A Survey of Mathematics Related Topics: Pg. 185, Example 1

This activity allows the user to investigate number patterns in Pascal's Triangle created by placement of multiples.

Pascal's Triangle is a triangle of numbers, each new number being the sum of the two above it. Here are a few rows:

                      1                   
                  1       1               
              1       2       1           
          1       3       3       1       
      1       4       6       4       1   
  1       5       10      10      5       1

We really should call it Zhu Shijie's Triangle, since Zhu, a Chinese mathematician from the fourteenth century, discovered it three hundred years before Pascal. Pascal's Triangle has many applications.

• In combinatorics and counting, we can use these numbers whenever we need to know the number of ways we can choose Y things from a group of X things.
• In algebra, we can use these numbers to figure out what a binomial raised to a power will be.

There are many interesting patterns in Pascal's triangle. Coloring the multiples of a given number -- like 2 or 3 -- yields interesting patterns. For example, let's work with the number 3. Checking everything in the fourth row (assuming that the first row -- the one with just a 1 in it -- is row "0") for multiples of 3 gives:

1 -- no
4 -- no
6 -- yes
4 -- no
1 -- no

So if we let no be blue and yes be red, we would have the pattern:

Doing this for all of the rows above this, too, gives: