# Tiling Problem with Patterns and Colors

My wife proposed an interesting tiling problem to me. The specific problem she proposed is:

I have tiles of 6 different patterns. Each pattern is in 3 different colors. I want to make a quilt of size 6 by 7, where "the patterns don't touch", meaning,

• The same pattern can not touch at all
• Like colors, but different patterns can touch only on a diagonal

I have trouble encoding those two rules into a relationship I can work with. Can someone help me solve this specific problem, and frame it in a general way?

• There are $18$ combinations of pattern and color. Do you want to use $2$ or $3$ of each? – Ross Millikan Jan 4 at 3:55
• One solution is to make a 2×2 square of {{A1, B2}, {C2, D1}} and tile it to any desired size, using only four patterns A, B, C, D and two colors 1, 2. This follows your rules but probably isn’t the solution you want; if so, can you make a rule that explains why it isn’t the solution you want? – Anders Kaseorg Jan 4 at 5:24

A B C D E F     1 2 3 1 2 3     A1 B2 C3 D1 E2 F3