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Christmas with a mathematical touch

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

(1)   \begin{equation*}  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 (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

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