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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 ? On the -th after Christmas, the singer receives presents. Thus, the amount of presents for each day can be calculated by

For example, on the 10th day, the singer receives 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 triangular number, you get the -th tetrahedral number. The -th tetrahedral number can be calculated by

Hence, on the 12th day after Christmas, the singer receives presents. One for every day of the year apart from Christmas day!

### Mathematics in a Christmas message

Suppose we have the equation

(1)

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 , so we get

If we raise both sides to the power of , we obtain

Adding to both sides and multiplying by m gives

Rewriting this equation gives

Merry Christmas and we wish you much love, health, and the greatest happiness!

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By Daniele Zedda • 18 February

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