  There are a number of results saying that for certain "path-following"
algorithms that solve PPAD-complete problems, the solution obtained by the
algorithm is PSPACE-complete to compute. We conjecture that these results are
special cases of a much more general principle, that all such algorithms
compute PSPACE-complete solutions. Such a general result might shed new light
on the complexity class PPAD.
  In this paper we present a new PSPACE-completeness result for an interesting
challenge instance for this conjecture. Chen and Deng~\cite{CD} showed that it
is PPAD-complete to find a trichromatic triangle in a concisely-represented
Sperner triangulation. The proof of Sperner's lemma --- that such a solution
always exists --- identifies one solution in particular, that is found via a
natural "path-following" approach. Here we show that it is PSPACE-complete to
compute this specific solution, together with a similar result for the
computation of the path-following solution of two-dimensional discrete Brouwer
functions.
