  A problem of optimally allocating partially effective ammunition $x$ to be
used on randomly arriving enemies in order to maximize an aircraft's
probability of surviving for time~$t$, known as the Bomber Problem, was first
posed by \citet{Klinger68}. They conjectured a set of apparently obvious
monotonicity properties of the optimal allocation function $K(x,t)$. Although
some of these conjectures, and versions thereof, have been proved or disproved
by other authors since then, the remaining central question, that $K(x,t)$ is
nondecreasing in~$x$, remains unsettled. After reviewing the problem and
summarizing the state of these conjectures, in the setting where $x$ is
continuous we prove the existence of a ``spend-it-all'' region in which
$K(x,t)=x$ and find its boundary, inside of which the long-standing, unproven
conjecture of monotonicity of~$K(\cdot,t)$ holds. A new approach is then taken
of directly estimating~$K(x,t)$ for small~$t$, providing a complete small-$t$
asymptotic description of~$K(x,t)$ and the optimal probability of survival.
