This is meant to encourage Kevin Costello and Roland Bacher to do more numerical simulations to bound a(3).

I was impressed that Kevin found a maximum as large as 9/50, and that with 5 ones.
I then computed the four essentially different forms that the determinant (or its negative) could take coming from a 3x3 matrix with 5 ones.  Except for one of them, I found I could simplify the expressions by using the substitutions z_i = a_i - 1.  I think when these forms are investigated, an explanation for the non-1 entries will be forthcoming. I also wonder if perhaps the non-one entries can be replaced with 4/9, -1/9, -5/9, and -1, which if it works would be explained by the heuristic of spacing out the non-1 values.  A similar analysis might be done using 4 ones instead of 5.

Gerhard "Ask Me About System Design" Paseman, 2011.12.06