Wang Tiles

Wang Tiles (or Wang dominoes) are a class of formal mathematical systems first proposed by Hao Wang in 1961. They consist of equal-sized square tiles with colored edges that are arranged side-by-side on a regular grid.

When placing tiles to cover an infinite plane, you must strictly follow three structural constraints:

• Orientation: Tiles must be placed with a fixed orientation and cannot be rotated or reflected.
• Edge Matching: Adjoining edges of touching tiles must share the exact same color.
• Grid Alignment: Every tile must occupy exactly one square in a standard square grid.

Busy Beaver function

Define the score of a set of Wang tiles as the largest d such that it can tile a dxd square, or 0 if there is no largest d.

Then BB_WT(n) = the maximum score over all n-tile sets.

The current format to describe a set of tiles is a list of tuples that have 4 nonegative numbers, each representing a side color. Example: [TOP,RIGHT,BOTTOM,LEFT]

n	Value	Champion
1	1	[1,0,0,0]
2	2	[1,0,2,0] [2,0,3,0]
3	3	[1,0,2,0] [2,0,3,0] [3,0,4,0]
4	4	[1,0,2,0] [2,0,3,0] [3,0,4,0] [4,0,5,0]
5	≥ 6	[1,0,2,0] [2,0,3,0] [3,0,4,0] [4,6,1,5] [4,5,2,6]
6	≥ 8	[1,0,2,0] [2,0,3,0] [3,0,4,0] [4,0,5,0] [5,7,1,6] [5,6,2,7]
7	≥ 10
8	≥ 12
9	≥ 14
10	≥ 212

Analysis

TODO