Consider a maximally constrained soluble 1-SAT problem with n variables. As described in §3, in such a problem each clause involves a separate variable and there is exactly one solution. To show that the algorithm works for all n, we evaluate Eq. (12).
For each assignment s, from Eq. (13). Then for each assignment r, . Each s in this sum can be characterized by
Substituting the values from Eq. (13) and (14), gives
This gives where if x=y and 0 otherwise by use of the identity
Thus, has all its amplitude in the state with no conflicts, i.e., the unique solution. A measurement made on this final state is guaranteed to produce a solution.