Pascal’ s Triangle

Today we are going to be looking into the different patterns in Pascal’s Triangle. I am not talking about Pascal from the Walt Disney Move Tangled. I am talking about Blaise Pascal, a famous French mathematician and philosopher. He was born on June 19, 1623, in Clermont-Ferrand, France. His mother died we he was just three years old, leaving behind his two sisters, his father, and himself. His father, Etienne Pascal, never remarried; instead he focused on educating his children, especially his son, Blaise Pascal. In 1653, Pascal released Traite du triangle arihmetique, which talked about binomial coefficients. This later became famous and  became known as Pascal’s Triangle.

Image: Hersfold (public domain), via Wikimedia Commons.

Image: Hersfold (public domain), via Wikimedia Commons.

Even though Blaise Pascal was the one to get all the credit for the triangle, the ancient Chinese actually developed it. The reason why he receives the credit for the triangle is because he discovered the patterns that are in the triangle. It could also be because the Europeans did not know about the previous discovery from China, and we have an Eurocentric math culture so we know it as Pascal’s Triangle.  Before we go into the different patterns that he discovered, we are going to review how Pascal’s Triangle is made. It starts with the number one at the very top, this is called row 0. Row 1 consists of 1 and 1 and the next row consists of the numbers 1, 2, and 3. This is determined by adding the two numbers above to the left and to the right to get the coefficients in the row. For example, in row 2 to get the first number we add 0+1=1, 1+1=2, 1+0=1. You will do this process for each of the rows to get the different coefficients.

Now that we know more about what Pascal’s Triangle lets look at the many different patterns that are present in Pascal’s Triangle. The first one is called the Hockey Pattern. If you start at any 1 in the triangle and go diagonally until you choose to stop, the sum of all the numbers in that diagonal is the number just below the last number, ensuring that you are looking at the number below and to the opposite side that the diagonal would have continued. If you start on the left side of the triangle you will go down to the right diagonally, then when you want to stop, you will go below and to the left of the last number to find the sum of the numbers within that diagonal. For the right side, you will go below and to the right of the diagonal for the sum. For example, look at the highlight red to see the pattern. 1+6+21+56=84.

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Two more patterns that are present in Pascal’s Triangle are called The Sum of Rows and Prime numbers. In the pattern, The Sum of Rows, the website All You Ever Wanted to Know About Pascal’s Triangle and More states: “The sum of the numbers in any row is equal to 2 to the nth power or 2n, when n is the number of the row.” This is saying if you pick the row 5 the sum of all the numbers in that row would equal 25. Remember that the first row is considered row 0. The next pattern, Prime numbers, states that if the second number in a row is prime, then all the other numbers in that row are divisible by that number.

These are just a few  of the patterns that are present in Pascal’s Triangle. There are many more patterns and I encourage you to do some more research and discover all the patterns in Pascal’s Triangle.

Sources:

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