**CAROLYN’S COMPOSITIONS**

**HOW TO COUNT THE GIFTS GIVEN IN THE TWELVE DAYS OF CHRISTMAS**

**A Simple Question…Right?**

A teacher assigned a project to my Whipple Elementary School class: Who can cut the longest strip of paper from a sheet 8 ½ by 11 inches?

Simple, I thought, and cut a strip 11 inches long before sitting back smugly.

When the results were shown, I discovered my naivety. Some students knew to cut around and around the paper, making lengthy strips. It was an eye opener for thinking creatively and out of the box.

I had the same sensation when I was reviewing information on the Twelve Days of Christmas for this year’s Christmas card/ornament. I came across a simple question: How many gifts would a person have to purchase if they bought every gift in the ditty’s list?

Simple, I thought. There are two ways of counting. One gift for each day, twelve total. Or, one gift the first day, two the second, three the third day, and so on. Just add the total. It’s 364.

Or, 376, if you count the pear tree separately from the first day’s gift, a partridge in a pear tree. But don’t forget the cows needed for the milking…did they need two cows, with the milking maids sharing, or did they each have their own cow? What about the piper’s pipes, the drummer’s drums? Does one count the eggs laid by the geese? It reaches ridiculosity when it is suggested that the count should include the microscopic life on both the animals and the people…

I began to see this question was not as easy to answer as I thought it was. In fact, the further I looked, the more complex the problem was.

Mathematicians have different methods of calculating the number of gifts in the Twelve Days of Christmas. For instance, Pascal’s Triangle: an arithmetical triangle you can use for some neat things in mathematics. Here’s how you construct it:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

Math has never been my high point, so I will abandon attempting how to use it. Except to say that one web site, http://dimacs.rutgers.edu/~judyann/LP/lessons/12.days.pascal.html, introduced me to the use of Pascal’s triangles to calculate the number of gifts in the Twelve Days of Christmas. (The diagrams are not being pasted into this site. If you want to see them you will have to click on the source site.)

The first (red) diagonal of Pascal’s triangle indicates the numbers of new gifts given on the consecutive days. 1 partridge in a pear tree 2 turtle doves 3 French hens 4 calling birds 5 gold rings 6 geese a-laying 7 swans a-swimming 8 maids a-milking 9 ladies dancing 10 lords a-leapin’ 11 pipers piping 12 drummers drumming |

The third (yellow) diagonal of Pascal’s triangle indicates the total number of gifts given.

1 = 1 partridge in a pear tree

4 = 2 turtle doves + 1 partridge in a pear tree

+ 1 partridge in a pear tree

10 = 3 French hens + 2 turtle doves + 1 partridge

in a pear tree + 2 turtle doves + 1 partridge

in a pear tree + 1 partridge in a pear tree

20 = 4 calling birds + 3 French hens + 2 turtle doves

+ 1 partridge in a pear tree + 3 French hens

+ 2 turtle doves + 1 partridge in a pear tree

+ 2 turtle doves + 1 partridge in a pear tree

+ 1 partridge in a pear tree

By the fourth day of Christmas you will have received a total of 20 gifts. At this point, the marked spots on Pascal’s triangle look like a Christmas stocking.

At this point, the math is going Beyond my capabilities, so I will proceed on to another site:

http://www.squarecirclez.com/blog/the-twelve-days-of-christmas-how-many-presents/1686

Notice that on each day there is one partridge (so I will have 12 partridges by the 12th day), and each day from the second day onwards there are 2 doves (so I will have 22 doves), and from the 3rd there are 3 hens (total of 30 hens), and so on. Mathematically speaking, my true love is giving me 1 + 2 + 3 + … +*n* presents on the *n*-th day after Christmas.

The number of presents each day is 1 on the 1st, then 3 on the 2nd, then 6 on the 3rd, then 10 on the 4th. We call this set of numbers the **triangular numbers**, because they can be drawn in a dot pattern that forms triangles. (Again, the diagrams are not pasting in so check the source site for illustrations.) To get the **total** number of presents, we need to add those triangular numbers, like this:

1 on the first day + 3 on the 2nd day + 6 + 10 + …

Another way of writing this is:

On the first day, 1 present.

On the 2nd day, 1 + 3 = 4 presents

On the 3rd day, 1 + 3 + 6 = 10 presents

On the 4th day, 1 + 3 + 6 + 10 = 20 presents.

These partial sums are called **tetrahedral numbers**, because they can be drawn as 3-dimensional triangular pyramids (tetrahedrons).

So how many dots (representing presents) will there be in the 12th tetrahedron?

Are you readers as confused as I am?

Another site explained it this way: The number of gifts on day *n* is the *n*th triangular number. The total number of gifts up to and including day *n* is the sum of the first *n* triangular numbers, known as the *n*th tetrahedral number. In the image below, the total number of balls is the fifth tetrahedral number. The number of balls in each layer are triangular numbers.

It doesn’t help. I’m satisfied with 12 or 364 gifts. How about you?

Perhaps I should digress and return to making the card/ornament based on the golden rings, the fifth day gift in the song.

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*Don’t forget to comment. The reader making the most comments during the month will receive a prize. For details click on: **http://carolyncholland.wordpress.com/monthly-prize-for-comments/*

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**ADDITIONAL READING:**

Christmas. Whose Season Is It?

How to Count the Gifts Given in the Twelve Days of Christmas

THE TWELVE DAYS OF CHRISTMAS 2007-A

THE TWELVE DAYS OF CHRISTMAS: 2008 STYLE

SANTAS, MRS. SANTAS, ELVES & REINDEER WANTED: Please apply—Application #1

SANTAS, MRS. SANTAS, ELF and REINDEER WANTED: Application #2

Carolyn,

Sorry that my explanation was confusing – I can see why you would say that.

I have re-written the first part of the article and I hope it makes more sense now.

http://www.squarecirclez.com/blog/the-twelve-days-of-christmas-how-many-presents/1686

Merry Christmas!

Comment by Zac — December 18, 2009 @ 9:46 am |

Thanks, Zac. I will check it out. Carolyn

Comment by carolyncholland — December 18, 2009 @ 2:19 pm |