At first, you may think that mathematics has nothing to do with Christmas. The only link I could think of was calculating whether you have enough budget to buy that all those present for your family. However, mathematicians will always think of a way to integrate mathematics and thus, also for Christmas. One of the most popular applications of mathematics to Christmas is analyzing the song “Twelve Days of Christmas”. In this song, a surprising mathematical pattern is used to find out how many presents the singer receives during the twelve days of Christmas. Furthermore, an ordinary equation turns out to send a special Christmas message, which will also be explained in this article. You will be amazed by how mathematics is involved in Christmas songs and messages.
Mathematics in a Christmas song
You may be familiar with the song Twelve Days of Christmas, where the singer receives presents from his/her love every day of Christmas. The 12 days refer to the period starting on Christmas and ending on January 6th, the day on which the 3 kings from the East brought gifts. The first lines of the song are as follows:
On the first day of Christmas,
my true love sent to me
A partridge in a pear tree.
On the second day of Christmas,
my true love sent to me
Two turtle doves,
And a partridge in a pear tree.
On the third day of Christmas,
my true love sent to me
Three French hens,
Two turtle doves,
And a partridge in a pear tree
Notice that the singer receives on each day 1 partridge, from the second day onwards he/she receives 2 doves every day and from the third day 3 hens are added and so on. This means that the number of presents each day is 1 on the first, 3 on the second, 6 on the third, then 10 on the fourth, etcetera. These set of numbers are called triangular numbers, since the amount of presents each day can be drawn in a dot pattern that forms triangles:
So what happens on day $n$? On the $n$-th after Christmas, the singer receives $1+2+…+n$ presents. Thus, the amount of presents for each day can be calculated by
\begin{equation*}
T_n = 1+2+…+n = \frac{n(n+1)}{2}
\end{equation*}
For example, on the 10th day, the singer receives $\frac{10 \cdot 11}{2} = 55$ presents. In order to get the total number of presents, we need to know the sum of the triangular numbers. If you add up the first $n$ triangular number, you get the $n$-th tetrahedral number. The $n$-th tetrahedral number can be calculated by
\begin{equation*}
T_1 + T_2 + …+ T_n = \frac{n(n+1)(n+2)}{6}
\end{equation*}
Hence, on the 12th day after Christmas, the singer receives $\frac{12 \cdot 13 \cdot 14}{6} = 364$ presents. One for every day of the year apart from Christmas day!
Mathematics in a Christmas message
Suppose we have the equation
\begin{equation} \label{eqn:1}
y = \frac{ln(\frac{x}{m} – sa)}{r^2}.
\end{equation}
Actually, this is a special Christmas equation since it contains a lovely Christmas message. But how is this possible? Let’s go through this transformation step by step.
First, we multiply both sides of equation (\ref{eqn:1}) by $r^2$, so we get
\begin{equation*}
y \cdot r^2 = ln(\frac{x}{m} – sa).
\end{equation*}
If we raise both sides to the power of $e$, we obtain
\begin{align*}
e^{yr^2} & = e^{ln(\frac{x}{m} – sa)}
& = \frac{x}{m} – sa
\end{align*}
Adding $sa$ to both sides and multiplying by m gives
\begin{align*}
e^{yr^2} + sa & = \frac{x}{m}
m \cdot (e^{yr^2} + sa) = x
\end{align*}
Rewriting this equation gives
\begin{align*}
m e^{yr^2} & = x – msa
me^{rry} = x\text{-}mas
\end{align*}
Merry Christmas and we wish you much love, health, and the greatest happiness!
This article was written by Renske Zijm
