Searching for Patterns in Pascal's Triangle

Faculty Note on Andy's Applets

These applets are provided for you and your students to experiment. If you would like to give your students some direction, the following questions make good prompts.

For the mod n triangles:

For prime n, what values for the numbers of rows lead to triangles with patterns that seem to be "complete"? Which values give triangles that seem to end in the middle of a pattern?
What about the same question for composite n?
Within the triangles there are downward-pointing subtriangles that stand out. What colors are in these subtriangles. Can you explain what causes this?

For the Z_n x Z_m triangles:

Compare and characterize the patterns for triangles with n = m as opposed to triangles with n and m relatively prime. Consider reflection about the central vertical axis.
What about when m = n²?
Within the triangles there are downward-pointing subtriangles that stand out. What colors are in these subtriangles. Can you explain what causes this?
What numbers of rows lead to the "best looking" triangles? How do these numbers relate to n and m?

For the D_n triangles:

Look at the triangles for n = 4 and n = 8. Compare them to the triangles for n = 3, 5, and 6.
Look at the mod n triangle for n = 2. How does that compare to the D_n triangles?
What numbers of rows lead to the "best looking" triangles? How do these numbers relate to n?
Within the triangles there are downward-pointing subtriangles that stand out. What colors are in these subtriangles. Can you explain what causes this?

For the Two Dimensional Automata Applet:

Experiment with different grid sizes, using the rule that relies on the elements directly above, below, to the right, and to the left of the center cell (first rule in the table below). Are there some grid sizes for which the evolution dies out (i.e., all cells take on the value of the identity and are white)? For example try D₄ with grid sizes of 32 and 64.
Try D₄ with grid sizes of 32 and 64 and the preceding rule. Now try it with the same rule except leave out one of the cells. What happens?
Experiment with different rules. How does information propagate through the grid? Try the following rules where x's represent the cells used:

For more ideas on how to use these see Bardzell and Shannon (2002) or http://faculty.salisbury.edu/~kmshannon/pascal/article/.

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