We solve two related extremal-geometric questions in the $n-$dimensional space $\mathbb{R}^n_{\infty}$ equipped with the maximum metric. First, we prove that the maximum size of a right-equidistant sequence of points in $\mathbb{R}^n_{\infty}$ equals $2^{n+1}-1$. A sequence is right-equidistant if each of the points is at the same distance from all the succeeding points. Second, we prove that the maximum number of points in $\mathbb{R}^n_{\infty}$ with pairwise odd distances equals $2^n$. We also obtain partial results for both questions in the $n-$dimensional space $\mathbb{R}^n_1$ with the Manhattan distance.