Patterns in Pascal's Triangle - with a Twist - Two More Patterns

Kathleen M. Shannon and Michael J. Bardzell

With which of the two groups would you place these two hexagons? Although 8 and 9 are both composite numbers, they are somewhat different in character from 6, 10, and 12 -- they each have only one prime factor. That is, 8 is a power of 2, and 9 is a power of 3, while 6 and 12 have both 2 and 3 as factors, and 10 has both 2 and 5 as factors.

What is really going on here? In the next section we look at these more closely.