  You play the following game: you start out with $n$ coins that all have
probability $p$ to land heads. You toss all of them and you then need to set
aside at least one of them, which will not be tossed again. Now you repeat the
process with the remaining coins. This continues (for at most $n$ rounds) until
all coins have been set aside. Your goal is to maximize the total number of
heads you end up with. In this paper we will prove that there exists a constant
$p_0 \approx 0.5495021777642$ such that, if $p_0 < p \le 1$ and the number of
remaining coins is large enough, then it is optimal to set aside exactly one
coin every round, unless all coins landed heads. When $\frac{1}{2} \le p \le
p_0$, it is optimal to set aside exactly one coin every round, unless at most
one coin came up tails. Let $v_{n,p}$ be the expected number of heads you
obtain when using the optimal strategy. We will show that $v_{n,p}$ is larger
than or equal to $v_{n-1,p} + 1$ for all $n \ge 3$ and all $p \ge \frac{1}{2}$.
When $p < \frac{1}{2}$ there are infinitely many $n$ for which this inequality
does not hold. Finally, we will see that for every $p$ with $0 < p \le 1$ the
sequence $n - v_{n,p}$ is convergent.
