I am a former maths teacher and owner of DoingMaths. I love writing about maths, its applications and fun mathematical facts.
The Twelve Days of Christmas: A Christmas Carol
'The Twelve Days of Christmas' is perhaps one of the most famous and most often sung Christmas carols of all.
It tells the tale of a person receiving gifts from their true love on each of the twelve days following Christmas. They start off on the first day with just the one gift—a partridge in a pear tree. On day two, as well as the partridge, they receive two turtle doves. On day three, they also get three French hens on top of the presents already mentioned.
This continues until day twelve, when we finish with the following verse:
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 gold rings
Four calling birds
Three French hens
Two turtle doves
and a partridge in a pear tree.
How Many Gifts Would I Receive Each Day?
One way to tackle the question of how many gifts you are receiving is to look at how many gifts are being received each day.
On day one this is simple; just the one gift.
On day two we get 1 + 2 = 3 gifts.
On day three we get an additional three gifts on top of day two's three and so receive 3 + 3 = 6 gifts.
On day four we add four to the previous day's total getting 6 + 4 = 10.
This continues all the way to twelve. For each day, we add that day's number to the tally from the previous day giving us the totals in the table below.
The Total Number of Gifts Received on Each Separate Day of Christmas
|Day Number||Total Number of Gifts Received on This Day|
The Total Number of Gifts Received
By adding up each of the numbers in the 'Total number of gifts received' column, we find we receive a whopping 364 presents in total. That's only one short of a gift a day for an entire year.
The Mathematics Behind the Totals
The number of gifts given each day follows an interesting mathematical pattern. Let's take a closer look.
Day 1 - 1
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Day 2 - 1 + 2 = 3
Day 3 - 1 + 2 + 3 = 6
Day 4 - 1 + 2 + 3 + 4 = 10 and so on.
This sequence; 1, 3, 6, 10, 15, 21, …, where the amount being added increases by one each time, is known as the triangular numbers, so called because we can create triangles by having 1 dot in the top row, 2 dots in the second row and so on as in the image below.
To find out more, read my article all about the Mathematics Behind the Triangular Numbers.
Tetrahedral Numbers and the Twelve Days of Christmas
We get another interesting sequence of numbers if we keep a running total of all of the gifts received up to and including each day. Take a look at the table below. In this table, as well as the number of gifts received each day, I have also put a running total in the right-hand column by adding together all of the numbers up to and including that day.
|Day Number||Total Number of Gifts Received on the Day||Total Number of Gifts Received by This Day|
The numbers appearing in the right-hand column form a sequence known as the tetrahedral numbers. Just like the triangular numbers are so called as they form triangles, the tetrahedral numbers are how many spheres (balls) are needed to make a tetrahedron (otherwise known as a triangular-based pyramid).
What About the Total Number of Each Gift?
We have seen so far that at the end of the twelve days we have a grand total of 364 gifts. We have also seen how many gifts we receive each day and how many gifts we will have received in total up to each day. But what about how many of each type of gift?
The partridge in the pear tree is easy. We get one of these on each of the twelve days, hence we end up with 1 × 12 = 12 partridges in pear trees.
We receive two turtle doves on each day apart from the first day, making 11 days, and so receive 2 × 11 = 22 turtle doves.
We receive three French hens on each day apart from the first two days, making 10 days and so receive 3 × 10 = 30 hens.
This continues following the same pattern; as the day number increases by one, the number of times it is gifted decreases by one. Mathematically this can be represented as so:
For a gift that is first received on the nth day;
Total number = n × (13 − n)
Using this formula, the total number of each gift has been calculated in the table below. You can see that the numbers form a pleasingly symmetrical pattern.
The Totals of Each Type of Gift
|Gift||Calculation||Total Number Received|
Partridges in pear trees
1 × 12
2 × 11
3 × 10
4 × 9
5 × 8
6 × 7
7 × 6
8 × 5
9 × 4
10 × 3
11 × 2
12 × 1
364 Gifts in 12 Days!
So there we have it. Over the 12 days of Christmas, your true love has given you a total of 364 gifts which includes a incredible 42 geese and, stranger still, 30 leaping lords!
This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.
© 2021 David