The Maths inside the Twelve Days of Christmas

Have you ever wondered about how many gifts the singer received in the famous Twelve Days of Christmas song?  The answer reveals a surprising mathematical pattern!

(If you need reminding about the lyrics they can be found at the bottom of this page.)

Let us look at how many gifts were received on each day.

 

Total Gifts

The first nice fact is that the singer receives 364 gifts in total – one for every day of the year apart from Christmas day!

There are also some other nice hidden mathematical patterns.

Gifts received each day

Let us look at what happens on day n.  On this day the singer receives 1+2+3+…+n gifts.  This is also the definition of the nth triangular number.  On day 7 for example the singer receives 28 gifts which is the 7th triangular number.

There is also a well known formula for adding up the first n natural numbers which is:

1 + 2 + 3 + ... +  \mathit{n}=\frac{1}{2} \mathit{n} (\mathit{n}+1)

If we imagine that the singer had continued singing for 20 days they would have received:

 \frac{1}{2} \times 20 \times 21 = 210 gifts on the 20th day!

Running total of gifts

Let us now look at the running total of gifts at the end of day  \mathit{n}

The running total will be equal to  \mathit{T_1} + \mathit{T_2}+...+\mathit{T_n} where  \mathit{T_i} stands for the  \mathit{i}th triangular number. We are basically adding up the first  \mathit{n} triangular numbers. . If you add up the first  \mathit{n} triangular numbers you end up with what is called the  \mathit{n}th tetrahedral number.

Luckily we have a formula for the nth tetrahedral number which is  \frac{n(n+1)(n+2)}{6}

Let’s check it works for our running total on day 12:  \frac{12 \times 13 \times 14}{6} = 364

Now imagine if the singer had continued receiving gifts from their true love for 20 days.  The singer will have:  \frac{20 \times 21 \times 22}{6} = 4620  gifts!  No wonder the true love stopped after 12 days!

xmas gifts istock

 

Lyrics of the Twelve Days of Christmas

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.

On the fourth day of Christmas,
my true love sent to me
Four calling birds,
Three French hens,
Two turtle doves,
And a partridge in a pear tree.

On the fifth day of Christmas,
my true love sent to me
Five golden rings,
Four calling birds,
Three French hens,
Two turtle doves,
And a partridge in a pear tree.

On the sixth day of Christmas,
my true love sent to me
Six geese a-laying,
Five golden rings,
Four calling birds,
Three French hens,
Two turtle doves,
And a partridge in a pear tree.

On the seventh day of Christmas,
my true love sent to me
Seven swans a-swimming,
Six geese a-laying,
Five golden rings,
Four calling birds,
Three French hens,
Two turtle doves,
And a partridge in a pear tree.

On the eighth day of Christmas,
my true love sent to me
Eight maids a-milking,
Seven swans a-swimming,
Six geese a-laying,
Five golden rings,
Four calling birds,
Three French hens,
Two turtle doves,
And a partridge in a pear tree.

On the ninth day of Christmas,
my true love sent to me
Nine ladies dancing,
Eight maids a-milking,
Seven swans a-swimming,
Six geese a-laying,
Five golden rings,
Four calling birds,
Three French hens,
Two turtle doves,
And a partridge in a pear tree.

On the tenth day of Christmas,
my true love sent to me
Ten lords a-leaping,
Nine ladies dancing,
Eight maids a-milking,
Seven swans a-swimming,
Six geese a-laying,
Five golden rings,
Four calling birds,
Three French hens,
Two turtle doves,
And a partridge in a pear tree.

On the eleventh day of Christmas,
my true love sent to me
Eleven pipers piping,
Ten lords a-leaping,
Nine ladies dancing,
Eight maids a-milking,
Seven swans a-swimming,
Six geese a-laying,
Five golden rings,
Four calling birds,
Three French hens,
Two turtle doves,
And a partridge in a pear tree.

On the twelfth day of Christmas,
my true love sent to me
Twelve drummers drumming,
Eleven pipers piping,
Ten lords a-leaping,
Nine ladies dancing,
Eight maids a-milking,
Seven swans a-swimming,
Six geese a-laying,
Five golden rings,
Four calling birds,
Three French hens,
Two turtle doves,
And a partridge in a pear tree!

 

Article by Hazel Lewis