Header 2: This is what a subsection would look like

Filler content is any nonsense or irrelevant content used in a graphic design setting to preview a style or design. Filler content can come in the form of text, images, or any other relevant media. One of the most well known filler texts is “Lorem ipsum.” While there is no single definitive version of this filler, it almost always starts with “Lorem ipsum” and continues with more Latin text which may or may not be grammatically correct. What you’re reading now is filler as well!

Header 3: This is what a subsubsection would look like

This link should take you to the definition for “Jekyll.”

Header 4: I’m not sure why I’d use this header

Words words words

Header 5: I don’t think I’ll need this one wither

Words words words

Header 6: Or this one

Words words words

Ok, moving on…

I included all those headers because I want this style test to be complete. Enjoy this overemphasized text. Look, there’s more! Here are some symbols: `_[\]{-!.}(*)#+

Here is some code:

#include <bits/stdc++.h>
using namespace std;

using ll = long long;

int main () {
    cout << "Hello, world!\n";
    return 0;
}

# another code block

Here is the same code with line numbers:

1
2
3
4
5
6
7
8
9
#include <bits/stdc++.h>
using namespace std;

using ll = long long;

int main () {
    cout << "Hello, world!\n";
    return 0;
}

1
2
3
4
5
6
7
8
9
10
11
#include <bits/stdc++.h>
using namespace std;

using ll = long long;

int main () {
    cout << "Hello, world!\n";
    return 0;
}

// I'm adding this comment to show double digit line numbers.

Type ll instead of long long. It’s shorter!

Look how even the inline code is highlighted! Here’s a more complex example: public class HelloWorld { public static void main (String [] args) { System.out.println("Hello, world!"); } }

Here’s a really long preformatted line:

s = "000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"

Here’s a long preformatted block with line numbers:

1
2
3
4
5
6
7
8
9
10
11
12
13
# This is just filler
s = "000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
c = s[3:10]
s = "000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
c = s[3:10]
s = "000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
c = s[3:10]
s = "000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
c = s[3:10]
s = "000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
c = s[3:10]
s = "000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
c = s[3:10]

Filler content is any nonsense or irrelevant content used in a graphic design setting to preview a style or design. Filler content can come in the form of text, images, or any other relevant media. One of the most well known filler texts is "Lorem ipsum." While there is no single definitive version of this filler, it almost always starts with "Lorem ipsum" and continues with more Latin text which may or may not be grammatically correct. What you're reading now is filler as well! Filler content is any nonsense or irrelevant content used in a graphic design setting to preview a style or design. Filler content can come in the form of text, images, or any other relevant media. One of the most well known filler texts is "Lorem ipsum." While there is no single definitive version of this filler, it almost always starts with "Lorem ipsum" and continues with more Latin text which may or may not be grammatically correct. What you're reading now is filler as well! Filler content is any nonsense or irrelevant content used in a graphic design setting to preview a style or design. Filler content can come in the form of text, images, or any other relevant media. One of the most well known filler texts is "Lorem ipsum." While there is no single definitive version of this filler, it almost always starts with "Lorem ipsum" and continues with more Latin text which may or may not be grammatically correct. What you're reading now is filler as well! Filler content is any nonsense or irrelevant content used in a graphic design setting to preview a style or design. Filler content can come in the form of text, images, or any other relevant media. One of the most well known filler texts is "Lorem ipsum." While there is no single definitive version of this filler, it almost always starts with "Lorem ipsum" and continues with more Latin text which may or may not be grammatically correct. What you're reading now is filler as well! Filler content is any nonsense or irrelevant content used in a graphic design setting to preview a style or design. Filler content can come in the form of text, images, or any other relevant media. One of the most well known filler texts is "Lorem ipsum." While there is no single definitive version of this filler, it almost always starts with "Lorem ipsum" and continues with more Latin text which may or may not be grammatically correct. What you're reading now is filler as well!

We can also combine emphasis with code: main main main ~~main~~ I’m not sure why I’d do this, though.

Code using kramdown’s tilde fences:

#include <bits/stdc++.h>
using namespace std;

using ll = long long;

string tildeTest = "~~~~";
string backquoteTest = "````";

int main () {
    cout << "Hello, world!\n";
    return 0;
}

Here is a horizontal rule.

