Here are the triangles for *n* = 2 through 12 drawn to more rows:

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 *n*? (Answer)

Journal of Online Mathematics and its Applications