How many total gifts in the twelve days of christmas if we extend 12 to any number?

Question

I got this question today in an interview: write a function to calculate the total number of gifts received for any day in the 12 days of christmas song. I wrote a simple function using a for() loop in c#'ish code that worked. Then the interviewer asked me to extend it to any number of days. The conversation then turned to how to optimize the loop. Apparently there's a cool math trick that will do this within the limits of whatever your integer is. Anyone know what it is and what it's called? Any language is ok and a reference to the algorithm would be fabuloso.

Answers that use recursion are NOT what I'm looking for.

EDIT: Answer for day 2 is 4 gifts total, not 3 since I will have 2 Trees (1 from today, 1 from yesterday) and 2 partridges. On day 12 I'll have received a total of 364. I want the formula that lets me input 12 and get 364.

Answer 1

On the n th day, we get 1 + 2 + 3 + ... + n gifts.

Or ... (1 + n) + (2 + n-1) + ...

In other words, (n + 1) * n/2.

Answer 2

• On the first day, you get 1.
• On the second day, you get 1 + 2.
• On the third day, you get 1 + 2 + 3.
• ...
• On nth day, you get 1 + 2 + 3 + ... + n.

The sum 1 + 2 + ... + n is n(n+1)/2. So the total number, T(N) is the sum of n(n+1)/2 for n in 1..N, where N is the number of days.

Now, n(n+1)/2 = n^2 / 2 + n / 2, and sum of n^2 for n in 1..N is N(N+1)(2N+1)/6, so you get:

T(N) = N(N+1)(2N+1)/12 + N(N+1)/4
     = N(N^2 + 3N + 2) / 6

No loops. No recursion.