Several patterns have been discovered within Pascal's triangle. The first pattern deals with odd and even numbers.
The figure below shows the pattern that occurs when all the odd integers in Pascal's triangle are highlighted (black)
and the remaining even integers are left white.
When looking at the triangle does there appear to be a greater percentage of white
compared to black? Or more black compared to white? The article "PASCAL'S TRIANGLE: 100% of the numbers are even" actually leads a discussion
to see which one is true. They discovered that the farther down the triangle goes, there appears to be more white, or even numbers.
In fact, they discovered that as the triangle approaches infinitely the percentage of odd numbers (black) becomes very low and the
percentage of even numbers (white) seems to approach 100%. Of course the percentage of even numbers can come very close to 100%,
but never actually reach it since there will always be odd integers in the triangle as well (Bhindi & McMenamin, 2010).
The next pattern is called the hockey stick pattern. The figure below shows what the hockey stick pattern looks like in the triangle.
One end of the hockey stick lies of the border of the triangle, while the handle of the stick consists of numbers inside the triangle
that are all located in different rows so that the hockey stick does not lie horizontally. The head of the hockey stick should not be
aligned in the same direction as the handle, but instead goes back in the direction of the end of the hockey stick. The sum of all the
numbers in the handle equals the number in the head of the hockey stick. For example, looking at the figure, the yellow hockey stick has
the integers 1, 6, 21, and 56. When you add all of these numbers together you get 84, which is the head of the hockey stick (Hilton & Pedersen, 1987).
A proof for this pattern can be found by following this link: Hockey Stick Proof
Another pattern that can be found in the triangle is called the Star of David pattern.
This pattern is shown below and can be found by first choosing any number (x) within the triangle.
Then connect the six neighboring cells to (x) using two equilateral triangles like shown in the figure.
The interesting thing is that the product of the integers in the vertices of the blue triangle is equal to the product of
the integers in the vertices of the red triangle. That is 7x36x56 = 8x21x84 (Hilton & Pedersen, 2010).
A proof of the Star of David can be found here: Star of David Proof
One last pattern found in Pascal's triangle is the Fibonacci sequence. The Fibonacci sequence is a series of numbers
in which each number is the sum of the two preceding numbers. This can be found in Pascal's triangle by adding all the
numbers in the diagonal together like shown in the figure below.
