# Personal Blogs

## If Christmas lasted all year

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A popular question in Christmas quizzes asks how many gifts were sent, and received, altogether in the carol The Twelve Days of Christmas.

So on day 1 (sing the song) it was 1

On day 2 (sing the song) it was 2 plus 1 = 3

On day 3 plus 2 plus 1 = 6

….

What is the total after 12 days: 1 plus 3 plus 6 …?

Now, what if there were 365 days of Christmas? What would the grand total be then?

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### New comment

If the pattern is,

1, 1+2, 1+2+3, 1+2+3+4, …,

n*(n+1)/2

after 12days, 12*13/2=78

after 365 days, 66795

But, I don’t know, because the number of gifts after 12 days is

1+3+6+…

not

1+2+…+12

### New comment

From Masami's work, a leap year is worth an extra 67,161 gifts!

We are stacking triangular numbers, and looking for the "volume" of the resulting shape:

                    *           *       * *    *     * *     * * **  * *   * * *   * * * *

sum(i * (i + 1) / 2) = 1/2 (sum(i^2) + sum(i)) = some faffing = n(n+1)(n+2)/6

For 365 days, the sum is 365 * 366 * 367 / 6 = 8,171,255 gifts -- which is going to need a lot of loft space! Humbug.

### New comment

Hi,
Thanks.
From the diagram, I understand now the pattern of
1, 3, 6, ...

Thanks, and a little bit early Merry Christmas!