Mysterious Patterns in Pascal’s Triangle
Hidden treasures of combinatorics
What do fractals have to do with open questions about mysterious primes and basic combinatorics?
In this article, we will explore patterns in one of the most interesting and well-studied mathematical objects. We will see how famous sequences of numbers, fractals, Fermat primes, and other amazing stuff is hidden in the triangle.
First encounter
When my dad was in high school, he was investigating ways of combining odds from a bookmaker to his advantage and found a remarkable symmetrical relationship among some of the numbers that showed up in the process of creating these betting systems.
He noticed that these numbers formed a particularly interesting pattern when putting them in a triangular-like shape after which he rushed to show his math teacher who immediately recognized the famous numbers. The triangle that he found is the following:
and continues like this to infinity!
To create the next row in the triangle, you add the numbers to the left and right above it (we imagine the blank spots beside the 1s are zeros). Despite the simple nature of this shape, amazing stuff emerges.
At the time, little did my father know that this object had been studied for more than a thousand years before him all over the world. In Iran, it is also known as Khayyam’s triangle, and in China, it is known as Yang Hui’s triangle.
In most of the “Western world”, it is known as Pascal’s triangle due to the study of the triangle by Blaise Pascal who used it to solve problems in probability theory.
The numbers comprising the triangle are known as binomial coefficients which are the number of ways you can choose k out of n objects. We are counting the entries in the triangle starting from 0 so for example, there are 6 ways of choosing 2 out of 4 objects which is the number on the 4'th row and 2'nd place in the row (counting from 0).
