How many gifts in "The Twelve Days of Christmas"?

Question

In the carol "The Twelve Days of Christmas," on day 1 your true love sends 1 gift; on day 2 they send 2 new gifts plus repeat the day-1 gift; on day 3 they send 3 new gifts plus the day-2 and day-1 ones; and so on. Each day $k$ , your true love gives $1 + 2 + \cdots + k$ presents.

How many gifts are given in total across all twelve days?

Accepted Answer

The grand total is 364 gifts. Gifts on a single day. On day k you receive 1 + 2 + 3 + \cdots + k = \dfrac{k(k+1)}{2} presents (a triangular number). So: Day Gifts that day --------------------- 1 1 2 3 3 6 4 10 5 15 6 21 7 28 8 36 9 45 10 55 11 66 12 78 Add across all twelve days. 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 = \mathbf{364}. A slicker count. Each type of gift is given on every day from when it first appears through day 12. The "partridge" (day 1 gift) appears all 12 days → 12; the "two turtle doves" appear on 11 days → 2 	imes 11 = 22; the "three French hens" on 10 …

How many gifts in "The Twelve Days of Christmas"?

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