238 lines
10 KiB
Markdown
238 lines
10 KiB
Markdown
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---
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layout: post
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title: Happy Trees
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---
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Source code related to this post is available [here](https://github.com/mediocregopher/happy-tree).
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This project was inspired by [this video](https://www.youtube.com/watch?v=_DpzAvb3Vk4),
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which you should watch first in order to really understand what's going on.
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My inspiration came from his noting that happification could be done on numbers
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in bases other than 10. I immediately thought of hexadecimal, base-16, since I'm
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a programmer and that's what I think of. I also was trying to think of how one
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would graphically represent a large happification tree, when I realized that
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hexadecimal numbers are colors, and colors graphically represent things nicely!
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## Colors
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Colors to computers are represented using 3-bytes, encompassing red, green, and
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blue. Each byte is represented by two hexadecimal digits, and they are appended
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together. For example `FF0000` represents maximum red (`FF`) added to no green
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and no blue. `FF5500` represents maximum red (`FF`), some green (`55`) and no
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blue (`00`), which when added together results in kind of an orange color.
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## Happifying colors
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In base 10, happifying a number is done by splitting its digits, squaring each
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one individually, and adding the resulting numbers. The principal works the same
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for hexadecimal numbers:
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```
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A4F
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A*A + 4*4 + F*F
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64 + 10 + E1
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155 // 341 in decimal
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```
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So if all colors are 6-digit hexadecimal numbers, they can be happified easily!
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```
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FF5500
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F*F + F*F + 5*5 + 5*5 + 0*0 + 0*0
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E1 + E1 + 19 + 19 + 0 + 0
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0001F4
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```
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So `FF5500` (an orangish color) happifies to `0001F4` (a darker blue). Since
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order of digits doesn't matter, `5F50F0` also happifies to `0001F4`. From this
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fact, we can make a tree (hence the happification tree). I can do this process
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on every color from `000000` (black) to `FFFFFF` (white), so I will!
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## Representing the tree
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So I know I can represent the tree using color, but there's more to decide on
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than that. The easy way to represent a tree would be to simply draw a literal
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tree graph, with a circle for each color and lines pointing to its parent and
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children. But this is boring, and also if I want to represent *all* colors the
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resulting image would be enormous and/or unreadable.
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I decided on using a hollow, multi-level pie-chart. Using the example
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of `000002`, it would look something like this:
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The inner arc represents the color `000002`. The second arc represents the 15
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different colors which happify into `000002`, each of them may also have their
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own outer arc of numbers which happify to them, and so on.
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This representation is nice because a) It looks cool and b) it allows the
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melancoils of the hexadecimals to be placed around the happification tree
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(numbers which happify into `000001`), which is convenient. It's also somewhat
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easier to code than a circle/branch based tree diagram.
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An important feature I had to implement was proportional slice sizes. If I were
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to give each child of a color an equal size on that arc's edge the image would
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simply not work. Some branches of the tree are extremely deep, while others are
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very shallow. If all were given the same space, those deep branches wouldn't
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even be representable by a single pixel's width, and would simply fail to show
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up. So I implemented proportional slice sizes, where the size of every slice is
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determined to be proportional to how many total (recursively) children it has.
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You can see this in the above example, where the second level arc is largely
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comprised of one giant slice, with many smaller slices taking up the end.
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## First attempt
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My first attempt resulted in this image (click for 5000x5000 version):
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[](/img/happy-tree/happy-tree-atmp1.png)
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The first thing you'll notice is that it looks pretty neat.
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The second thing you'll notice is that there's actually only one melancoil in
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the 6-digit hexadecimal number set. The innermost black circle is `000000` which
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only happifies to itself, and nothing else will happify to it (sad `000000`).
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The second circle represents `000001`, and all of its runty children. And
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finally the melancoil, comprised of:
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```
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00000D -> 0000A9 -> 0000B5 -> 000092 -> 000055 -> 00003 -> ...
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```
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The final thing you'll notice (or maybe it was the first, since it's really
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obvious) is that it's very blue. Non-blue colors are really only represented as
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leaves on their trees and don't ever really have any children of their own, so
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the blue and black sections take up vastly more space.
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This makes sense. The number which should generate the largest happification
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result, `FFFFFF`, only results in `000546`, which is primarily blue. So in effect
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all colors happify to some shade of blue.
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This might have been it, technically this is the happification tree and the
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melancoil of 6 digit hexadecimal numbers represented as colors. But it's also
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boring, and I wanted to do better.
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## Second attempt
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The root of the problem is that the definition of "happification" I used
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resulted in not diverse enough results. I wanted something which would give me
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numbers where any of the digits could be anything. Something more random.
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I considered using a hash instead, like md5, but that has its own problems.
