TITLE: Conjecture on NP-completeness of tesselation of Wang Tile up to finite size
QUESTION [9 upvotes]: Motivated by these following questions on tessellation:
coloring in lattice
Reference for Wang Tile
Computational approach deciding whether a set of Wang Tile could tile the space up to some size
Could we actually show that the problem of trying to tile a finite size square with a given set of Wang tile to be NP-complete?

REPLY [6 votes]: The problem is also trivially NP-complete without an anchor tile. The trick is that we can easily force any tile to become an anchor tile. First, take a set of tiles that have a unique way of tiling an $n\times n$ square, e.g., by putting (respectively matching) $(i,j)$'s on their sides where $1\le i,j\le n$. Then take the "direct product" of this tile set with, e.g., the tile set described by Joel except for the position where you want your anchor tile, there keep only that. This gives a suitable tile set.
ps. I used to give this as an exercise for undergrads and some of them always solved it.