Pascal’s Triangle


Pascals Triangle is one of the most incredible cheat sheets, in my opinion. The brilliance behind this work is magnificent! The triangle was named after Blaise Pascal, but it was first used and studied by the Persians and Chinese long before Pascal was born. Somehow, Pascal gained the credit for the triangle. I decided to explain some of its interesting patterns that occur in the triangle. Once you learn the rules, you find yourself trying to find an error in the triangle, but the triangle will prove its efficiency every time. It’s almost too good to be true.  

The Basics:

The very  top point of the triangle is known as the zeroth row. This row starts with the number 1. As we move onto row two, the numbers are 1 and 1. The second row is made by adding the two numbers to the left above the number and to the right above the number together. Anything outside the triangle is a zero. A good easy example of this pattern in pascals triangle is if you look at the number two. The top left number above 2 is 1 and the top right number above 2 is 1. If you do the math, 1 + 1 = 2. If you look at any number in the triangle, this rule stands true for all of them. I found this cool animation to help the confusion!

The Triangle is Symmetrical:

The graph is symmetrical. The numbers on the right side correspond to the numbers on the left side.

Pascal's Triangle Symmetry

Sum of The Rows:

If you add the numbers in each row together, each row will equal to 2^{n} . Where n is equal to the row number. You could also look at this rule as simply just the next row in the sequence is a double of the row before. Example: row 4 is double the sum of row 3. You should try it yourself! Here is some proof:

20 = 1
21 = 1+1 = 2
22 = 1+2+1 = 4
23 = 1+3+3+1 = 8
24 = 1+4+6+4+1 = 16

 Awesome Elevens:

11 is a unique number in the triangle, because similar to the number 2, each row answers what 11^ {n} equals. N is equal to the row.  The trick is you have to ignore the 1’s along the outside of the triangle and focus on the number between the 1s. Here is proof of the pattern:

Row # Formula = Multi-Digit number Actual Row
Row 0 110 = 1 1
Row 1 111 = 11 1 1
Row 2 112 = 121 1 2 1
Row 3 113 = 1331 1 3 3 1
Row 4 114 = 14641 1 4 6 4 1
Row 5 115 = 161051 1 5 10 10 5 1
Row 6 116 = 1771561 1 6 15 20 15 6 1
Row 7 117 = 19487171 1 7 21 35 35 21 7 1
Row 8 118 = 214358881 1 8 28 56 70 56 28 8 1

Hockey Stick Sequence:

If you start at a one of the number ones on the side of the triangle and follow a diagonal line of numbers. You can find the sum of the certain group of numbers you want by looking at the number below the diagonal, that is in the opposite direction of the diagonal you made. The picture below shows three good examples to make this pattern easier to understand.

1+6+21+56 = 84
1+7+28+84+210+462+924 = 1716
1+12 = 13

I hope you enjoyed this blog about Pascals Triangle! Once I was introduced to this triangle, I was very fascinated and I hope you are too now. There is many more patterns to the triangle, but some are complicated to explain. Some of them I am trying to figure out on my own.


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