The popular Christmas carol starts innocuously enough. On day one, the character in the song gets a single present (a partridge in a pear tree). But on day two, the beneficiary receives a new present (a pair of turtle doves) plus another partridge in a pear tree. Day three brings a second helping of day two’s gifts, plus more new items (three French hens). By the twelfth day, the narrator is an undeniable pack rat – and maybe in violation of local zoning – after receiving 12 drummers drumming, and new copies of all the previous day’s gifts.
By the twelfth verse of the song, there are a lot of gifts. How do you count it all up? Pure arithmetic provides the easiest, though longest, way to do it:
The total, 364 gifts, means that our true love gets a present for almost every day of the year!
Greeks bearing gifts – triangular numbers
Here’s where some neat mathematical patterns come in, according to Bill Butterworth, professor of mathematics at Barat College of DePaul University in Illinois. The individual sums on each row-(1+2), (1+2+3) and so on-are what the ancient Greeks called “triangular numbers” because they make a triangle if you replace the numbers with X’s.
The triangular numbers form a sequence: 1, 3, 6, 10, and so on.
Instead of doing drawn-out arithmetic, you can simply use the triangular numbers to find out the number of gifts bestowed on any day of the song. For example, the 12th triangular number, 78, corresponds to the number of gifts granted in day 12. How many gifts were furnished on day 10? You just look up the 10th triangular number: 55.
Adding up the first 12 triangular numbers gives you the total number of gifts mentioned in the song: 364.
Pascal’s Triangle
The daily and 12-day totals for the gifts show up in another, more remarkable math pattern, discovered over a millennium ago and shaped coincidentally like a Christmas tree. It’s called Pascal’s Triangle, named after the Renaissance French mathematician Blaise Pascal, who developed it after its discovery by Arab, Chinese, and Persian mathematicians.
Today, Pascal’s Triangle has a wide range of uses in probability theory, fractals, calculus, and many other areas of math. “I’ve found Pascal’s Triangle to be one of the richest and most accessible mathematical ‘objects’ over the years,” says Butterworth. Indeed, Pascal’s triangle contains an astounding bag of mathematical tricks that includes more than one holiday connection.
The first fourteen rows of Pascal’s Triangle look like this:
To identify any number on the triangle, mathematicians specify its horizontal “row” and its diagonal “column,” with both column and row numbers starting at 0. For example, the third row is 1, 2, 1, and the first column (in either direction) is 1, 2, 3, 4, and so on.
How are the numbers chosen? In the triangle, every number, such as 10 (in row 6, column 3), is the sum of the two numbers in the previous row diagonally above it – 4 and 6:
This simple rule yields many powerful numerical patterns-including one that we’ve seen before.
“The triangular numbers – 1, 3, 6, 10, and so on – appear in the second column of Pascal’s triangle,” Butterworth points out. That’s the first ingredient for “The Twelve Days of Christmas” connection.
Twelve days in the triangle
The second ingredient is a math result known – coincidentally once again – as the Christmas Stocking Theorem.
The Christmas Stocking Theorem says this: Go to the top of any column, and select a diagonal string of as many numbers as you’d like down that column. To find the sum of those numbers, you don’t have to add them up. You can find the sum nearby in the triangle. Just put your finger on the last number in the string; move your finger to the next number in the column; then slide your finger over to the next column. That number provides the sum of the string.
To apply this to the “Twelve Days,” let’s choose column 2, the one with the triangular numbers. Then, choose the first 12 numbers.
According to the Christmas stocking theorem, you can find the sum of those numbers by putting your finger on the last number of the twelve-78-then going to the next number in the column – 91 – and then sliding over to the next column. What number do you find there? 364!
This same pattern works for any column of the triangle. As you may have noticed, there is actually a connection to Christmas stockings: When highlighted, the string of numbers down the column resembles the shank of a Christmas stocking hanging from the tree, while the sum appears in the toe! Because of this shape, the Christmas Stocking Theorem is also known as the “Hockey Stick Theorem,” another popular winter pastime.
