  A task is randomly drawn from a finite set of tasks and is described using a
fixed number of bits. All the tasks that share its description must be
performed. Upper and lower bounds on the minimum $\rho$-th moment of the number
of performed tasks are derived. The case where a sequence of tasks is produced
by a source and $n$ tasks are jointly described using $nR$ bits is considered.
If $R$ is larger than the R\'enyi entropy rate of the source of order
$1/(1+\rho)$ (provided it exists), then the $\rho$-th moment of the ratio of
performed tasks to $n$ can be driven to one as $n$ tends to infinity. If $R$ is
smaller than the R\'enyi entropy rate, this moment tends to infinity. The
results are generalized to account for the presence of side-information. In
this more general setting, the key quantity is a conditional version of R\'enyi
entropy that was introduced by Arimoto. For IID sources two additional
extensions are solved, one of a rate-distortion flavor and the other where
different tasks may have different nonnegative costs. Finally, a divergence
that was identified by Sundaresan as a mismatch penalty in the Massey-Arikan
guessing problem is shown to play a similar role here.
