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### Twelve Days of Christmas Gifts

```
Date: 12/06/2001 at 18:26:25
From: Katie

How many presents did the person receive in the song "The 12 Days of
Christmas"?
```

```
Date: 12/07/2001 at 15:58:16
From: Doctor Ian

Hi Katie,

On the first day, there is one gift:

1

On the second day, there is another one of that gift, and two more:

1
1 2

On the third day,

1
1 2
1 2 3

and so on, up to 12 days:

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
^ ^ ^
| | |
| | +---------------------------- 10 * 3
| +------------------------------ 11 * 2
+-------------------------------- 12 * 1

So that's

12*1 + 11*2 + 10*3 + 9*4 + ... 4*9 + 3*10 + 2*11 + 1*12

Note that some of the terms are the same:

12*1 + 11*2 + 10*3 + 9*4 + ... 4*9 + 3*10 + 2*11 + 1*12
\      \      \     \_________/     /      /      /
\      \      \___________________/      /      /
\      \_______________________________/      /
\___________________________________________/

That's convenient, because it means you only have to add up half the
terms:

2 * [12*1 + 11*2 + 10*3 + 9*4 * 8*5 * 7*6]

Can you take it from here?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Puzzles
Middle School Word Problems

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