  We provide direct proofs of product and coproduct formulae for Schur
functions where the coefficients (Littlewood--Richardson coefficients) are
defined as counting puzzles. The product formula includes a second alphabet for
the Schur functions, allowing in particular to recover formulae of
[Molev--Sagan '99] and [Knutson--Tao '03] for factorial Schur functions. The
method is based on the quantum integrability of the underlying tiling model.
