Wang stated following conjecture about Wang tiles (which was proven false by R. Berger):
A finite set of plates [Wang's tiles] is solvable if and only if it has at least one periodic solution.
Also Wang said that if this conjecture holds, then there is an algorithm which decides whether finite set of Wang tiles is solvable. Here is a citation from Wang's publication (1961):
Thus, we proceed to build all possible rectangles from copies of the plates of different sizes, using smaller ones first. If 4.1.2 [the Wang's conjecture] is true, the process will always terminate in one of two ways: either at some stage we arrive at a cyclic rectangle and, therefore, the original set is solvable; or else we arrive at a size such that there is no rectangle of that size in which adjoining edges always have the same colors. The latter alternative is in fact a necessary and sufficient condition under which the original set is not solvable.
I don't understand how this statement is true. We can try enumerate all possible rectangle with increasing area. If set is solvable, then we will eventually find suitable ("cyclic") rectangle. However, if set is not solvable, how do we know when to stop? How we differentiate between case that rectangle does not exist and case that such rectangle has not been found yet?
Of course, I am missing something important... A lot of papers are referencing this fact, so it cannot be false.