  The triangle packing number $\nu(G)$ of a graph $G$ is the maximum size of a
set of edge-disjoint triangles in $G$. Tuza conjectured that in any graph $G$
there exists a set of at most $2\nu(G)$ edges intersecting every triangle in
$G$. We show that Tuza's conjecture holds in the random graph $G=G(n,m)$, when
$m \le 0.2403n^{3/2}$ or $m\ge 2.1243n^{3/2}$. This is done by analyzing a
greedy algorithm for finding large triangle packings in random graphs.
