Mathematics, models, and Magz. I: Patterns in Pascalâ€™s triangle and tetrahedron. (English)

Math. Mag. 85, No. 2, 97-109 (2012).

Summary: This paper describes how the authors used a set of magnetic toys to discover analogues in 3 dimensions of well known theorems about binomial coefficients. In particular, they looked at the Star of David theorem involving the six nearest neighbors to a binomial coefficient $\tbinom{n}{r}$. If one labels the vertices of the bounding hexagon with the numbers $1, 2, 3, 4, 5, 6$, consecutively (in either direction), then the product of the coefficients with even labels is the same as the product as the coefficients with odd labels. Furthermore the two figures formed by connecting the odd and even vertices are both equilateral triangles arranged so that a rotation of $\pm 60^\circ$ exchanges the triangles. There is a generalized Star of David theorem concerning a semi-regular hexagon with similar results. The paper describes analogous results for trinomial coefficients involving, sometimes but not always, tetrahedra instead of triangles.