Here are the triangles mod n for n = 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 n = 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?
Faculty Notes Now look at the n = 6 triangle. Can you see a color identification that would transform it to the n = 2 triangle? How about to the n = 3 triangle?

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

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

You might also like to see the triangles drawn to more rows:
Pascal's triangle mod 2 with 125 rows Pascal's triangle mod 3 with 125 rows Pascal's triangle mod 4 with 125 rows Pascal's triangle mod 5 with 125 rows
Pascal's triangle mod 2 with 125 rows Pascal's triangle mod 3 with 125 rows Pascal's triangle mod 4 with 125 rows Pascal's triangle mod 5 with 125 rows
Pascal's triangle mod 2 with 125 rows Pascal's triangle mod 3 with 125 rows Pascal's triangle mod 4 with 125 rows colors assigned to 0 through 12

You probably noticed immediately the downward-pointing royal blue triangles in all of the patterns. Since royal blue corresponds to the number zero, you can see that, whenever you get a string of zeros on a row, this will generate a downward pointing blue triangle -- any entry below two zeros will be zero. But the non-zero entries on either end of the string will encroach one position on each side for each row.

For example, if you start with a row {1,0,0,0,0,0,1} and generate the following rows, you get:

1

 

0

 

0

 

0

 

0

 

0

 

1

 

1

 

0

 

0

 

0

 

0

 

1

 

 

 

1

 

0

 

0

 

0

 

1

 

 

 

 

 

1

 

0

 

0

 

1

 

 

 

 

 

 

 

1

 

0

 

1

 

 

 

 

 

 

 

 

 

1

 

1

 

 

 

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

The downward pointing triangles containing subsets of the colors for a triangle occur for similar reasons. For example, look at one of the blue triangles for the mod 2 triangle. Now look at the same area in the mod 4 and mod 8 triangles. What do you see? Look at the same region in the mod 6 triangle. What can you say about the colors in this region in the mod n triangles for any even mod?

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