Puzzles in Schubert calculus were originally developed by Knutson and Tao as combinatorial objects for computing the expansion of the product of two Grassmannian Schubert classes. I will present a puzzle rule for restricting type A Schubert classes for the Grassmannian to type C in equivariant cohomology. The proof uses the machinery of quantum integrable systems. I will then describe work in progress toward an extension of this rule to cotangent bundles in the context of Lagrangian correspondences between symplectic resolutions, and Maulik—Okounkov classes. This joint work with A. Knutson and P. Zinn-Justin.