PASCAL’S TRIANGLE IN FORMAL MATH LANGUAGE
Right and left sides of the triangle are formed by one’s. To find each inside number, add the two numbers above it. Rows are numbered beginning with the top row which is zero.
Other interesting patterns I found while researching Pascal’s Triangle:
- The sums of the diagonals form the Fibonacci sequence. 1, 1, 2, 3, 5, 8. 13 …
- The diagonal rows next to the row of one’s are counting numbers.
- The product of six number’s surrounding any number (a ring) is always a perfect square. The six numbers’ I worked with were 1,1,5,10,6,3 that surround the number 4 from the 4th row. The product is 900 with a square root of 30.
- The multiples of a number form inverted triangles. When two numbers next to each other have the same factor, the number below them will also have the same factor. For example: 4 and 6 each has a factor of 2. The 10 below also has a factor of 2. The shape these three numbers make, if shaded in, would be an inverted triangle.
- Pascal’s Triangle can be used to answer questions involving combinations. For Example: Find the number of combinations of 4 things taken 2 at a time. Refer to row 4 of Pascal’s Triangle. (Remember the 0 element of every row is 1) There are 6 combinations of 4 things taken 2 at a time.
Number Taken at a Time
- Prime numbers: If the first element in a row is a prime number, all of the numbers in that row are divisible by it. Example: Row 5 is 1, 5, 10, 10, 5, 1.
- Hockey Sticks: If a diagonal of numbers of any length is selected starting at any of the 1’s bordering the sides of the triangle and ending on any number inside the triangle, the sum of the number selected is equal to the number below the diagonal. 1+3+6=10. This number combination is circled on Pascal’s Triangle below and looks like a hockey stick.
- Sums of the Row: The sum of the numbers in any row is equal to 2 to the nth power with n = the number of row. I looked at the 3rd row. 2³ = 1+3+3+1=6
- The most fascinating pattern to me is that when all the odd numbers on Pascal’s Triangle are colored in it forms a Sierpinski Triangle.