Look at this pretty math! $x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$. This solves the equation $0 = ax^2 + bx + c$. This is a fraction: $\frac{1}{2x+1}$.

Here’s a long line of math:

$$ \int \sqrt[3]{\tan\,x}\,dx = -\frac{1}{2}\log\left(1 + \tan(x)^{2/3}\right) +\frac{1}{4}\log\left(1 - \tan(x)^{2/3} + \tan(x)^{4/3}\right) +\frac{\sqrt{3}}{2}\tan^{-1}\left(\frac{2\tan(x)^{2/3}-1}{\sqrt{3}}\right) + C$$

Here’s a semi-long line of math:

$$\int \sqrt[3]{\tan\,x}\,dx = -\frac{1}{2}\log\left(1 + \tan(x)^{2/3}\right) +\frac{1}{4}\log\left(1 - \tan(x)^{2/3} + \tan(x)^{4/3}\right) +\dots + C$$

Math expressions aren’t automatically broken into multiple lines. For example, notice how $\int \sqrt[3]{\tan\,x}\,dx = -\frac{1}{2}\log\left(1 + \tan(x)^{2/3}\right) +\frac{1}{4}\log\left(1 - \tan(x)^{2/3} + \tan(x)^{4/3}\right) +\dots + C$ is on a single line even though it is in a paragraph, and the normal text is broken up into multiple lines.

Notice how <div>s are centered, but <span>s are left-aligned.

$\int \sqrt[3]{\tan\,x}\,dx = -\frac{1}{2}\log\left(1 + \tan(x)^{2/3}\right) +\frac{1}{4}\log\left(1 - \tan(x)^{2/3} + \tan(x)^{4/3}\right) +\dots + C$

Markdown images don’t float.

But HTML images can! All we need to do is use flex! Notice how <div>s expand, but <img>s don’t. However, when using flex, we need to tell Jekyll to render the <div>s as Markdown. Notice how the quotes are angled as well: “won’t.”

Here’s an image that takes up the page:

Here’s an image that overflows:

Enjoy this block quote. Filler content is any nonsense or irrelevant content used in a graphic design setting to preview a style or design. Filler content can come in the form of text, images, or any other relevant media. One of the most well known filler texts is “Lorem ipsum.” While there is no single definitive version of this filler, it almost always starts with “Lorem ipsum” and continues with more Latin text which may or may not be grammatically correct. What you’re reading now is filler as well!

Here’s a nested quote!

End nest

Another

Nested

Quote

This is a shorter block quote.

One dash - Two dash – Three dash —

This is the same line with no spaces: One dash-Two dash–Three dash—

HTML is a DSL. This is a link Notice how those terms are defined. Hmm… I don’t actually like this format. Footnotes seem better, but I want to style everything just in case. Maybe it’ll look better with CSS

Here is an unordered list.

Item
Filler content is any nonsense or irrelevant content used in a graphic design setting to preview a style or design. Filler content can come in the form of text, images, or any other relevant media. One of the most well known filler texts is “Lorem ipsum.” While there is no single definitive version of this filler, it almost always starts with “Lorem ipsum” and continues with Latin text which may or may not be grammatically correct. What you’re reading now is filler as well!
Here’s an image

Here is an ordered list.

Item
Filler content is any nonsense or irrelevant content used in a graphic design setting to preview a style or design. Filler content can come in the form of text, images, or any other relevant media. One of the most well known filler texts is “Lorem ipsum.” While there is no single definitive version of this filler, it almost always starts with “Lorem ipsum” and continues with more Latin text which may or may not be grammatically correct. What you’re reading now is filler as well!
1. Here is a list within a list!
2. Neat isn’t it?
  - We can nest even more if we wanted to!
  - Look!
    - More nesting!
      - Even more nesting!
Item

Here’s a definition list:

Markdown: The markup language that this page is written in; Also popular on GitHub
Jekyll
Jekyllrb: The static website generator used for this website
Filler content: Filler content is any nonsense or irrelevant content used in a graphic design setting to preview a style or design. Filler content can come in the form of text, images, or any other relevant media. One of the most well known filler texts is “Lorem ipsum.” While there is no single definitive version of this filler, it almost always starts with “Lorem ipsum” and continues with more Latin text which may or may not be grammatically correct. What you’re reading now is filler as well!

