# 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

## 1 Answer

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