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Let's say you're me and you're in math class and you're supposed to be learning about factoring.

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Trouble is, your teacher is too busy trying to convince you that factoring is a useful skill

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for the average person to know with real-world applications ranging from passing your state exams

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all the way to getting a higher SAT score

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and unfortunately does not have the time to show you

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why factoring is actually interesting.

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It's perfectly reasonable for you to get bored in this situation.

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So like any reasonable person, you start doodling.

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Maybe it's because your teacher's sophorific voice reminds you of a lullaby

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but you're drawing stars.

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And because you're me, you quickly get bored of the usual 5-pointed star

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and get to wondering: why five?

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So you start exploring.

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It seems obvious that a 5-pointed star is the simplest one-

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the one that takes the least number of strokes to draw.

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Sure you can make a star with 4 points but that's not really a star

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the way you're defining stars.

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Then there's the 6-pointed star which is also familiar

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but totally different from the 5-pointed star because

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it takes 2 seperate lines to make.

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And then you're thinking about how

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much like you can put 2 triangles together to make a 6-pointed star,

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you can put two squares together to make an 8-pointed star.

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And any even numbered star with "p" points can be made of 2 "p over 2" gons.

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It is at this point that you realize if you wanted to avoid thinking about factoring

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maybe drawing stars was not the brightest idea.

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But wait! 4 would be an even number of points

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but that would mean you could make it out of 2 "2-gons"

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Maybe you were taught polygons with only two sides can't exist

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but for the purposes of drawing stars it works out rather well.

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Sure, the 4-pointed star doesn't look too star-like

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But then you realize that you can make a 6-pointed star out of 3 of these things

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and you've got an asterisk, which is definitely a legitimate star.

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In fact, for any star with a number of points that is divisible by 2

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you can draw it asterisk style.

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But that's not quite what you're looking for

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what you want is a doodle game, and here it is:

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draw "p" points in a circle, evenly spaced.

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Pick a number "q".

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Starting at one point, go around the circle and connect to the point q places over.

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Repeat.

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If you get to the starting place before you've covered all the points

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jump to a lonely point and keep going.

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That's how you draw stars.

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And it's a successfull game in that previously you were considering

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running, screaming, from the room

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or the window is open so that's an option too.

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But now you're not only entertained,

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but beginning to become curious about the nature of this game.

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The interesting thing is that the more points you have,

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the more different ways there is to draw the star.

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I happen to like 7-pointed stars because there's two really good ways to draw them.

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but they're still simple.

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I would like to note here that I have never actually left a math class via the window,

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not that I can say the same for other subjects.

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8 is interesting too, because not only are there a couple nice ways to draw it,

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but one's a composite of two polygons

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while another can be drawn without picking up the pencil.

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Then there's 9,

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

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And, because you're me, and you're a nerd, and you like to amuse yourself,

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you decide to call this kind of star a Square Star.

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because that's kind of a funny name.

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So you start drawing other square stars.

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4 4-gons,

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2 2-gons,

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even the completely degenerate case of 1 1-gon.

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Unfortunately 5 pentagons is already difficult to discern,

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and beyond that it's very hard to see and appreciate the structure of square stars.

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So you get bored and move on to 10 dots and a circle,

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which is interesting because this is the first number where you can make a star

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as a composite of smaller stars,

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that is, 2 boring old 5-pointed stars.

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Unless you count asterisk stars, in which case 8 was 2 4s, or 4 2's. or 2 2's and a 4.

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But 10 is interesting, because you can make it as a composite in more than one way.

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because it's divisible by five which itself can be made in 2 ways.

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Then there's 11, which can't be made out of seperate parts at all, because 11 is prime.

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Though here you start to wonder how to predict how many times around the circle

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it will go before getting back to the start.

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But instead of exploring the exciting world of modular arithmetic, you move on to 12

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which is a really cool number

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because it has a whole bunch of factors.

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And then something starts to bother you:

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Is a 25 pointed star composite made of 5 5-pointed stars a Square Star?

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You've been thinking only of pentagons because the lower numbers didn't have this question.

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How could you have missed that?

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Maybe your teacher said something interesting about prime numbers

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and you accidentally lost focus for a moment.

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And, oh no.

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It gets even worse.

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6 squared would be a 36 pointed star, made of 6 hexagons.

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but if you allow use of 6 pointed stars, then it's the same as

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a composite of 12 triangles.

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And that doesn't seem in keeping with the spirit of square stars.

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You'll have to define square stars more strictly.

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But you do like the idea that there's three ways to make the 7th square star.

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Anyway, the whole theory of what kind of stars can be made with what numbers

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is quite interesting

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and I encourage you to explore this during your math class.