  In 1956, Tutte proved the celebrated theorem that every 4-connected planar
graph is hamiltonian. This result implies that every more than
$\frac{3}{2}$-tough planar graph on at least three vertices is hamiltonian and
so has a 2-factor. Owens in 1999 constructed non-hamiltonian maximal planar
graphs of toughness arbitrarily close to $\frac{3}{2}$ and asked whether there
exists a maximal non-hamiltonian planar graph of toughness exactly
$\frac{3}{2}$. In fact, the graphs Owens constructed do not even contain a
2-factor. Thus the toughness of exactly $\frac{3}{2}$ is the only case left in
asking the existence of 2-factors in tough planar graphs. This question was
also asked by Bauer, Broersma, and Schmeichel in a survey. In this paper, we
close this gap by constructing a maximal $\frac{3}{2}$-tough plane graph with
no 2-factor, answering the question asked by Owens as well as by Bauer,
Broersma, and Schmeichel.
