## The Nth Day of Christmas

Today’s the Nth day of Christmas (The tenth day, to be precise) — as a function of N, about how many gifts total has my true love given me since the first day of Christmas?

It’s cooler to have a quick estimate of the rate at which the number of gifts grows, rather than the exact formula; there’s a quick way to get a rough sense of the right answer.

Does the total number of gifts grow exponentially? Factorially? As the square of N? Something else?

Don’t peek until you want the answer!

[spoiler]

Ok: on the Nth day, my true love gives to me 1+2+3+…+N gifts; you might know the formula for that sum, and it’s a good exercise to work it out again, but the really important thing is that when you sum up consecutive integers, the total grows about like N^2 (ignoring pesky constants)

In fact, if we sum up consecutive kth powers, the total grows about like N^(k+1). This is really a kind of counting version of integration (and in fact, is exactly one of the tools ancient mathematicians, such as Archimedes, used to work out certain integral calculus problems 2000 years before Newton and Leibnitz invented calculus!) So if on day n, we receive about n^2 gifts, by day N, we’ve received about 1^2 + 2^2 + 3^2 + … + N^2 gifts, for a total of roughly N^3.

Thus the total grows about like N^3, ignoring pesky constants and lower level terms. The exact formula, should you need it for checking your true love’s love, is going to be N^3/6 + N^2/2 + N/3; on day twelve you should be expecting a total of 364 gifts!

[/spoiler]

## Pete said,

January 4, 2011 at 1:35 pm

Clarification (spoiler free):

What threw me off on this was that I forgot that the gifts “re-total” every day. For instance, on the 3rd day of Christmas, the true love in question gifts three french hens, two turtle doves, and a partridge in a pear tree. This is in addition to the two french hens and partridge (+ tree) from day 2 and the partridge (+ tree) from day 1, bringing the total accumulated gifts on day 3 to 3 french hens, 4 turtle doves, and 3 partridges (plus pear-producing perennials).

Otherwise the math is too easy and not as much fun. :-)

## Stephen Morris said,

January 8, 2011 at 1:29 am

I found the answer using an old friend [spoiler]Pascal’s Triangle[/spoiler]

## JK said,

January 20, 2011 at 12:09 am

We can take this a step further and determine the derivative for the equation. This would be: (3/6)*N^2+N+(1/3)

## Andrew Austern said,

February 5, 2011 at 5:28 pm

I didn’t know that you repeat the previous gifts for each day. I just thought it was re-describing what had already been given. That’s what I get for being Jewish. :)