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There's no gaurantee that any number would actually reach `000001`, which isn't
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required but it's a nice feature that I wanted. It also would be unlikely that
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there would be any melancoils that weren't absolutely gigantic.
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I ended up redefining what it meant to happify a hexadecimal number. Instead of
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adding all the digits up, I first split up the red, green, and blue digits into
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their own numbers, happified those numbers, and finally reassembled the results
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back into a single number. For example:
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```
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FF5500
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FF, 55, 00
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F*F + F*F, 5*5 + 5*5, 0*0 + 0*0
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1C2, 32, 00
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C23200
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```
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I drop that 1 on the `1C2`, because it has no place in this system. Sorry 1.
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Simply replacing that function resulted in this image (click for 5000x5000) version:
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[](/img/happy-tree/happy-tree-atmp2.png)
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The first thing you notice is that it's so colorful! So that goal was achieved.
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The second thing you notice is that there's *significantly* more melancoils.
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Hundreds, even. Here's a couple of the melancoils (each on its own line):
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```
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00000D -> 0000A9 -> 0000B5 -> 000092 -> 000055 -> 000032 -> ...
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000D0D -> 00A9A9 -> 00B5B5 -> 009292 -> 005555 -> 003232 -> ...
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0D0D0D -> A9A9A9 -> B5B5B5 -> 929292 -> 555555 -> 323232 -> ...
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0D0D32 -> A9A90D -> B5B5A9 -> 9292B5 -> 555592 -> 323255 -> ...
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...
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```
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And so on. You'll notice the first melancoil listed is the same as the one from
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the first attempt. You'll also notice that the same numbers from the that
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melancoil are "re-used" in the rest of them as well. The second coil listed is
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the same as the first, just with the numbers repeated in the 3rd and 4th digits.
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The third coil has those numbers repeated once more in the 1st and 2nd digits.
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The final coil is the same numbers, but with the 5th and 6th digits offset one
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place in the rotation.
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The rest of the melancoils in this attempt work out to just be every conceivable
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iteration of the above. This is simply a property of the algorithm chosen, and
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there's not a whole lot we can do about it.
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## Third attempt
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After talking with [Mr. Marco](/members/#marcopolo) about the previous attempts
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I got an idea that would lead me towards more attempts. The main issue I was
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having in coming up with new happification algorithms was figuring out what to
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do about getting a number greater than `FFFFFF`. Dropping the leading digits
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just seemed.... lame.
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One solution I came up with was to simply happify again. And again, and again.
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Until I got a number less than or equal to `FFFFFF`.
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With this new plan, I could increase the power by which I'm raising each
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individual digit, and drop the strategy from the second attempt of splitting the
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number into three parts. In the first attempt I was doing happification to the
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power of 2, but what if I wanted to happify to the power of 6? It would look
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something like this (starting with the number `34BEEF`):
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```
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34BEEF
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3^6 + 4^6 + B^6 + E^6 + E^6 + E^6 + F^6
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2D9 + 1000 + 1B0829 + 72E440 + 72E440 + ADCEA1
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1AEB223
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1AEB223 is greater than FFFFFF, so we happify again
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1^6 + A^6 + E^6 + B^6 + 2^6 + 2^6 + 3^6
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1 + F4240 + 72E440 + 1B0829 + 40 + 40 + 2D9
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9D3203
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```
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So `34BEEF` happifies to `9D3203`, when happifying to the power of 6.
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As mentioned before the first attempt in this blog was the 2nd power tree,
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here's the trees for the 3rd, 4th, 5th, and 6th powers (each image is a link to
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a larger version):
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3rd power:
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[](/img/happy-tree/happy-tree-atmp3-pow3.png)
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4th power:
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[](/img/happy-tree/happy-tree-atmp3-pow4.png)
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5th power:
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[](/img/happy-tree/happy-tree-atmp3-pow5.png)
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6th power:
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[](/img/happy-tree/happy-tree-atmp3-pow6.png)
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A couple things to note:
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* 3-5 are still very blue. It's not till the 6th power that the distribution
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becomes random enough to become very colorful.
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* Some powers have more coils than others. Power of 3 has a lot, and actually a
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lot of them aren't coils, but single narcissistic numbers. Narcissistic
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numbers are those which happify to themselves. `000000` and `000001` are
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narcissistic numbers in all powers, power of 3 has quite a few more.
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* 4 looks super cool.
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Using unsigned 64-bit integers I could theoretically go up to the power of 15.
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But I hit a roadblock at power of 7, in that there's actually a melancoil which
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occurs whose members are all greater than `FFFFFF`. This means that my strategy
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of repeating happifying until I get under `FFFFFF` doesn't work for any numbers
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which lead into that coil.
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All images linked to in this post are licensed under the [Do what the fuck you
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want to public license](http://www.wtfpl.net/txt/copying/).
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