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[SINGING] On the first day of Christmas, my true love gave to me, the multiplicative identity. I always hated the song the "12 days of Christmas" when I was younger. Not that the tune or the words are any worse than any other Christmas song, but it's just so long and repetitive. Singing it sucks too, because like math class, it seems no matter how hard you try to pay attention, you lose focus out of the sheer tedium of it all and forget where you are. Unless you keep vigorous records by drawing complicated graphs. [SINGING] On the second day of Christmas my true love gave to me, the only even prime, and the absolute value of e to the i pi. See? It doesn't have to be repetitive, but the "12 days of Christmas" is more fun to think about, then to actually sing or listen to. I do like these kinds of visualizations. And you can think of other ways to visualize the process of singing the "12 Days." Which might come in useful next time you're at a family Christmas party, and someone insists on singing through the whole thing. [SINGING] On the third day of Christmas my true love gave to me, the number of spatial dimensions, at least macroscopically-- don't yell at me string theorists-- the number of points that define a line, and the limit of sine x over x as x goes to zero. Unlike a normal, reasonable songs, adding 12 more verses wouldn't just make it twice as long, because the stupid thing just grows and grows. Even 99 bottles of beer on the wall has the promise of getting to zero, unless you believe in anti-beer. But "12 Days" is just disheartening, because the closer you think you're getting to the end, the longer the verses get. Dragging it out to the bitter end. [SINGING] On the fourth day of Christmas my true love gave to me, the smallest possible number of sides on a polyhedron, the number of points that define a plane, the divisor of even numbers, and any other number to the power of 0. If I had a time machine, and was not bitterly anti-time travel-- and yes I've actually protested with a sign-- one of the things on my list would be to go back to the year 0, pick your era, and be like, hey three kings of orient, hurry it up, will ya? Also what's myrrh? Because it sounds like the noise a camel makes. Really though, five days of Christmas would be more reasonable. Even eight I could live with. [SINGING] On the fifth day of Christmas, my true love gave to me, five golden ratio producing pentagons, the number of sides on a square, the number of sides on that rigid, functional, and beautiful creature called the triangle, and I guess a two, and the number of sides on a Mobius strip. Graphing the numbers like this may seem trivial, but I think it's nice to be reminded that these numbers aren't just squiggles on a page, symbols to be manipulated, but actually represent something. And I think it's nice that it results in such a lovely triangle. You know how I feel about triangles. [SINGING] On the sixth day of Christmas, my true love gave to me, my name in Roman numerals, number of feet in iambic pentameter, my name in Roman numerals backwards, the first Mersenne prime, the number of syllables in a foot of iambic pentameter, and sine x squared plus cosine x squared. Right. So if you want to know how many times you're going to have to sing a line about whatever stuff your true love got you, it's like counting up all the things in one of these triangles. That has however many layers, in this case 12. The answer will be what's called a triangular number, for obvious reasons. You can also shift the things around to get an equilateral triangle, which is how triangular numbers are usually explained. So the first triangular number is 1, the second is 3, the third is 6, the fourth is 10. [SINGING] On the seventh day of Christmas, my true love gave to me, the most common lucky number, the first perfect number, the only prime ending in five. The number of colors sufficient for coloring in a map, the only prime triangular number, the highest number that is its own factorial, and 1/2 plus 1/4 plus 1/8 plus 1/16 and so on. Almost. Here's a famous story that I've heard told a few different ways, and the actual facts are fuzzy, but basically, here's the gist of it. Carl Gauss was bored in his math class. I imagine why, because not only was Gauss a pretty smart guy, but math classes sucked 300 years ago, just as much as they suck today. And Gauss would get himself in trouble when he was bored. Maybe he also liked to escape via the window. So his teacher got fed up and was like, Gauss-- I mean, he probably called him Carl, but that's not the point-- Guass, if you're so freaking bored with my class, how about you go sit in the corner and add up all the numbers between 1 and 100. That ought to keep you busy for a while. So Gauss goes to the corner, but he's just sitting there. And the teacher gets all mad and he's like, hey Gauss. I guess that means you've already added up all those numbers, right? And Gauss is like, sure, it's 5,050. [SINGING] On the eighth day of Christmas, my true love gave to me, the only perfect cube that's a Fibonacci number besides one, the number of Frieze patterns, the number of sides on a cube, the number of platonic solids, the first composite number, the number of regular polytropes in all dimensions greater than four, the Euler characteristic of polyhedra homeomorphic to a sphere, and the number where if it's the base of a logarithm it's undefined. His teacher of course, didn't believe him. I think the teacher spent the next 10 minutes adding up the numbers by hand in an effort to catch Gauss in a lie, and when he saw that Gauss was right, he probably gave him detention any way. Or more likely whacked him with a ruler a few times for having the gall to be smarter than him. Or it could be that the story's mostly made up. I don't know. Nevertheless, here's how he did it. Instead of adding up the numbers individually, like his teacher did, which would have been super boring, he realized this fact. The numbers 1 through 100 come in pairs that add up to 101. 1 plus 100. 2 plus 99. 3 plus 98. 4 plus 97. And so on. There's 50 such pairs of numbers, and 50 times 101 is really easy to do in your head, because it's 50 times 100 plus 50, or 5050. [SINGING] On the ninth day of Christmas my true love gave to me, an upside down six, infinity sideways, flipped over L, an upside down nine, a funny looking S, a sail for a boat, backwards E, half a heart on a plate, and a simple, boring, short and straight line. So yeah. If you want to add up all the numbers between 1 and anything, this trick works. For example, all the numbers between 1 and 12. 1 plus 12 is 13. 2 plus 11 is 13. All the way in to 6 plus 7. That's 6 times 13, which I do my head as 60 plus 18 equals 78. So 78 is the 12th triangular number, while 5,050 is the 100th triangular number. At least it's not "100 Days of Christmas." I'll stick with "Bottles of Beer." [SINGING] On the tenth day of Christmas, my true love gave to me, the base of our Arabic numeral system, the base of a nonary numeral system, the base used in octal, and the base of septinary, and the base of senary, and a number five, quaternary's base, ternary's base also, and binary two, and the base of unary. Here's my favorite way of visualizing the triangular numbers. Say you've got them in this configuration where they make a nice right equilateral triangle, or half a square. Finding the area of a square is easy, because you just square the length of it. In this case, 12 times 12. And the triangle is half of that. Only not really, because half the square means you only get half of this diagonal, so you've got to add back in the other half. But that's easy because there's 12 things in the diagonal, and 12/2 is 6. So to get the n-th triangular number, just take n squared over 2 plus n over 2, or n squared plus n over 2. [SINGING] On the eleventh day of Christmas, my true love gave to me the number my amp goes up to, the number of fingers on my hands, the German word for no, what I did after I eat, the number of heads on a hydra, at least until you start cutting them off, the number of strings on a guitar, the number I like to do high, the amount of horsemen of the Apocalypse, the number of notes in a triad, the number of pears in a pair of pears, and the number of partridges in a pear tree. [SINGING] On the twelfth day of Christmas, my true love gave to me-- Actually, enough is enough. Merry Christmas.