## Holiday Math Special: In the 12 Days of Christmas song, which of the presents do you get the most?

Intended Audience:  Teachers, prospective teachers, and parents (public, private, homeschool).

Recently Autumn asked, “In the twelve days of Christmas, which of the presents do you get the most?”  This is the type of question you hope your child asks you, because it can lead everyone in the family on a great adventure where math just “happens” in the course of thinking through the answer.

In this video, we take you only part way on that adventure.  The adventure can continue in many different ways.  Here are some follow-up questions you can ask:

1. How many total presents do you get on each day?
2. How many total presents do you get for all 12 days?
3. If it was the “The $M$ days of Christmas” song instead, how many presents do you get for each type of present?  Write an expression in terms of the total number of days $M$ and the day $n$ the first time the present is given.  (The answer, $n(M+1-n)$, can almost be read off of the table at the end of the video.)  Use your expression to find the present you get the most for 100 days of Christmas, i.e., when $M=100$.
4. If it was the “The $M$ days of Christmas” song instead, how many total presents do you get after all $M$ days?  Write an expression in terms of $M$. (Hint: it will be a cubic expression in $M$.)  Use your expression to find the total number of presents you would get if $M=100$.

The answers to Questions 3 and 4 are very doable high school (9th grade) problems that lead to all sorts of interesting side questions. If you need to, you can use Wikipedia to help find close formulas for finite series such as $1^2+2^2+3^2+\cdots+M^2$ (sum of squares) and the pyramidal numbers.

Because of the amount of writing in this video, I’m holding the light pen, but Autumn makes up for this by singing and just being herself.  You may also note that I accidentally cover up Autumn a bit–we are still getting the hang of the light board!