  We study the best coapproximation problem in the Banach space $ \ell_1^n, $
by using Birkhoff-James orthogonality techniques. Given a subspace
$\mathbb{{Y}}$ of $\ell_1^n$, we completely identify the elements $x$ in
$\ell_1^n,$ for which best coapproximations to $x$ out of $\mathbb{{Y}}$ exist.
The methods developed in this article are computationally effective and it
allows us to present an algorithmic approach to the concerned problem. We also
identify the coproximinal subspaces and co-Chebyshev subspaces of $\ell_1^n$.
