How much crap do you think I need?

Everyone knows the song “12 days of Christmas”. But do you know exactly how many things “my true love gave to me”?

Let’s clarify the lyrics as receiving one gift (a partridge in a pear tree – we won’t count the tree separately) on “the first day of Christmas”, and THREE gifts on the second day (two turtle doves AND a partridge in a pear tree), etc. etc.

On day x, I have

d(x) = 1 + 2 + 3 + ... + x = x(x+1)/2

gifts, by the formula for the sum of the first x natural numbers. On the nth day, I get

S(n) = d(1)+d(2)+...+d(x) = 1(1+1)/2 + 2(2+1)/2 + ... + n(n+1)/2

 gifts, or:

S(n) = (1² + 2² + ... + n²) /2 + (1 + 2 + 3 + ... + n)/2.

Simplifying - using the formula for the sum of the first n squares and the first n numbers, we have:

S(n) = n(n+1)(2n+1)/12 + n(n+1)/4

Further simplifying we get:

S(n) = n³/6 + n²/2 + n/3

For n=12 or 12 days we get:

S(12) = 12³/6 + 12²/2 + 12/3
1728/6 + 144/2 + 12/3
288 + 72 + 4
364

You can prove this the hard way:
One
1 +
Two
(1 + 2) +
Three
(1 + 2 + 3) +
Four
(1 + 2 + 3 + 4) +
Five
(1 + 2 + 3 + 4 + 5) +
Six
(1 + 2 + 3 + 4 + 5 + 6) +
Seven
(1 + 2 + 3 + 4 + 5 + 6 + 7) +
Eight
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) +
Nine
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) +
Ten
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) +
Eleven
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11) +
Twelve
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12)=
364
That’s 364 birds, musicians, dancers, and rings.
Who needs that much stuff?
On the bright side, in today’s market all those gifts would be worth more than $20,000.00. So if you returned all of it you could buy yourself a car or something.