- Region-specific density. Some regions can only have a certain number of colors (green + blue).
- Blue ratio. total number of blue on the grid cannot exceed some number.
- Be symmetric-ish. For each coloring move, aim for symmetry. Like color the next point as opposite as possible to the last (like opposite edge).
I am smart enough to realize this is basically a coloring problem. Apparently not smart enough to solve it in a mathematically glorious way. So here's the coding solution most likely to offend mathematicians. For my math friends please check out this note to learn more about proper q-coloring https://www.math.tau.ac.il/~peledron/homepage_files/Proper%20colorings%20IMU%20meeting.pdf You will not learn it here.
