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Patterns in Pascal's Triangle Project Description Prerequisites Warm up Problems Hints Resources Teaching Notes Extension Problems Results

Extension Problems for the Patterns in Pascal’s Triangle Problem

  1. What is the ratio of the number of odds to the number of evens in Pascal’s triangle? What does it mean to ask about the ratio for an infinite object such as this?
  2. What portion of the triangle is each remainder for a given modulus (e.g., what portions are 0, 1, or 2 mod 3)?
  3. Show that a plus b to the p is congruent to a to the p plus b to the p mod p for prime p. What does this result tell us about the distribution of zeroes in Pascal’s triangle mod p?
  4. Explore “Pascal’s 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 Pascal’s triangle, related to this array of numbers?

    Layers of Pascal's tetrahedron

    A three-dimensional array of numbers

    “Pascal’s tetrahedron”


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