id: 06074899
dt: j
an: 2012e.00903
au: Hilton, Peter; Pedersen, Jean
ti: Mathematics, models, and Magz. I: Patterns in Pascalâ€™s triangle and
tetrahedron.
so: Math. Mag. 85, No. 2, 97-109 (2012).
py: 2012
pu: Mathematical Association of America (MAA), Washington, DC
la: EN
cc: U60 K20
ut: Magz; binomial coefficients; trinomial coefficients; Star of David theorem;
homologues; multinomial coefficients; Hoggat-Alexanderson theorem
ci:
li: doi:10.4169/math.mag.85.2.097
ab: 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.
rv: