How to Find Patterns in Pascal's Triangles

Pascal’s triangle is a triangular mathematical pattern that mathematician Blaise Pascal studied. Pascal’s triangle often appears in probability equations. Many patterns exist in the triangle. Finding these patterns makes great exercises for students.

Things You'll Need

• Pencil
• Paper
• Calculator
• Crayon

Instructions

1. Find the triangle pattern by beginning with the number 1 at the top and placing numbers underneath in a triangle shape. Begin each row with the number one, then add the numbers in the row above it together and write down the sum, then end each row with the number one. For example:
   1 (row 0)
   1 1 (row 1)
   1 2 1 (row 2)
   1 3 3 1 (row 3)
   1 4 6 4 1 (row 4)
   1 5 10 10 5 1 (row 5)
   1 6 15 20 15 6 1 (row 6)
   Add 1+1=2 to find the second row (1 2 1). Add 1+2=3 and also add 2+1=3 to find the third row (1 3 3 1). Add 1+3=4, 3+3=6, and 3+1=4 to find the fourth row (1 4 6 4 1) and so on. This pattern continues infinitely.

2. Draw a line down the center of the triangle to find a symmetrical pattern in the triangle. The numbers on one side have a match on the opposite side.

3. Find the pattern 2 to the N power for each row by adding all the numbers in each row. For example, the second row (1+2+1=4) equals 2 to the power 2, or 2 squared (2x2=4). The third row (1+3+3+1=8) equals 2 to the power of 3, or 2 cubed (2x2x2=8). The fourth row (1+4+6+4+1=16) equals 2 to the power of 4 (2x2x2x2=16), and so on for infinity.

4. Identify the diagonal patterns in the triangle by starting at the top and going diagonally down to the left. Begin with row 0 at the number 1 and go diagonally left to find the first diagonal contains only ones (1 1 1 1 1 1…). Start at the ending 1 in row 1 to find the second diagonal contains counting numbers (1 2 3 4 5…). Begin at the ending 1 in row 2 to find that the third diagonal container triangular numbers (1 3 6 10 15…). Continue this pattern to find the tetrahedral numbers in the fourth diagonal (1 4 10 20 35…).

5. Color the even numbers in the triangle one color and the odd numbers another color to find the Sierpinski Triangle, an infinitely repeating triangular pattern.

References

• Montclair State University: Pascal's Triangle: Patterns
• Math in Fun: Pascal's Triangle
• Wolfram Math World: Pascal's Triangle