Twelve Days of Christmas

There’s a well-known Christmas song called The Twelve Days of Christmas.

It’s a recursive song in which, each day, for twelve days, a lover is gifted presents. On the first day, one gift is given. On the second, two new gifts are given plus the first gift repeated. On the third day, three new gifts, plus a repeat of the two, and a repeat of the one. Each day, the cardinality of the number of gifts is increased by one.

There are regional variants of the song, but here are the commonly sung Western lyrics:

The Twelve Days of Christmas

On the first day of Christmas my true love gave to me
A partridge in a pear tree

On the second day of Christmas my true love gave to me
Two turtle doves, and a partridge in a pear tree

On the third day of Christmas my true love gave to me
Three French hens, two turtle doves, and a partridge in a pear tree

On the fourth day of Christmas my true love gave 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 gave to me
Five gold 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 gave to me
Six geese a laying, five gold 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 gave to me
Seven swans a swimming, six geese a laying, five gold 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 gave to me
Eight maids a milking, seven swans a swimming
Six geese a laying, five gold rings
Four calling birds, three French hens
Two turtle doves, and a partridge in a pear tree

On the ninth day of Christmas
Nine ladies dancing, eight maids a milking
Seven swans a swimming, six geese a laying, five gold 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 gave to me
Ten lords a leaping, nine ladies dancing, eight maids a milking
Seven swans a swimming, six geese a laying, five gold 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 gave 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 gold 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 gave 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 gold rings
Four calling birds, three French hens
Two turtle doves, and a partridge in a pear tree

How many gifts?

A commonly asked question is how many gifts are given in total?

You might have quickly noticed that, on any particular day, the number of gifts given is that corresponding triangular number. On the first day, there is one gift given. On the second day there is 1+2 = 3 gifts given. On the third day there are 1+2+3 = 6 gifts given. On the fourth day there are 1+2+3+4 = 10 gifts given …

To find the total of all the gifts given, up to that day, we need to sum up all the previous triangular numbers. This is given the name a Tetrahedral Number because of it’s shape. Imagine a triangular number of pool balls, onto which we stack a one smaller triangle, and again, until we reach the top.

	Day	Total Gifts
	1	1
	2	4
	3	10
	4	20
	5	35
	6	56
	7	84
	8	120
	9	165
	10	220
	11	286
	12	364

A total of 364 gifts are given over the twelve days.

There is a simple formula to calculate the tetrahedral numbers:

In no surprise to anyone, Triangular and Tetrahedral Numbers are easy to find in Pascal’s Triangle.

Most Popular Gift

What is the most modal gift given over the season? Yes, on the twelfth day of Christmas you get given twelve drummers, but you only get that once. Conversely, you get a partridge in a pair tree a dozen times, but there is only one of them. The answer must lay somewhere inbetween.

The total number of each type of gift given is n(13-n), where n is the day. We can see that it’s a tie between receiving a total of 42 laying geese, and 42 swimming swans.

	Day	Total of that gift
	1	12
	2	22
	3	30
	4	36
	5	40
	6	42
	7	42
	8	40
	9	36
	10	30
	11	22
	12	12

Sing Along

Here is a non-traditional arrangement of the song by Pentatonix.