Let the Post Correspondence problem with input
$$K = ((x_1, y_1), . . . ,(x_n, y_n)), x_i, y_i ∈ \Sigma^*$$
for $$ i = 1, . . . , n $$
Find a concrete solution for the input
$$K = ((001,0), (01, 011),(01, 101),(10, 001))$$
The Post Correspondence problem is undecidable so it's really hard to find a concrete solution. I've tried brute forcing it but it gets frustrating after a few tries.
Is there a logical way to deduce that K is a solution, rather than finding a pattern that works?