I briefly spoke about the infamous Bomber Problem. If you would like to read more, see ABCs of the Bomber Problem and its Relatives, Weber, 2011. There is also a talk about this here. This problem has been open for over 35 years, and has been said by researchers who know it to be highly additive. It is very infuriating that for a problem with such a very simple dynamic programming equation an "obvious" fact cannot be proved or disproved.
$F(n,t)$ is the probability that a bomber plane, which is potentially attacked by enemy fighters $t$ times, can successfully defend itself against all such attacks, when starting with a supply of $n$ air-to-air missiles. The probability of surviving an attack by using $k$ missiles is $(1-\theta^k)$, where $0<\theta<1$.
$F(n,t) = q F(n,t-1) + p \max_{k\in\{1,\ldots,n\}}(1-\theta^k)F(n-k,t-1)$
$F(n,0)=1$
Conjecture B is that the maximizing $k$ is nondecreasing in $n$.
Many problems very similar to this are easy to solve.
Klinger and Brown's 1968 paper is here.
$F(n,t)$ is the probability that a bomber plane, which is potentially attacked by enemy fighters $t$ times, can successfully defend itself against all such attacks, when starting with a supply of $n$ air-to-air missiles. The probability of surviving an attack by using $k$ missiles is $(1-\theta^k)$, where $0<\theta<1$.
$F(n,t) = q F(n,t-1) + p \max_{k\in\{1,\ldots,n\}}(1-\theta^k)F(n-k,t-1)$
$F(n,0)=1$
Conjecture B is that the maximizing $k$ is nondecreasing in $n$.
Many problems very similar to this are easy to solve.
Klinger and Brown's 1968 paper is here.
