Pascal’s Triangle Patterns

Pascal's_Triangle1Pascal’s triangle is a triangular array of the binomial coefficients. (But don’t worry, it isn’t important to understand binomial coefficients to do this activity.) Pascal’s Triangle is named after mathematician Blaise Pascal. Pascal did not invent this array, but he was the first to develop many of its uses and was the first to organize all the information about it together in a treatise.

To create your own Pascal’s Triangle, place a 1 in the first row (which contains only 1 number). In subsequent rows, each number is the sum of the two numbers above it, treating space outside the triangle as zero. Therefore, the next row, contains 0+1 or 1 and 1+0 or 1. The next row is 0+1, 1+1, and 1+0, as shown in the animation below.

PascalTriangleAnimated2

The cool thing about Pascal’s Triangle is that there are a ton of patterns you can find. Here are just a few:

  1. The first diagonal is all ones.
  2. The second diagonal is the counting numbers.
  3. The third diagonal is the triangular numbers.
  4. The horizontal sums (adding across a row) are the powers of 2 (20=1, 21=2, 22=4, 23=8, 24=16, etc.).
  5. Each row is a power of 11 (110=1, 111=11, 112=121, 113=1331, etc.).
  6. And my favorite of all: if you color the odd numbers one color and the even numbers another color, you get a pattern that looks like the Sierpinski Triangle fractal.

Pascal's_Triangle_with_even_numbers_shaded.svg

 

Can you find any other patterns?

For more patterns, check out this cool page at Math is Fun. If you would like to challenge your kids or students to fill in Pascal’s Triangle and find some of the patterns, I have a worksheet here.

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