We show that interlacing triangular arrays, introduced by Aggarwal-Borodin-Wheeler to study certain probability measures, can be used to compute structure constants for multiplying Schubert classes in the $K$-theory of Grassmannians, in the cohomology of their cotangent bundles, and in the cohomology of partial flag varieties. Our results are achieved by establishing a splitting lemma, allowing for interlacing triangular arrays of high rank to be decomposed into arrays of lower rank, and by constructing a bijection between interlacing triangular arrays of rank 3 with certain proper vertex colorings of the triangular grid graph that factors through generalizations of Knutson-Tao puzzles. Along the way, we prove one enumerative conjecture of Aggarwal-Borodin-Wheeler and disprove another.