Patterns in Pascal's Triangle - Counting Stick Software

Explore patterns in Pascal's Triangle!

In 1653, a french mathematician named Blaise Pascal described a triangular arrangement of numbers corresponding to the probabilities involved in flipping coins, or the number of ways to choose n objects from a group of m indistinguishable objects.

The first seven rows of Pascal's Triangle look like:

              1                  n=0

            1   1                n=1

          1   2   1              n=2
                     
        1   3   3   1            n=3

      1   4   6   4   1          n=4

    1   5  10   10  5   1        n=5

  1   6   15  20  15  6   1      n=6

Note that every number in the interior of the triangle is the sum of the two numbers directly above it.

It turns out that Pascal's triangle holds many interesting numeric patterns. One way of seeing some of these patterns is to pick a number x and color all numbers in the triangle that are evenly divisible by x with one color, and all the other numbers in the triangle with a second color. To see as much of the pattern as possible, you need to be able to see as many rows of the triangle as possible, but coloring a large number of rows like this by hand is very time consuming. A computer can color 128 rows of the triangle in less than a second, so we can use it to look at the results using many different divisors. The application below lets you choose the number that you want to use as a divisor. Then it colors the first 128 rows of Pascal's triangle, coloring a square black if the number that square represents is evenly divisible by the divisor you have selected and red if it is not. To change the divisor, type in a new number and click the "Set Divisor" button.

If you have a large monitor, here is a version which displays 256 rows of the triangle.

Here's something to investigate:

Look at the triangle using the divisors 3, 5, and 7.

Do you see a pattern? What do 3, 5, and 7 have in common?

Now try 9. Does the pattern continue?

Can you figure out what it is about 3, 5, and 7 that causes this pattern, and why it doesn't continue at 9? Does it continue with other numbers larger than 9?