Aperiodvent, Day 17: Christmas Stockings in Pascal’s Triangle

If you’re familiar with Pascal’s triangle, you’ll know it has a lot of brilliant hidden patterns and features. One of my favourites is the Christmas Stocking Identity, also more prosaically called the Hockey Stick Identity. The identity states:

$$\sum_{i=r}^n \binom{i}{r} = \binom{n+1}{r+1} \qquad \textrm{ for } n, r \in \mathbb{N}, n > r$$

This means that if you follow a diagonal line downwards into the triangle and add the terms you encounter, the sum will be equal to the term just off the diagonal wherever you stop. This is shown in this diagram, where you can see that:

$1 + 6 + 21+ 56 + 126 + 252 = 462$

To celebrate this fun and festive fact, I’ve put together a PDF you can print and cut up to demonstrate this, by sliding the holes around over the triangle. Enjoy!

This post is part of the Aperiodical’s 2018 Aperiodvent Calendar.