I’m pretty sure there’s a Peanuts comic strip in which Linus works out how many of each gift was given in the “Twelve Days of Christ

And now it’s all this

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2024-11-25 18:30:02

I’m pretty sure there’s a Peanuts comic strip in which Linus works out how many of each gift was given in the “Twelve Days of Christmas.” I’ve been unable to Google it, because the search results are overwhelmed by links to “A Charlie Brown Christmas.” So I decided to work it out myself.

It’s obvious that there are 12 partridges in a pear tree, one for each of the 12 days. And it follows that there are 22 turtle doves, two for each of the 11 days after the first. By induction, we can say that the total number of the kth gift is

Adding up twelve numbers is easy, especially since the symmetry lets you add just the first six and double it. But let’s work out a formula. We start with

The name comes from building a pyramid of balls with a square base. The top row has just one ball; the row under it has four; the row under that has nine, and so on. The kth pyramidal number is the total number of balls in a pyramid with n rows. For n=12, the square pyramidal number is

This turns out to be a tetrahedral number. Tetrahedral numbers are also called triangular pyramidal numbers; they’re similar to the square pyramidal number but with a triangular base instead of a square.