  We answer a question posed by Makowsky and Pnueli and show that the logic
$(\pm\mbox{HEX})^\ast[\mbox{FO}_s]$, where HEX is the operator (i.e., uniform
sequence of Lindstr\"om quantifiers) corresponding to the well-known {\bf
PSPACE}-complete decision problem Generalized Hex, collapses to the fragment
$\mbox{HEX}^1[\mbox{FO}_s]$ and, moreover, that this logic has a particular
normal form which results in the problem HEX being complete for {\bf PSPACE}
via quantifier-free projections with successor (HEX is the first ``natural''
problem to be shown to have this property). Our proof of this normal form
result is remarkably similar to Immerman's original proof that transitive
closure logic, $(\pm\mbox{TC})^\ast[\mbox{FO}_s]$, has such a normal form;
which is surprising given that $(\pm\mbox{HEX})^\ast[\mbox{FO}_s]$ captures
{\bf PSPACE} and $(\pm\mbox{TC})^\ast[\mbox{FO}_s]$ captures {\bf NL}. We also
show that $(\pm\mbox{HEX})^\ast[\mbox{FO}]$ does not capture {\bf PSPACE} and
that this logic does not have a corresponding normal form.