This is a footnote¹². Look down for the definitions³.

This	is
a	table.
This is a very long cell followed by an empty cell

Here’s a table with that is more organic.

Trial	Energy	Height	Mass	Duration
1	12.2	1.2	350	1.2
2	10.1	2.7	470	0.9
3	17.5	1.2	210	0.9
4	9.0	0.5	1020	0.8
5	14.5	1.5	490	1.0
6	10.2	2.0	380	1.5

Here is a more abnormal table. Notice the column alignment.

Trial	Energy	Height	Mass	Duration
1	12.2	1.2	350	1.2
2	10.1	2.7	470	0.9
3	17.5	1.2	210	0.9
4	9.0	0.5	102000000	0.88888888888888888888888888
5	14.5	1.5	490	1.0 2
6		2.0	380	33

Here is a table that is very long.

Data	Data	Data	Data	Data	Data	Data	Data	Data	Data	Data	Data	Data	Data	Data	Data	Data	Data	Data	Data	Data	Data	Data
2.4	2.4	2.4	2.4	2.4	2.4	2.4	2.4	2.4	2.4	2.4	2.4	2.4	2.4	2.4	2.4	2.4	2.4	2.4	2.4	2.4	2.4	2.4

This table	has no
header so	it looks
like	this.

Header	Header
This table	has
a footer	so it
looks	this.
Second	footer row

This table	has
multiple headers	so it
looks	this.

Here is a todo list:

This item is done!
This one is not.
very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very
very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very long.

Compared to a bulleted list:

This item is done!
This one is not.
This one is not done, either.

This bullet is actually another list!

Here’s a list with no separation:

This item is done!
This one is not.
This one is not done, either.

This is a header

This is a subheader

The headers above were made with a different markdown notation.

This link should take you to the subsection at the top.

Here’s the 404 page. And here is a link to an example site! Notice that the 404 link has a hint when you hover over it. Here is another link: https://github.com Email: me@example.com. Here is yet another link: Homepage.

Here are some emojis and symbols: ✓ ✗ ✔ ✘ 🇺🇸 📅

These are emojis in a header ✔ 🇺🇸 📅

¡ʇɹoddns ǝpoɔᴉun ǝʌɐɥ ǝʍ ǝɹns ǝʞɐɯ oʇ sᴉ sᴉɥ⊥

Now for some mathy filler:

Googology

Googology is the study of large numbers. If you are interested in math, you have probably come across a few of these large numbers:

Googol, equal to $10^{100}$.
Googolplex, equal to $10^{10^{100}}$.
Graham’s number, defined as $g_{64}$, where $g_1 = 3 \uparrow\uparrow\uparrow\uparrow 3$ and $g_n = 3 \uparrow^{g_{n-1}} 3$.
Rayo’s number, the smallest number larger than any finite number describable by at most $10^{100}$ symbols using first-order set theory.

Rayo’s number has a rather unsavory property: it’s not computable. Compare this to Graham’s number. Even though there isn’t enough matter in the universe to represent Graham’s number, there is still a theoretical program which, if given enough time and space, would compute Graham’s number. Also, there are still some properties of Graham’s number which we can prove, even in our tiny universe. For example, the we know the last digit of Graham’s number is a $7$.

For the rest of the article, I want to talk about really large, but computable, numbers. Even though Graham’s number is unimaginably large, there are infinitely many numbers which are larger, such as $g_{64} + 1$. Other larger numbers include $g_{65}$, $g_{10^{100}}$, and $g_{g_{64}}$. We can even define a new sequence $\left(h_n\right)$ by $h_1 = g_1$ and $h_n = g_{h_{n-1}}$ and easily create much, much larger numbers. We can continue this by making sequences $\left(i_n\right)$, $\left(j_n\right)$, and so on, each one iterating the previous to create even faster growing sequences.

