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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If the pattern is,
1, 1+2, 1+2+3, 1+2+3+4, …,
then, answer may be
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
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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.
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Hi,
Thanks.
From the diagram, I understand now the pattern of
1, 3, 6, ...
Thanks, and a little bit early Merry Christmas!