2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/216102We 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.29 pages, color figures. v2: corrected minor misprints and added short appendix. v3: fig 16 fixedMathematical PhysicsCombinatoricstext