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How Many Gifts Were Given in the 12 Days of Christmas?
One of the intriguing questions that often comes up during the holiday season is, "How many gifts were given in the 12 Days of Christmas?" This question pertains to the classic Christmas carol "The Twelve Days of Christmas," which is believed to detail the gifts received by the singer from their "true love" each day over a period of 12 days.
First, let’s recount what gifts are received on each day in the song:
- 1st day: A partridge in a pear tree
- 2nd day: Two turtle doves
- 3rd day: Three French hens
- 4th day: Four calling birds
- 5th day: Five golden rings
- 6th day: Six geese a-laying
- 7th day: Seven swans a-swimming
- 8th day: Eight maids a-milking
- 9th day: Nine ladies dancing
- 10th day: Ten lords a-leaping
- 11th day: Eleven pipers piping
- 12th day: Twelve drummers drumming
On each subsequent day, the gifts from the previous days are repeated along with a new gift. To find the total number of gifts given by the end of the 12 days, we need to calculate the cumulative total of all the presents mentioned in the song.
The easiest way to do this is to break it down day by day and keep a running total.
Day-by-Day Breakdown
- On the 1st day: 1 partridge in a pear tree (1 gift)
- On the 2nd day: 2 turtle doves + 1 partridge in a pear tree (3 gifts total)
- On the 3rd day: 3 French hens + 2 turtle doves + 1 partridge in a pear tree (6 gifts total)
- On the 4th day: 4 calling birds + 3 French hens + 2 turtle doves + 1 partridge in a pear tree (10 gifts total)
- On the 5th day: 5 golden rings + 4 calling birds + 3 French hens + 2 turtle doves + 1 partridge in a pear tree (15 gifts total)
- On the 6th day: 6 geese a-laying + 5 golden rings + 4 calling birds + 3 French hens + 2 turtle doves + 1 partridge in a pear tree (21 gifts total)
- On the 7th day: 7 swans a-swimming + 6 geese a-laying + 5 golden rings + 4 calling birds + 3 French hens + 2 turtle doves + 1 partridge in a pear tree (28 gifts total)
- On the 8th day: 8 maids a-milking + 7 swans a-swimming + 6 geese a-laying + 5 golden rings + 4 calling birds + 3 French hens + 2 turtle doves + 1 partridge in a pear tree (36 gifts total)
- On the 9th day: 9 ladies dancing + 8 maids a-milking + 7 swans a-swimming + 6 geese a-laying + 5 golden rings + 4 calling birds + 3 French hens + 2 turtle doves + 1 partridge in a pear tree (45 gifts total)
- On the 10th day: 10 lords a-leaping + 9 ladies dancing + 8 maids a-milking + 7 swans a-swimming + 6 geese a-laying + 5 golden rings + 4 calling birds + 3 French hens + 2 turtle doves + 1 partridge in a pear tree (55 gifts total)
- On the 11th day: 11 pipers piping + 10 lords a-leaping + 9 ladies dancing + 8 maids a-milking + 7 swans a-swimming + 6 geese a-laying + 5 golden rings + 4 calling birds + 3 French hens + 2 turtle doves + 1 partridge in a pear tree (66 gifts total)
- On the 12th day: 12 drummers drumming + 11 pipers piping + 10 lords a-leaping + 9 ladies dancing + 8 maids a-milking + 7 swans a-swimming + 6 geese a-laying + 5 golden rings + 4 calling birds + 3 French hens + 2 turtle doves + 1 partridge in a pear tree (78 gifts total)
Now, to find the total, we sum up all the gifts accumulated over the 12 days. Adding them together:
1 (first day) + 2+1 (second day) + 3+2+1 (third day) + 4+3+2+1 (fourth day) + 5+4+3+2+1 (fifth day) + 6+5+4+3+2+1 (sixth day) + 7+6+5+4+3+2+1 (seventh day) + 8+7+6+5+4+3+2+1 (eighth day) + 9+8+7+6+5+4+3+2+1 (ninth day) + 10+9+8+7+6+5+4+3+2+1 (tenth day) + 11+10+9+8+7+6+5+4+3+2+1 (eleventh day) + 12+11+10+9+8+7+6+5+4+3+2+1 (twelfth day).
This calculation essentially sums up a sequence of triangular numbers:
1 + (1+2) + (1+2+3) + (1+2+3+4) + (1+2+3+4+5) + (1+2+3+4+5+6) + (1+2+3+4+5+6+7) + (1+2+3+4+5+6+7+8) + (1+2+3+4+5+6+7+8+9) + (1+2+3+4+5+6+7+8+9+10) + (1+2+3+4+5+6+7+8+9+10+11) + (1+2+3+4+5+6+7+8+9+10+11+12).
Each triangular number can be calculated using the formula \( \frac{n(n+1)}{2} \), so the sum of the triangular numbers from 1 to 12 is:
\( \sum_{n=1}^{12} \frac{n(n+1)}{2} \)
This can be simplified to:
\( \sum_{n=1}^{12} \frac{n^2 + n}{2} = \frac{1}{2} \left( \sum_{n=1}^{12} n^2 + \sum_{n=1}^{12} n \right) \)
Using the formula for the sum of the first \( n \) squares (\( \sum_{n=1}^{k} n^2 = \frac{k(k+1)(2k+1)}{6} \)) and the formula for the sum of the first \( n \) numbers (\( \sum_{n=1}^{k} n = \frac{k(k+1)}{2} \)), we get:
\( \sum_{n=1}^{12} n^2 = \frac{12 \cdot 13 \cdot 25}{6} = 650 \)
\( \sum_{n=1}^{12} n = \frac{12 \cdot 13}{2} = 78 \)
So the total number of gifts is:
\(\frac{1}{2} \left( 650 + 78 \right) = 364 \)
Therefore, the total number of gifts given in the "Twelve Days of Christmas" is 364.
This result is interesting because it also represents nearly the number of days in a year, minus one day, perhaps symbolically suggesting year-round giving minus just a single day.
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