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.”