This problem drove me crazy for hours.

I came across this problem a couple weeks ago over at Dan Meyer’s blog.

When I gave my students this, most of them quickly said 78. We talked about which problem they had just solved and how the song went. They came up with tables and other crazy ways of pairing numbers, it was fun.

There was moaning when I asked how many total gifts there would be if it was The 25 Days of Christmas. Some students were already working towards The X Days of Christmas, by pairing. Most classes ended after finding the 12 days….

But I was on a mission to find The X Days of Christmas.

I pitched this problem to a few colleagues, I threw together the following table:

Constant third difference indicates cubic…

One colleague put the ordered pairs of (Days, Total Gifts) into a calculator and fit a cubic regression to it. Another colleague came up with the following (through some magical hand waving). They both ended up simplifying to almost the same equation.

Long ago (like two years), I had a modeling class in college where we messed with these sort of things. I pulled out my book and found it… its called Newton’s Difference Formula, the powers at Wolfram Alpha say this about it.

I am not quite sure where this comes from… but the deltas are the differences, which is exactly what my colleague came up with. This problem turned out to be DEEP. I always try and rethink a problem if I find myself going into calculus, but this time I needed it to come up and understand The X Days of Christmas.

I dug around some and came across these fun resources relating to this as well.

Vi Hart

Spiked Math

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*Related*

My favorite way to see it is in Pascal’s Triangle. Given a North to Southwest “column” that contains a sequence you find interesting, the column below will be the cumulative sum. Proceeding from the left side, we have all 1s, then their cumulative sum is consecutive positive integers, then their cumulative sum are triangular numbers, then their running sum are the tetrahedral numbers (aka the x days of christmas total presents), etc.

Give yourself 5 minutes (at most) and I bet you’ll see how to translate x-days into the coordinates you need in the triangle and, as a consequence, the binomial coefficient that gives you the total number of presents. Just remember that the 1 at the very top is best considered the 0th row and the numbers in each row start with 0 from the edge.

Of course, anything Vi Hart does is pure gold. Make sure to take a look at her Parable of the Polygons.

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