I first encountered Pascal’s triangle on a worksheet my older brother brought home from school when I was in eighth grade. At the time, I was taking geometry (with a teacher who had a cool pair of shoes with math equations all over them) and the phrase “binomial expansion” meant nothing to me yet — although the pattern of numbers triangularly cascading down the page my brother brought home mesmerized me. In classes, when I was bored, I’d try to construct the triangle on the backs of worksheets, silently trying to add numbers in my head and being careful to write the numbers small enough so as to fit more lines on a single page.
Pascal’s Triangle was glossed over in my high school math classes, so it wasn’t until freshman year of college in a combinatorics class when I became more formally acquainted with it. Combinatorics is a branch of math that, at heart, is all about counting. The class was fascinating to me because it was unlike any other math class I’d taken previously. Most often, the goal when proving a statement in that introductory combinatorics class was not to fill pages and pages with complicated calculations, untangling and then tying back together one side of an equation to make it identical to the other. Instead, the goal was usually to step back and think about the narratives packaged inside two different mathematical phrases, and then to consider why these narratives should coincide.
If you aren’t familiar with Pascal’s triangle, it’s a way of arranging numbers in rows. The first row only has a single number: 1. The second row has two numbers: 1 and 1. From then onwards, each row is created by writing numbers that are the sum of the two numbers above in the previous row, with a 1 on each end of the row. Since each row has an additional number in it than the previous, writing out the numbers in such a way results in a triangular shape, as pictured below.
It was in that combinatorics class where I learned about some of the patterns inside Pascal’s triangle — and the narratives they encapsulated. One of the patterns in Pascal’s triangle is that the sums of the numbers in each row correspond to powers of 2. For example, the sum of the numbers in Row 2 is 1+2+1=4, which is the second power of 2 (i.e., 2 times 2 equals 4).
A “combinatorial” proof of this would be to think about the sum of numbers and the power of 2 as two different ways to count the same thing. For example, if we’re still in Row 2, we can think about trying to find the number of ways to choose some number of objects out of 2 possible objects. Let’s consider the Stanford vs. South Carolina women’s basketball game tonight and the UConn vs. Arizona game tonight as our two objects. How many different possibilities are there for what games you watch? There are exactly four: you can watch none of the games, you can watch both games, you can watch just the Stanford vs. South Carolina game, or you can watch just the UConn vs. Arizona game. I won’t go into the particulars, but other than just listing out these possibilities we could have counted them in two other ways, which are hidden inside the mathematical expression for summing the numbers in the row and taking powers of 2.
The actual mathematical statement to prove this pattern is to the right of the figure below. The algebra to prove it is tricky, but it’s not the only way to prove the statement. Lurking inside each expression is a story about counting — and while the details of the stories differ, they have the same end result.
Pascal’s triangle, for me, is an example of a problem or structure that I have returned to many times since I first encountered it in middle school. There are many other patterns inside Pascal’s triangle aside from the one described above — I only just learned of a new one this week while writing this. In math, and elsewhere, are phenomena whose complexities and intricacies we are unaware of at first encounter. Yet something of their rhythm or scope draws our attention, and over time, through experience, questioning and sometimes even surprise, the patterns and mysteries begin to unfold.