Projective Plane of Order 12
(More a comment than answer:)

Finite projective planes exist for values of n which are powers of a prime, and there are infinitely many values of n which are ruled out by a theorem of R.H. Bruck and H. Ryser, which was generalized to block designs by Chowla:

http://en.wikipedia.org/wiki/Bruck%E2%80%93Chowla%E2%80%93Ryser_theorem

n = 10, as was stated, was solved (no plane exists) by a computer search so the first value of n not ruled out by Bruck-Ryser is n = 12. However, the computer work did not seem to give new insights as to whether or not there are only the prime power planes. What seems to be needed are new mathematical methods for insight into the commonly made conjecture that only the prime power planes exist.