This question came to my mind when I was contemplating on the tilling of bathroom's floor:
What is the maximum number of colors we can use for an optimum coloring of $N^2$ square sub-tiles of $N×N$ square block tile, which for every tiling of whole plane with these identical tile blocks(only four states of rotating $90$ degree blocks can be used) there exist a two-head infinite path for at least one color of the sub-tiles?
Note: A two-head infinite path of one coloured sub-tiles consist of elements of sub-tiles with the same color which for each element in the path there exist two another non-adjacent elements of the path from its 8 adjacent sub-tiles squares.(also it does not intersect with itself for example does not make circle)