12/22/08 - Advanced Math and the 12 Days of Christmas

For those of you who don't know, I have a bachelor's degree in Computer Science with a minor in math. I'm 1/3 of the way to getting my master's in Applied Computing (but have no plans on going back to finish...that's a story for a different post). At one point, my desire was to get my master's in math, but UCA did not offer the degree path I wanted to take. I say that, only to explain why this next posts makes sense on my blog. I LOVE math, problems, numbers and any combination of those.

At my work Christmas party, there were several games and prizes and one of the questiosn during one of the games was to calculate how many total gifts were received during the 12 Days of Christmas. For those not familar with the song, here are the lyrics.

Before I go any further, I want to explain that I will be using fairly complex mathematical equations, so if you are geniunly terrified of the entire subject of math (which I've met several people who are), I recommend that you stop reading now. Also, as a disclaimer, some purists would say that two gifts are received during the first day of giving, both the partridge and the pear tree, but I do not view it this way, it is considered all one gift in my book. Other versions of this problem state that the maids a milking had to be milking something and therefore you have to count those items as well. The drummers included a drum and drumsticks, so on and so forth. For simplicity, I am considering each item to be the actual number as stated in the snog.

At first thought, it seems simple enough, just add 1 for the first day, 2 for the 2nd day, and so on until you add 12 for the 12th and final day. Or basically, a summation of the numbers 1-12. This can be denoted as follows:

The answer to this equation, is of course 78, but that answer would be incorrect, because we simply calculated how many gifts were received on the 12th day. If you go back and look at the lyrics, by the end of the song, the receiver of gifts has been given a total of 12 partridges in a pear tree. This means that on day 1 they receive 1 gift, day 2 they receive a total of 3 gifts, and on day 3 they receive 6 gifts. This is a double summation problem as denoted in the following equation:
\sum_{n=1}^{12} ({\sum_{m=1}^{n} m})

This is not quite as easy to solve, but with the help of simple summation rules, we can get it down to a simplied version. Those rules I'm referring to can be found, summarized and explained on these two websites: yongyoon.net PDF File and UCDavis Summation Rules page.

Summation Rule 2 on the UCDavis page says that we can simplify to this equation:
\sum_{n=1}^{12} \frac{n(n + 1)}{2}} = \sum_{n=1}^{12} \frac{n^2 + n}{2}}

By using Rule 1 and Rule 2 found in the yongyoon.net PDF file, the following equation jumps back to something a little more complicated, as follows:
\frac{\sum_{n=1}^{12}n^2 + \sum_{n=1}^{12}n}{2}}

Now using Rules 2 and 3 from the UCDavis page again, we can simplify further, removing the summation sybol (sigma) completely:
\frac{\frac{n(n+1)(2n+1)}{6} + \frac{n(n+1)}{2}}{2}}

When n=12, as we are solving for, we simplify this even further to the following, easily solved equation, which as you can see gives the magical answer of 364.
\frac{\frac{{12}({12}+1)(2*{12}+1)}{6} + \frac{{12}({12}+1)}{2}}{2}} = 364

In summary, the whole point of this was to solve mathematically, the total number of gifts received during the song "12 Days of Christmas". The basic premise is that the base equation is a double summation. Every day, the receiver gets that days presents as well as a repeat of all the previous days. For simplicity, here is the "final answer":
\sum_{n=1}^{12} ({\sum_{m=1}^{n} m}) = 364

*To create the fancy LaTex images, I used Roger's Online Equation Editor. It appears that it is a free resource, and he even offers hosting of the images provided. Although I do not generate much traffic at all, I didn't want to steal any of this bandwidth and therefore am hosting the images on my own site.


4 Comments - Leave one yourself

You lost me but I'm not a math person. I don't hate it but don't love it either.

By Blogger sdhorton, at December 22, 2008 at 1:20 PM  

Very interesting. Doesn't surprise me that you probably REALLY enjoyed this. I followed it, actually understood it, but I would have never been able to figure that out on my own. I guess you love math, I guess I love audits. We are both pretty much nerds...I just look better than you!

By Anonymous Anonymous, at December 22, 2008 at 3:39 PM  

I'm getting back into math so that I can start college in the fall. It's been fun.

I plan to use math to put objects into space.

By Anonymous Anonymous, at December 22, 2008 at 4:48 PM  

Nerd. :P

By Blogger babibootiful, at December 22, 2008 at 8:08 PM  

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