Maximal partial spreads and the modular n-queen problem

We prove that for any integer n in the interval there is a maximal partial spread of size n in PG (3, q) where q is odd and q7. We also prove that there are maximal partial spreads of size (q²+3)/2 when gcd(q+1,24)=2 or 4 and of size (q²+5)/2 when gcd(q+1,24)=4.