## Sunday, December 11, 2011

### The 12 Days of Christmas

Holiday Cheer!

So it is that time of the year: most of us are in a rush to purchase Christmas presents for our loved ones, perhaps taking advantage of the vast of amount of discounts stores are offering to entice us to buy. This is also the time where Christmas Carols are sung, even played 24/7 on some radio stations. Where I live (Azusa, CA which is about 25 miles east of Los Angeles, or Dodgertown for the baseball fans), I know of two stations that are all-Holiday music all the time until Christmas.

Chances are that you have heard about the "12 Days of Christmas" sung by about every recording artist that's ever lived (at least in America). The song was originally published around 1780 and came to the United States about 1910. The song tells about a very generous person give his/her true love presents for each day of Christmas. The days of Christmas starts on December 25 (or 26 for some cultures), and the giving lasts for 12 days.

How Many Presents?

How may presents did the generous lover give his/her true love? It depends on how the gifts are counted.

Take the second day for example:

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

Taking a literal count, we can say the true love received a total of three gifts: the doves AND another partridge in a pear tree.

However, some view each day as a summary of all the gifts after the new set of gifts were received. Hence, the true love received only the two doves because the true love was already given the partridge in a pear tree.

Some even count the partridge as one gift and the pear tree as another. For simplicity, I'll count the partridge in a pear tree as one gift.

The Math

To calculate the number of gifts given during the 12 Days of Christmas, it is only a matter of addition.

However, we can use the shortcut if we are adding consecutive numbers from 1 to N:

S = ∑(K = 1, N, K) = ((N + 1) × N) ÷ 2

How do we get this?

Let S represent the sum:

S = 1 + 2 + 3 + ... + (n-2) + (n-1) + n

2 × S = 1 + 2 + 3 + ... + (n-2) + (n-1) + n + 1 + 2 + 3 + .... + (n-2) + (n-1) + n

Note that addition is communicative. That is for any two numbers a and b, a + b = b + a. This will allow us to rearrange the terms on the right side of the equation like so:

2 × S = 1 + n + 2 + (n-1) + 3 + (n-2) + .... + (n-2) + 3 + (n-1) + 2 + n + 1

Add each pair of terms on the right side. Note that 2 + (n-1) = n+1, 3 + (n-2) = n+1, and so on.

2 × S = (n+1) + (n+1) + (n+1) + .... + (n+1) + (n+1) + (n+1)

Observe that there are n terms of (n+1) on the right side.

2 × S = n × (n+1)

Divide both sides by 2 and we get:

S = (n × (n + 1)) / 2

We'll use this formula to help us count the number of presents.

Summary Count

If each day the lucky recipient merely recaps what he/she has received to that day, then the number of gifts the person gets is 78.

S = (12 × (12 + 1)) / 2 = (12 × 13) / 2 = 78 which consists of:

One partridge in a pear tree, (received on the first day)
Two turtle doves, (received on the second day)
Three French hens, (etc...)
Four calling birds,
Five gold rings,
Six geese-a-lying,
Seven Swans-a-swimming,
Eight Maids-a-milking,
Ten Lords-a-leaping,
Eleven Pipers pipping, and
Twelve Drummers drumming

Note there is no mention to what exactly the maids were milking but I digress.

Literal Count

A popular view to find the number of gifts is to make a literal count. That means each time the singer person sings about what he/she got from his/her true love actually was received. From this perspective, after the 12 Days of Christmas were over, the person actually got a total of 12 partridges in pear trees, along with 22 turtle doves, and so on. Hopefully the person has a big yard to fit all the pear trees, the turtle doves, a lake for all the geese, and many rooms to house the maids, ladies, lords, pipers, and drummers.

Let's calculate the total shall we?

First Day:
A Partridge in a Pear Tree

This is easy:
S = 1
T = 1

Second Day:
Two Turtle doves and
A Partridge in a pear tree

S = 1 + 2 = (2 × 3) / 2 = 3
T = 3 + 1 = 4 (1 from Day 1 and 3 from Day 2)

Third Day:
Three French hens,
Two Turtle doves, and
A Partridge in a pear tree

S = 1 + 2 + 3 = (3 × 4) / 2 = 6
T = 4 + 6 = 10 (6 from Day 3 and 4 from the past two days)

Fourth Day:
Four calling birds,
Three French hens,
Two Turtle doves, and
A Partridge in a pear tree

S = 1 + 2 + 3 + 4 = (4 × 5) / 2 = 10
T = 10 + 10 = 20

Fifth Day:
Five golden rings,
Four calling birds,
Three French hens,
Two Turtle doves, and
A Partridge in a pear tree

S = 1 + 2 + 3 + 4 + 5 = (5 × 6) / 2 = 15
T = 20 + 15 = 35

Sixth Day:
Six geese-a-laying,
Five golden rings,
Four calling birds,
Three French hens,
Two Turtle doves, and
A Partridge in a pear tree

S = 1 + 2 + 3 + 4 + 5 + 6 = (6 × 7) / 2 = 21
T = 35 + 21 = 56

If the true love stopped here, the person has already accumulated a nice haul of 56 presents! But we know the true love kept going, so...

Seventh Day:
Seven Swans-a-swimming and everything else from the first six days

S = 1 + 2 + 3 + 4 + 5 + 6 + 7 = (7 × 8)/2 = 28
T = 56 + 28 = 84

Eighth Day:
Eight Maids-a-milking and everything else from the first seven days

S = 1 + ... + 8 = (8 × 9)/2 = 36
T = 84 + 36 = 120

This has got to cost the true love a fortune!

Ninth Day:
Nine Ladies dancing and everything else from the first eight days

S = 1 + ... + 9 = (9 × 10)/2 = 45
T = 120 + 45 = 165

Tenth Day:
Ten Lords-a-leaping and everything else from the first nine days

S = 1 + ... + 10 = (10 × 11)/2 = 55
T = 165 + 55 = 220

Eleventh Day:
Eleven Pipers pipping and everything else from the first ten days

S = 1 + ... + 11 = (11 × 12)/2 = 66
T = 220 + 66 = 286

Twelfth Day:
Twelve Drummers drumming and everything else from the first eleven days

S = 1 + ... + 12 = (12 × 13)/2 = 78
T = 286 + 78 = 364

The grand total of all the gifts were 364. I can only imagine that the last few days busy the person and the true love.

Here is a link that has the lyrics of "The Twelve Days of Christmas".

Twelve Days of Christmas

Happy Holidays everyone! Merry Christmas! May this month (and all months thereafter) find you peace, love, warmth, and gratitude!

Eddie

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