  The simplest cubic fields $\mathbb{Q}(\rho)$ are generated by a root $\rho$
of the polynomial $x^3-ax^2-(a+3)x-1$ where $a\geq -1$. In this paper, we will
show that the Pythagoras number of the order $\mathbb{Z}[\rho]$ is equal to $6$
for $a\geq 3$.
