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Patterns in Pascal's Triangle - with a Twist - Patterns for n = 2 through 12

Author(s): 
Kathleen M. Shannon and Michael J. Bardzell

 

Here are the triangles mod n for = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.

 

Pascal's triangle mod 2 with 50 rows Pascal's triangle mod 3 with 50 rows Pascal's triangle mod 4 with 50 rows
Pascal's triangle mod 5 with 50 rows Pascal's triangle mod 2 with 50 rows Pascal's triangle mod 3 with 50 rows
Pascal's triangle mod 4 with 50 rows Pascal's triangle mod 5 with 50 rows Pascal's triangle mod 2 with 50 rows
Pascal's triangle mod 3 with 50 rows Pascal's triangle mod 4 with 50 rows colors assigned to 0 through 12

Look at the triangles for = 2 and n = 4. See how similar they are? If you were to recolor the yellow circles as royal blue and the light blue circles as black, the n = 4 triangle would be the same as the n = 2 triangle. Why do you think that might be?

Now look at the = 6 triangle. Can you see a color identification that would transform it to the = 2 triangle? How about to the = 3  triangle? Answers

Next look at the = 8 triangle. Can you make an identification that would transform it to the = 4 triangle? You could then make another identification to transform it to the n = 2 triangle. Answers

Can you see identifications between other triangles? What is the pattern here? Answers

Faculty Notes

JOMA

Journal of Online Mathematics and its Applications

Dummy View - NOT TO BE DELETED