  The set of possible spectra (\lambda,\mu,\nu) of zero-sum triples of
Hermitian matrices forms a polyhedral cone. We give a complete determination of
its facets, finishing a long story with recent highlights by [Helmke-Rosenthal,
Klyachko, Belkale]. We introduce_puzzles_, which are new combinatorial gadgets
to compute Grassmannian Schubert calculus, and will probably be the main point
of interest for many readers. As the proofs indicate, the Hermitian sum problem
is very naturally studied using puzzles directly, and their connection to
Schubert calculus is quite incidental to our approach. In particular, we get
new, puzzle-theoretic, proofs of the results in [H,Kly,HR,Be]. Along the way we
give a characterization of ``rigid'' puzzles, which we use to prove a
conjecture of W. Fulton: ``if for a triple of dominant weights \lambda,\mu,\nu
of GL(n,C) the irreducible representation V_\nu appears exactly once in
V_\lambda tensor V_\mu, then for all N\in \naturals, V_{N\lambda} appears
exactly once in V_{N\lambda} tensor V_{N\mu}.''
