A bstract Using a one-way Monte Carlo algorithm from several different starting points, we get an approximation to the distribution of toric threefold bases that can be used in four-dimensional F-theory compactification. We separate the threefold bases into “resolvable” ones where the Weierstrass polynomials ( f, g ) can vanish to order (4 , 6) or higher on codimension-two loci and the “good” bases where these (4 , 6) loci are not allowed. A simple estimate suggests that the number of distinct resolvable base geometries exceeds 10 3000 , with over 10 250 “good” bases, though the actual numbers are likely much larger. We find that the good bases are concentrated at specific “end points” with special isolated values of h 1,1 that are bigger than 1,000. These end point bases give Calabi-Yau fourfolds with specific Hodge numbers mirror to elliptic fibrations over simple threefolds. The non-Higgsable gauge groups on the end point bases are almost entirely made of products of E 8 , F 4 , G 2 and SU(2). Nonetheless, we find a large class of good bases with a single non-Higgsable SU(3). Moreover, by randomly contracting the end point bases, we find many resolvable bases with h 1,1 ( B ) ∼ 50-200 that cannot be contracted to another smooth threefold base.