  We study a "pen testing" problem, in which we are given $n$ pens with unknown
amounts of ink $X_1, X_2, \ldots, X_n$, and we want to choose a pen with the
maximum amount of remaining ink in it. The challenge is that we cannot access
each $X_i$ directly; we only get to write with the $i$-th pen until either a
certain amount of ink is used, or the pen runs out of ink. In both cases, this
testing reduces the remaining ink in the pen and thus the utility of selecting
it.
  Despite this significant lack of information, we show that it is possible to
approximately maximize our utility up to an $O(\log n)$ factor. Formally, we
consider two different setups: the "prophet" setting, in which each $X_i$ is
independently drawn from some distribution $\mathcal{D}_i$, and the "secretary"
setting, in which $(X_i)_{i=1}^n$ is a random permutation of arbitrary $a_1,
a_2, \ldots, a_n$. We derive the optimal competitive ratios in both settings up
to constant factors. Our algorithms are surprisingly robust: (1) In the prophet
setting, we only require one sample from each $\mathcal{D}_i$, rather than a
full description of the distribution; (2) In the secretary setting, the
algorithm also succeeds under an arbitrary permutation, if an estimate of the
maximum $a_i$ is given.
  Our techniques include a non-trivial online sampling scheme from a sequence
with an unknown length, as well as the construction of a hard, non-uniform
distribution over permutations. Both might be of independent interest. We also
highlight some immediate open problems and discuss several directions for
future research.
