Many Questions, the Same Answer

Why should candy schedules, binary matrices, graph matchings, domino tilings, lattice points, and a statistic on colored polyominoes lead to the same numbers? We explain these connections through the central Delannoy numbers. Explicit bijections translate individual solutions between most of the models, while a generating-function argument treats the weighted polyomino identity. The emphasis is on transferring a solution: a recurrence, formula, or refinement proved in one setting becomes useful in several others. We also make precise several statements that are incomplete as commonly phrased, concerning zebras, Aztec diamonds, comb matchings, and hyperoctahedra, and we distinguish the unrestricted candy rule from the restricted rule studied in [1]. The result is a connected account of ten problems, written for readers comfortable with binomial coefficients and elementary algebra, with a more advanced section on generating functions.