I just watched a TV commercial that showed laying hexagonal tile in a bathroom, and the tile had a gold border. While the border on the tile in the commercial was completely surrounding the circumference of the tile, I was wondering how many edges of the hexagonal tile would need to be gold, in order to form a pattern of tiles that had at least one gold edge on each of it's neighboring tiles, without the neighboring tile having a corresponding gold edge.
Assuming gold is expensive, and the gold edge on the hexagonal tile is solid, how few edges of a hexagonal tile could have a gold edge, and still be able to make a pattern with other tiles, with a fully golden border between all tiles, starting from a center tile and not counting room borders, etc?
Note that to have all gold borders, all edges must have at least 1 hexagon edge to be gold colored. Since each edge is only bordered by two hexagons, at least half the hexagon edges must be gold. Since there are six hexagon edges per hexagon, on average, each hexagon must have at least 3 gold edges. I'll leave the proof that a 3 gold edged hexagon can indeed tile the way you want it to.