Let
X
X
denote the rational curve with
n
+
1
n+1
nodes obtained from the Riemann sphere by identifying 0 with
∞
\infty
and
ζ
j
\zeta ^j
with
−
ζ
j
-\zeta ^j
for
j
=
0
,
1
,
…
,
n
−
1
j=0,1,\dots ,n-1
, where
ζ
\zeta
is a primitive
(
2
n
)
(2n)
th root of unity. We show that if
n
n
is even, then
X
X
has no smooth Weierstrass points, while if
n
n
is odd, then
X
X
has
2
n
2n
smooth Weierstrass points.