  We address an analog of a problem introduced by Erd\H{o}s and Fishburn,
itself an inverse formulation of the famous Erd\H{o}s distance problem, in
which the usual Euclidean distance is replaced with the metric induced by the
$\ell^1$-norm, commonly referred to as the $\textit{taxicab metric}$.
Specifically, we investigate the following question: given $d,k\in \mathbb{N}$,
what is the maximum size of a subset of $\mathbb{R}^d$ that determines at most
$k$ distinct taxicab distances, and can all such optimal arrangements be
classified? We completely resolve the question in dimension $d=2$, as well as
the $k=1$ case in dimension $d=3$, and we also provide a full resolution in the
general case under an additional hypothesis.
