  We demonstrate that $\sim10\,\textrm{s}$ after the core-collapse of a massive
star, a thermonuclear explosion of the outer shells is possible for some
(tuned) initial density and composition profiles, assuming that the neutrinos
failed to explode the star. The explosion may lead to a successful supernova,
as first suggested by Burbidge et al. We perform a series of one-dimensional
(1D) calculations of collapsing massive stars with simplified initial density
profiles (similar to the results of stellar evolution calculations) and various
compositions (not similar to 1D stellar evolution calculations). We assume that
the neutrinos escaped with a negligible effect on the outer layers, which
inevitably collapse. As the shells collapse, they compress and heat up
adiabatically, enhancing the rate of thermonuclear burning. In some cases,
where significant shells of mixed helium and oxygen are present with
pre-collapsed burning times of $\lesssim100\,\textrm{s}$ ($\approx10$ times the
free-fall time), a thermonuclear detonation wave is ignited, which unbinds the
outer layers of the star, leading to a supernova. The energy released is small,
$\lesssim10^{50}\,\textrm{erg}$, and negligible amounts of synthesized material
(including $^{56}$Ni) are ejected, implying that these 1D simulations are
unlikely to represent typical core-collapse supernovae. However, they do serve
as a proof of concept that the core-collapse-induced thermonuclear explosions
are possible, and more realistic two-dimensional and three-dimensional
simulations are within current computational capabilities.
