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.

Aperiodvent, Day 15: Mathematical Present Wrapping video

In the viral YouTube hit of Christmas 2015, Katie Steckles demonstrates some of the most mathematically satisfying ways you can wrap your Christmas presents.

Aperiodvent, Day 13: Fold-and-Cut Christmas Tree

The fold and cut theorem, which states that, after sufficient folding, any shape made of straight lines can be cut out of a piece of paper in one cut, is probably the most crafts-friendly result in all of maths.  Inspired by The Aperiodical’s very own Katie Steckles’ video on the subject, Sam Hartburn has created a handy PDF with instructions for folding and cutting a festive Christmas tree shape.

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

Aperiodvent, Day 12: Fractal Christmas Trees

If you’re looking for a fun hands-on project that’s mathematical and Christmassy, look no further than Think Maths‘ classic Fractal Christmas trees – building a Sierpinski tetrahedron tree, Menger Sponge base and Koch Snowflake star.