Once again Sarthak Parikh (Class VIII, Sardar Patel) gave interesting stories, and made nice pictures in response to our last column. This week’s column is about Pascal’s Triangle and the patterns that can be found in it. Pascal’s triangle has an infinite number of rows, and it begins as follows.
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Each row has one more entry than the previous row. The row begins and ends with 1. The other entries in the middle are the sum of the two entries above. For example, in the fourth row: $4=1+3; 6= 3+3; 4=3+1$.
Pascal’s triangle is a storehouse of patterns. Make 16 rows of Pascal’s triangle, and find as many patterns as you can. For example, can you find $1, 2, 3, 4,\dots$ in the triangle? What is the sum of entries in each row? If we replace all the even numbers in the triangle by $0$, and all the odd entries by $1$, we get the Fractal known as Sierpinski Triangle. This computer-generated picture shows this pattern.