  Vakil studied the intersection theory of Schubert varieties in the
Grassmannian in a very direct way: he degenerated the intersection of a
Schubert variety X_mu and opposite Schubert variety X^nu to a union {X^lambda},
with repetition. This degeneration proceeds in stages, and along the way he met
a collection of more complicated subvarieties, which he identified as the
closures of certain locally closed sets.
  We show that Vakil's varieties are _positroid varieties_, which in particular
shows they are normal, Cohen-Macaulay, have rational singularities, and are
defined by the vanishing of Pl\"ucker coordinates [Knutson-Lam-Speyer]. We
determine the equations of the Vakil variety associated to a partially filled
``puzzle'' (building on the appendix to [Vakil]), and extend Vakil's proof to
give a geometric proof of the puzzle rule from [Knutson-Tao '03] for
equivariant Schubert calculus.
  The recent paper [Anderson-Griffeth-Miller] establishes (abstractly; without
a formula) three positivity results in equivariant K-theory of flag manifolds
G/P. We demonstrate one of these concretely, giving a corresponding puzzle
rule.
