Once there was a song. It was maddeningly repetitive and endlessly parodied.

It endorsed “quantity over quality” in gift-giving. It advocated buying for our loved ones not a good book, or a nice pair of pajamas, or even a scented candle, but teeming masses of geese, each one actively laying eggs.

The song proposed, dubiously, that *human people* could be given as gifts—maids, pipers, drummers, even lords and ladies. The song’s human-trafficking aspect has received little attention to date.

The song pegged the length of Christmas at a mystifying 12 days. The religious holiday, so far as I know, lasts 24 hours, and the secular capitalist holiday lasts roughly two-thirds of the year. But I guess “The One Day of Christmas” wouldn’t make much of a song, and “The 247 Days of Christmas” might bring about the slow, irritating end of human civilization.

The song invited a lot of questions, but one in particular leapt out to the mathematical mind. (Or to the bored mind seeking stimulation—which is fundamentally the same thing.) The question: How many gifts, in all, did my true love give to me?

I first heard this question asked during bar trivia. There were 12 of us, spread across two tables—10 UC Berkeley mathematicians (including my wife), little-old-high-school-teacher me, and an engineer named Neel.

Advantage: mathematicians. Or so you’d think.

After a few moments, the 10 mathematicians (okay, PhD students, but close enough) threw down their pencils in triumph, and showed the following solution. It entails finding the total number of gifts for each day, and then adding up those figures:

It gets you the answer, sure. But it requires sigma notation, a formula for finding the n-th triangular number, and after all that, a fair bit of messy arithmetic.

Neel and I glanced at each other, then shared the solution we’d arrived at independently. It begins by finding the number given of each type of gift:

Then, noting that the second set of 6 gifts has exactly the same total as the first set of 6, we do a little arithmetic to arrive at our final answer:

No sigma notation, no formulas, and no nasty arithmetic. Even the mathematicians agreed it was a better method for finding the number.

“On the twelfth day of Christmas, my true love gave to me,” we later sang, “10 mathematicians, and 2 people who could actually do math efficiently.”

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The religious holiday does indeed last (or at least used to last, or something) 12 days! http://en.wikipedia.org/wiki/12_Days_of_Christmas

Ah, that’s right! Bringing back some Spanish-class memories of “Dia de Los Reyes,” which I think corresponds to the 12th day of Christmas (where Dec. 25th is the first).

I gave this problem to my 4th graders in my first year teaching. (Minimal prep! I was just barely keeping my head above water planning wise, and I was pretty certain this could fill at least 30-45 minutes on one of our final sugar rush days before December break. I doubt I had even bothered to solve for the exact answer before handing it to my students.)

Some of the kids took 30 minutes, and benefited from some prompting. “Hmm… how can we organize our thinking? What have we used before?” (COME ON, KIDS, WE USE TABLES EVERY DAY.)

My more precocious 9-year-olds solved the problem the same way you did, and did so quickly.

Every 4th grader solved it. Not a single one used sigma notation. Sometimes I really do think the young ones benefit from not being hindered by trained formalities. I have seen 3rd graders solve systems of equations faster than 8th graders. (That said, my 8th graders are MUCH better about using generalizable strategies…)

Yeah, well said. I think that’s exactly the tradeoff: Trained formalities can hinder our speed and intuition, but they give us certainty and generalizability.

I guess, broadly speaking, the best learning path is (1) build local intuition; (2) generalize to an algorithmic procedure; (3) learn when to apply the algorithm, and when to employ a shortcut method.

And yeah, Michael, I fear sigma notation in 4th grade is a dystopia not too far from our own…

Hell yes we kicked ass that trivia! Perhaps this is a sign that we should resuscitate that tradition. Happy holidays my friend!

I’d totally be down for some Red Tomato trivia next semester. (Kipp’s is a little late for me, as it is for all sane citizens of the world.)

The 12th day of Christmas is when epiphany falls, which is the last day of Christmas in the religious calender and the reason that it’s considered unlucky to leave your Christmas decorations up past January 6th

Whoa, it’s unlucky past January 6th? That explains some things about my childhood…

It is indeed a nice and simple way to have young students understand! I wouldnt say its more efficient than using summations though, cuz if you are calculating for 247 days of christmas then your method will be tiring =( summations will still give the ans with just the board you use =)

Definitely! My and Neel’s method is simpler for a small number of days, but for a large number of days, you’d eventually want to resort to sigma notation. (Our method can be written using sigma notation as well, but I’m pretty sure it will wind up being computationally equivalent.)

Has anyone else ever seen this the way I do, that the SONG is recursive but not gift giving? I guess that would eliminate the need to do all this math and wouldn’t allow the media to rehash, every Christmas season as if it was the first time anyone ever thought of the idea, the “actual” number of presents given by the True Love. 78 gifts in total has been my take on this since pretty much forever, ONE set of each item.

This is from a guy who opted out of his 6th grade Secret Santa, so take me for the Scrooge I so obviously am.

Your reading makes a lot more sense. I think that was the way I first interpreted it, but with time the commercialistic, gift-maximizing spirit of the holiday must have overtaken me.

Reblogged this on tglennb and commented:

Try this at 11:00pm on New Year’s Eve!

I’m surprised the mathematicians were so slow.

It’s just Sigma(n(13-n)), so that’s 13*Sigma(n) – Sigma(n^2), which any mathematician ought to be able to read off as (13×78) – 630 = 384.

Perhaps you would also be surprised at how many mathematicians are terrible with numbers.

I know I was!