Tiling Problem with Patterns and Colors

Question

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?

Answer 1

You do not need to consider both rules at the same time. Tile the patterns (A,B,C,D,E,F) so that the first rule is satisfied. Tile the colors (1,2,3) so that the second rule is satisfied. Then combine the two to get the desired result.

A B C D E F     1 2 3 1 2 3     A1 B2 C3 D1 E2 F3
C D E F A B  +  2 3 1 2 3 1  =  C2 D3 E1 F2 A3 B1
E F A B C D     3 1 2 3 1 2     E3 F1 A2 B3 C1 D2