 So let's say you've got some leftover Halloween candy, or let's just admit it, you bought it half off on November 1st, and you justified spending 8 bucks on 4 gallons of smarties because you can just feel the mathematical potential reeking off of them. One of the first things you might notice is that you can open them by pulling the ends, and the smartie magically spins around in a magic circle that makes candy appear. And of course that's because the ends are twisted, but if you really want to revel in the mathematical joy of smarties, you'll look at just how those ends are twisted and why it makes them so joyously open-up-able. So let's use the rigorous mathematical notation of drawing an arrow on candy. One side twists clockwise, the other side twists counterclockwise, and yet both arrows point in the same direction, and when you pull the ends, the opposite twists unravel in perfect symmetric harmony. Clockwise is the mirror image of counterclockwise, and the whole thing is just deliciously beautiful. But of course mathematics is about making things up and seeing what happens, so let's say you invent a clockwise smartie where both ends twist clockwise, and when you pull the ends, well, the smartie doesn't know which way to twist, it's just kind of stuck. And of course you could make a counterclockwise smartie that still has the same problem, or, as a mathematician would say, it maintains the property of unopen up ability. And of course we must make all possible smarties, the regular smartie, the clockwise smartie, the counterclockwise smartie, and an anti-smartie where both twists are flipped from the original, which maintains the property of open up ability, and from the outside it might seem the exact same as the original smartie, but it makes it so the twists go against the direction the wrapper rolls, instead of along with the direction the wrapper rolls. Okay, so now that you can open up a smartie, you have to figure out how to eat it. I mean, I don't know about you, but back in my day we made up rules for how to eat smarties because if you simply just shove the whole thing in your mouth you might as well go eat a couple spoons full of sugar straight out of the bag. So one rule we had was for different power-ups, the different color smartie bits could give you based on the rarity of the color. If you go ahead and make a bar graph of the different colors, you can see that pink is the most common, with six entire smartie bits, whereas this purpley color is the rarest, most special, magical smartie bit color, and I'm just gonna make sure these add to 15, so I know I didn't lose any off the side of my notebook. Okay, let's try another smartie. This one has six of the rare special purpley bits, so did we get super lucky, or maybe they're more common than we thought? And apparently there's white smartie bits, which we didn't get before, and what if there's another color that we haven't seen yet? So if someone can rope their math class into doing a statistical analysis of smartie colors, I'd appreciate it. Anyway, one of the smartie eating rules that I use is that you must eat three at a time in sandwich form, meaning they must be stacked and the top and bottom must be the same color. Sometimes you can find natural sandwiches in your smartie pack, meaning three smartie bits in correct sandwich arrangement straight out of the wrapper. I like to eat those first and then see if I can finish the pack eating sandwiches only. And I usually can, so that's five sandwiches per pack, although I wonder whether it's always possible, or what if there were five or seven colors, and how would you prove it's always possible if smarties only had two or three colors, and if the middle has to be a different color than the outside pair, then that changes everything. Okay, time to make shapes! Here is my smartie friend, smartie friend, smartie friend, my smartie friend, has a smartie friend, blah blah blah, smartie friend. It would be nice if you could just twist the ends together to connect smarties, and it doesn't quite hold, but it can make the glue look better and get in there better. And of course, since some twist clockwise and some twist counterclockwise, you've got to make the ends match up correctly. But for a triangle, if this end is clockwise, this has to be clockwise, and so the other ends have to be Why does counterclockwise also start with C? Let's do W for witter shins, which means clockwise and is one of many words to remind us words are made up by people. But then to match the witter shins ends, both ends of this smartie have to go witter shins, and that's not how real smarties go, whereas if they were arranged more square-like, then the ends can match up in a twistically correct manner. And if each twist connection has to be either clockwise flavor or witter shins flavor, then really we're asking a good old-fashioned graph theory question about the two colorability of vertices. More shapes! We can make a twistically correct square, and a twistically correct cube should be possible. Oh, but the corners aren't rigid, so the square is flop. Sounds like a job for triangles! In fact, I can convert this cube into a square antiprism, though I know anything with triangles can't be twistically correct, but it still flops this way along the square, so really the solution is to open it up and add more triangles, and more triangles, until I have an icosahedron of 20 triangles. Or maybe to keep the squares from flopping, they can support each other. There's a mathematical weaving of six squares that I'm quite fond of, but it looks like smarties are not the right ratio to fit into that weaving. Oh, but there's another great one, with four triangles that has a wider ratio. It won't be twistically correct, but if I weave four triangles just right, we should get a beautiful symmetric regular polylink. Yes. And even though it's untwistically correct, it does have a twist-like property. If we follow the flow of the smarties down, we see that the squary bits all flow counterclockwise, and the triangle bits flow witter shins. So there's a mirror image version, where the squary bits flow witter shins, and the triangle bits flow clockwise, just like how unopenable smarties have a mirror image version. Shapes! Okay, that's all the time we have for today. This video was not sponsored by smarties, but was sponsored by viewers like you, thanks to your incredible and enthusiastic response to my call for support. I am now full-time on Patreon, so check that out if you haven't already. I will see you again soon.