Today, I'll begin to share the number patterns used to color the triangles. The first step is building Pascal's triangle. Blaise Pascal, a French child prodigy, dabbled in many fields: physics, writing, mathematics, inventions, and Christian philosophy. While ancient cultures knew this pattern, Pascal was the first to write about its application in the real world as well as neat patterns within the pattern in one treatise. Considering he died young (age 39), he accomplished much. Near the end of his life, he turned away from most of his interests and wrote about his faith.
The numbers in the triangle below are quite predictable. Attached is a document you can download and fill out. (I'll post the answers in the spoilers.) An obvious pattern is along the left and right edges. It's all the same number. The adjacent triangles shouldn't be hard to figure out if you look down the diagonal. It might take you some time to figure out how to calculate three of the numbers inside the triangle, namely the 6 and two 10s. Once you know the trick, the rest of the triangle is easy. I'll give one hint at the very end of the post. Another hint is the first comment.
I filled out one of these myself. The numbers grow rather large as you work your way down the triangle. I found myself writing with smaller and smaller letters until I labeled the five-digit numbers A, B, C, D, and E and identified them on the sheet.
Scroll down to see.
The number 6 is calculated from two numbers in the row above it. The same calculation applies to the 10s. Then, you continue the pattern as you work row by row.