In an article describing the twelve balls weighing problem, the author mentions a solution that involves the finite projective plane of order 3, discovered by Rick Wilson. Does anyone know what this solution could have been?

Will Orrick is right, the problem is solved by exhibiting a matrix $3\times 12$ with entries in $\{1,0,1\}$ where all columns are pairwise independent and the row sums are zero, as mentioned in Wilson's book. In general you can solve the $\frac{3^r1}{2}1$ coin problem using $r$ weighings. You need to use one of the generator matrices of the simplex code, so the columns are given by the points in the projective space $P(r1,3)$ (you throw out the point at infinity) and by induction show that the choices can be made to arrange the zero sum rows. For the case of 12 coins it suffices to consider the projective plane, and you get a constant weight code and thus end up weighing groups of 4 coins each time. 

