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 16: Relatively Prime Christmas Special

Remember this time last year, when we had a chat with Samuel Hansen for their amazing Relatively Prime podcast? We discussed our collective hobby of getting angry at stupid formulae in the news, and how it sometimes crosses over with the festive season. Listen again below.

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 14: Dodecahedron star lantern

If you manage to dismiss all the ads, the blog Happiness is Homemade has a post which shows you how to make a cool dodecahedral star lantern out of paper (and glue, and you’ll need a light source too if you want it to actually function as a lantern).

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

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.

Aperiodvent, Day 11: MathsJam Shout

Today is the regular monthly MathsJam date for people to meet in pubs and do maths all over the world (some are meeting next week instead – check MathsJam.com to find your nearest). Their internal puzzle sharing sheet is snowflake themed this month, and is shared online so you can join in from home.

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