## Faculty Notes for

Searching for Patterns in Pascal's Triangle

With a Twist

We introduce Pascal's triangle
with a brief historical note and a description of how it is constructed. We
explain how one can read off the coefficients of the expansion of (*a + b*)^{n}
as well as the number of different ways to choose *k* objects from a
set of* n* objects. It is not our purpose to delve into the many combinatorial
results that can be related to the numerical patterns in Pascal's Triangle.
Rather we look at patterns that can be seen in Pascal's triangle when we modify
the algorithm for constructing it.

First, we use addition modulo *n* to construct
the triangle -- then we use other forms of (finite) group multiplication. Naturally
we include pages explaining these operations. In each of our modifications,
we have a finite number of possible entries for the triangle, which we then
represent as different color dots. Thus the patterns we are searching for are
visual.

In many cases the mathematics behind the patterns is
beyond the scope of the paper, and so we merely hint at it. When this occurs,
we include faculty notes with somewhat more explanation for those familiar with
abstract algebra. We also hope that these visual patterns may provide visual,
intuitive representations of some of the more abstract patterns found within
group theory itself. For more information on using these images to create visualization
exercises for abstract algebra, please visit our Web
site or see Bardzell and
Shannon (2002).

We also include notes in places where we can suggest
activities for different mathematics classes related to the discussions. All
of these additional pages are linked through buttons with musical notes, just
like the ones at the top of this page, and these "mathematician pages" are marked
at the top with the same notes.

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