  Let $L$ be a finite-dimensional real normed space, and let $B$ be the unit
ball in $L$. The sign sequence constant of $L$ is the least $t>0$ such that,
for each sequence $v_1, \ldots, v_n \in B$, there are signs $\varepsilon_1,
\ldots, \varepsilon_n \in \{-1, +1\}$ such that $\varepsilon_1 v_1 + \ldots +
\varepsilon_k v_k \in t B$, for each $1 \leq k \leq n$.
  We show that the sign sequence constant of a plane is at most $2$, and the
sign sequence constant of the plane with the Euclidean norm is equal to
$\sqrt{3}$.