So, in a sense, no number is “large” since there are so many larger numbers, and no sequence is “fast-growing” since iterating an increasing sequence produces a faster-growing sequence. Ok, what about this: Is there a sequence of computable functions $f_1, f_2, \dots$ such that for any sequence $\left(a_n\right)$, some $f_m$ grows faster than $\left(a_n\right)$? Note that these functions to not have to be total. One candidate for this such a sequence of functions might be $f_m(x) = x \uparrow^m x$. However, that does not work since $\left(g_n\right)$ grows faster than any $f_m$.

Actually, such a sequence of functions does not exist. If we are given any sequence $\left(f_m\right)$, we can define $\left(M_n\right)$ as follows:

$$ M_n = 1 + n\max_{1 \leq i \leq n} f_i(n) $$

If all $f_i(n)$ are undefined, then let $M_n = 1$. Either way, $\left(M_n\right)$ grows faster than any $f_m$.

One simple “counterexample” to this theorem seems to be enumerating all computable functions and letting $f_m$ be the $m$th of these functions. Since for any computable sequence $\left(a_n\right)$, there exists a faster-growing computable function, some $f_m$ must outpace $\left(a_n\right)$. This counterexample does not work since $\left(M_n\right)$ is not necessarily computable. In order to compute $\left(M_n\right)$ for general $\left(f_m\right)$, we’d need a procedure that solves the halting problem to know whether or not a certain $f_i(n)$ is defined or not. In fact, this past few paragraphs can be rearranged into a proof that the halting problem is unsolvable.

Now for some code:

$><<"eval ['#{gets(p).unpack1('b*').tr('01',' 	')}'].pack'b*'"

#include <bits/stdc++.h>
using namespace std;

#define F for (i = 0; i < n; i++) 
long n, s, v [100000], u [100000], i, t;

main () {
    cin >> n;
    F cin >> v[i];
    F cin >> u[i];
    F s += v[i] - u[i], v[i] = s;
    sort(v, v + n);
    s = v[n/2];
    F t += abs(s - v[i]);
    cout << t;
}

from PIL import Image
import numpy as np

def imageToArray(image):
    return np.array(list(image.convert("L").getdata()))

def imageDist(a, b, adjustment):
    b = imageToArray(b) 

    # return abs(np.sum(a) - np.sum(b) + adjustment)
    return np.sum(abs(a - b) ** 2)

import java.util.*;
import java.math.*;

public class FactorQueue {
    private PriorityQueue<Pair> pq;
    private int x;

    private void put (BigInteger b, long j) {
        long next = (j - b.mod(BigInteger.valueOf(j)).longValue()) % j;

        pq.add(new Pair(next, j));
    }

    public boolean next () {
        if (x == pq.peek().x) {
            while (x == pq.peek().x) {
                Pair p = pq.poll();
                p.x += p.y;
                pq.add(p);
            }
            x++;
            return false;
        }
        else {
            x++;
            return true;
        }
    }

    public FactorQueue (BigInteger b, int n) {
        n++;
        boolean [] isPrime = new boolean [n];

        for (int i = 2; i < n; i++) {
            isPrime[i] = true;
        }
        pq = new PriorityQueue<>();
        x = 0;

        for (int i = 2; i < n; i++) {
            if (isPrime[i]) {
                for (long j = (long)i*i; j < n; j += i) {
                    isPrime[(int) j] = false;
                }

                put(b, i);
            }
        }
    }
}

class Pair implements Comparable<Pair> {
    public long x, y;

    public Pair (long x, long y) {
        this.x = x;
        this.y = y;
    }

    @Override
    public int compareTo (Pair other) {
        return Long.compare(x, other.x);
    }
}

Sample Footnote ↩
Notice how the footnotes are at the bottom! ↩
I’m probably gonna write some very long footnotes some times so here’s a test of that! Filler content is any nonsense or irrelevant content used in a graphic design setting to preview a style or design. Filler content can come in the form of text, images, or any other relevant media. One of the most well known filler texts is “Lorem ipsum.” While there is no single definitive version of this filler, it almost always starts with “Lorem ipsum” and continues with more Latin text which may or may not be grammatically correct. What you’re reading now is filler as well! ↩