Extension Problems for the Patterns in Pascals Triangle Problem
- What is the ratio of the number of odds to the number of evens
in Pascals triangle? What does it mean to ask about the
ratio for an infinite object such as this?
- What portion of the triangle is each remainder for a given
modulus (e.g., what portions are 0, 1, or 2 mod 3)?
- Show that
for prime p. What does this result tell us about the
distribution of zeroes in Pascals triangle mod p?
- Explore Pascals tetrahedron: a three-dimensional
version of the triangle in which each number is the sum of the
three neighbors above it: PT(l, r,
c) = PT(l 1, r, c)
+ PT(l 1, r 1, c)
+ PT(l 1, r, c
1). Can you find settings, parallel to those that generate Pascals
triangle, related to this array of numbers?