  Tsirelson's norm $\|\cdot \|_T$ on $c_{00}$ is defined as the supremum over a
certain collection of iteratively defined, monotone increasing norms $\|\cdot
\|_k$. For each positive integer $n$, the value $j(n)$ is the least integer $k$
such that for all $x \in \mathbb{R}^n$ (here $\mathbb{R}^n$ is considered as a
subspace of $c_{00}$), $\|x\|_T = \|x\|_k$. In 1989 Casazza and Shura asked
what is the order of magnitude of $j(n)$. It is known that $j(n) \in
\mathcal{O}(\sqrt{n})$. We show that this bound is tight, that is, $j(n) \in
\Omega(\sqrt{n})$. Moreover, we compute the tight order of magnitude for some
norms being modifications of the original Tsirelson's norm.
