  We study the existence of Neumann eigenfunctions which do not change sign on
the boundary of some special domains. We show that eigenfunctions which are
strictly positive on the boundary exist on regular polygons with at least 5
sides, while on equilateral triangles and cubes it is not even possible to find
an eigenfunction which is nonnegative on the boundary.
  We use analytic methods combined with symmetry arguments to prove the result
for polygons with six or more sides. The case for the regular pentagon is
harder. We develop a validated numerical method to prove this case, which
involves iteratively bounding eigenvalues for a sequence of subdomains of the
triangle. We use a learning algorithm to find and optimize this sequence of
subdomains, making it straightforward to check our computations with standard
software.
