  The Belkale-Kumar product on H*(G/P) is a degeneration of the usual cup
product on the cohomology ring of a generalized flag manifold. In the case
G=GL_n, it was used by N. Ressayre to determine the regular faces of the
Littlewood-Richardson cone.
  We show that for G/P a (d-1)-step flag manifold, each Belkale-Kumar structure
constant is a product of d(d-1)/2 Littlewood-Richardson numbers, for which
there are many formulae available, e.g. the puzzles of [Knutson-Tao '03]. This
refines previously known factorizations into d-1 factors. We define a new
family of puzzles to assemble these to give a direct combinatorial formula for
Belkale-Kumar structure constants.
  These "BK-puzzles" are related to extremal honeycombs, as in
[Knutson-Tao-Woodward~'04]; using this relation we give another proof of
Ressayre's result.
  Finally, we describe the regular faces of the Littlewood-Richardson cone on
which the Littlewood-Richardson number is always 1; they correspond to nonzero
Belkale-Kumar coefficients on partial flag manifolds where every subquotient
has dimension 1 or 2.
