Let.s say you.re me and you.re in math class, and you.re supposed to be learning about factoring. Trouble is, your teacher is too busy trying to convince you that factoring is a useful skill for the average person to know, with real-world applications ranging from passing your state exams all the way to getting a higher SAT score, and unfortunately does not have the time to show you why factoring is actually interesting.

It is perfectly reasonable for you to get bored in this situation, so like any reasonable person you start doodling. Maybe it.s because your teacher.s soporific voice reminds you of a lullaby, but you.re drawing stars. And because you.re me, you quickly get bored of the usual 5-pointed star and get to wondering. Why 5?

So you start exploring. It seems obvious that a 5-pointed star is the simplest one, the one that takes the least number of strokes to draw. Sure you can make a star with four points, but that.s not really a star the way you.re defining stars.

Then there.s the six-pointed star, which is also pretty familiar, but totally different from the 5-pointed star because it takes two separate lines to make. And then you.re thinking about how, much like you can put two triangles together to make a six-pointed star, you can put two squares together to make an eight-pointed star, and any even-numbered star with P points can be made out of two P/2-gons.

It is at this point that you realize that if you wanted to avoid thinking about factoring, maybe drawing stars was not the brightest idea.

But wait. Four would be an even number of points, but that would mean you can make it out of two 2-gons. Maybe you were taught polygons with only 2 sides can.t exist, but for the purposes of drawing stars, it works out rather well. Sure the 4-pointed star doesn.t look too star-like, but then you realize you can make the six-pointed star out of three of these things, and you.ve got an asterisk, which is definitely a legitimate star. In fact, for any star where the number of points is divisible by two, you can draw it asterisk-style.

But that.s not quite what you.re looking for. What you want is a doodle-game, and here it is: Draw P points in a circle, evenly spaced. Pick a number Q. Starting at one point, go around the circle and connect to the point Q places over. Repeat. If you get to the starting place before you.ve covered all the points, jump to a lonely point and keep going.

That.s how you draw stars. And it.s a successful game in that previously you were considering running screaming from the room, or the window is open so that.s an option too, but now you.re not only entertained but beginning to become curious about the nature of this game.

The interesting thing is that the more points you have, the more different ways there is to draw the star. I happen to like 7-pointed stars because there.s two really good ways to draw them but they.re still simple.

I would like to note here that I have never actually left a math class via the window, not that I can say the same for other subjects.

8 is interesting too because not only are there a couple nice ways to draw it, but one.s a composite of two polygons while another can be drawn without picking up the pencil.

Then there.s 9, which, in addition to a couple other nice versions, you can make out of 3 triangles.

And because you.re me and you.re a nerd and you like to amuse yourself, you decide to call this kind of star a Square Star, because that.s kind of a funny name. So you start drawing other square stars. Four four-gons. Two 2-gons. Even the completely degenerate case of one 1-gon.

Unfortunately five pentagons is already difficult to discern and beyond that it.s very hard to see and appreciate the structure of square stars, so you get bored and move on to ten dots in a circle, which is interesting because this is the first number where you can make a star as a composite of smaller stars.that is, two boring old 5-pointed stars.unless you count asterisk stars, in which case eight was two fours, or four twos, or two twos and a four. But Ten is interesting because you can make it as a composite in more than one way, because it.s divisible by 5, which itself can be made in two ways.

Then there.s 11 which can.t be made out of separate parts at all because 11 is prime, though here you start to wonder how to predict how many times around the circle you.ll go before getting back to start.

But instead of exploring the exciting world of modulo arithmetic you move on to 12, which is a really cool number because it has a whole bunch of factors.

And then something starts to bother you. Is a 25-pointed star composite made of five five-pointed stars a square star? You had been thinking only of pentagons because the lower numbers didn.t have this question. How could you have missed that? Maybe your teacher said something interesting about prime numbers, and you accidentally lost focus for a moment. And oh no, it gets even worse. 6 squared would be a 36-pointed star made of six hexagons, but if you allow use of six-pointed stars, then it is the same as a composite of 12 triangles, and that doesn.t seem in keeping with the spirit of square stars. You.ll have to define Square Stars more strictly. But you do like the idea that there.s three ways to make the 7th Square Star.

Anyway, the whole theory of what kinds of stars can be made with what numbers is quite interesting and I encourage you to explore this during your math class.