Star Theorem Patterns Relating to 2n-gons in Pascal's Triangle - and More

We first pose a Sudoku-type puzzle, involving lattice points in Pascal's Triangle, and lines passing through them. Then, through the use of an expanded notation for the binomial coefficient ((n)(r)), we exploit the geometry of Pascal's Triangle and produce a non-computational solution to the well-known Star of David pattern about products of two subsets of the 6 nearest neighbors to a given binomial coefficient. We generalize and obtain analogous patterns, with similar geometric proofs, for 2n-gons where the entries at the lattice points of the symmetric (but not necessarily triangular) arrays are what we call separable functions (e.g., binomial coefficients, Leibnitz Harmonic coefficients, and Gaussian Polynomials, or q-analogues).