A well-known algorithmically unsolvable decision problem. Given a finite set of “dominoes” of the form shown in Fig. a, with u and v being strings, the question is whether or not one can form a sequence, as shown in Fig. b, such that reading all the us in order gives the same string as reading all the vs. Fig. c shows such a sequence, where Λ is the empty string. Even though there are only finitely many different dominoes given, there is an infinite supply of duplicates for each one; the same domino can thus be used more than once in the sequence. Dominoes cannot be inverted.
Depending on the dominoes given, it is sometimes obvious that the answer to the question is “no”. However there is no algorithm that can discover this in all cases.
Post's correspondence problem.