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index.html
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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="stylesheet" href="styles/fonts.css">
<link rel="stylesheet" href="styles/global.css">
<link rel="stylesheet" href="styles/mathinput.css">
<link rel="stylesheet" href="styles/checkpoint.css">
<link rel="stylesheet" href="styles/mobile.css">
<title>Tricky</title>
<link rel="stylesheet" href="lib/katex/katex.min.css"
integrity="sha384-n8MVd4RsNIU0tAv4ct0nTaAbDJwPJzDEaqSD1odI+WdtXRGWt2kTvGFasHpSy3SV" crossorigin="anonymous">
<script defer src="lib/katex/katex.min.js"
integrity="sha384-XjKyOOlGwcjNTAIQHIpgOno0Hl1YQqzUOEleOLALmuqehneUG+vnGctmUb0ZY0l8"
crossorigin="anonymous"></script>
<script defer src="lib/katex/contrib/auto-render.min.js"
integrity="sha384-+VBxd3r6XgURycqtZ117nYw44OOcIax56Z4dCRWbxyPt0Koah1uHoK0o4+/RRE05"
crossorigin="anonymous"></script>
<script src="scripts/autotex.js"></script>
<script defer type="module" src="scripts/mathinput.js"></script>
</head>
<body>
<div id="math-input-template" class="math-input">
<textarea class="math-input-textarea monospaced" spellcheck="false"
placeholder="Type your answer here..."></textarea>
<div class="math-input-output-wrap">
<div class="math-input-tex-area"></div>
<div class="math-input-error-area monospaced" style="display: none;">
<span class="math-input-error-box"></span>
<br>
<span class="math-input-error-msg"></span>
</div>
</div>
</div>
<div id="menu-background" class="mobile-only hidden"></div>
<div id="main-wrap">
<div id="question-sidebar" tabindex="-1">
<div id="question-sidebar-wrap" class="invisible">
<button data-target-elem="welcome-page" class="sidebar-button">
<span class="material-icons inline-big">
waving_hand
</span>
<span><strong>Welcome</strong></span>
</button>
<button data-target-elem="tutorial-page" class="sidebar-button">
<span class="material-icons inline-big">
import_contacts
</span>
<span><strong>Tutorial</strong></span>
</button>
<hr class="mo-mobile">
<button data-question-name="easy" class="question-sidebar-button sidebar-button">
<span class="inline-circle"></span>
<span class="question-name">Simple Trigonometry</span>
</button>
<button data-question-name="unixtime" class="question-sidebar-button sidebar-button">
<span class="inline-circle"></span>
<span class="question-name">Detroit: Become Creepy</span>
</button>
<button data-question-name="profile" class="question-sidebar-button sidebar-button">
<span class="inline-circle"></span>
<span class="question-name">Garlic Phone Animation</span>
</button>
<button data-question-name="empirical" class="question-sidebar-button sidebar-button">
<span class="inline-circle"></span>
<span class="question-name">Empirically Empirical</span>
</button>
<button data-checkpoint-name="cp1" class="checkpoint-sidebar-button sidebar-button">
<span class="inline-big material-icons">flag</span>
<span><strong>Checkpoint 1</strong></span>
</button>
<hr class="mo-mobile">
<button data-question-name="leapyear" class="question-sidebar-button sidebar-button">
<span class="inline-circle"></span>
<span class="question-name">Leap of Faith</span>
</button>
<button data-question-name="passant" class="question-sidebar-button sidebar-button">
<span class="inline-circle"></span>
<span class="question-name">Google En Passant</span>
</button>
<button data-question-name="quadratic" class="question-sidebar-button sidebar-button">
<span class="inline-circle"></span>
<span class="question-name">Cos why not.</span>
</button>
<button data-checkpoint-name="cp2" class="checkpoint-sidebar-button sidebar-button">
<span class="inline-big material-icons">flag</span>
<span><strong>Checkpoint 2</strong></span>
</button>
<hr class="mo-mobile">
<button data-question-name="panda" class="question-sidebar-button sidebar-button">
<span class="inline-circle"></span>
<span class="question-name">ren pamdy</span>
</button>
<button data-question-name="rotation" class="question-sidebar-button sidebar-button">
<span class="inline-circle"></span>
<span class="question-name">spinspinspinspin</span>
</button>
<button data-checkpoint-name="cp3" class="checkpoint-sidebar-button sidebar-button">
<span class="inline-big material-icons">flag</span>
<span><strong>Checkpoint 3</strong></span>
</button>
<hr class="mo-mobile">
<button data-question-name="recursion" class="question-sidebar-button sidebar-button">
<span class="inline-circle"></span>
<span class="question-name">HHHHHHHHHHHHHHH!</span>
</button>
<button data-question-name="hastring" class="question-sidebar-button sidebar-button">
<span class="inline-circle"></span>
<span class="question-name">HAHAHAHAHAHAHAH!</span>
</button>
<button data-question-name="hagoteem" class="question-sidebar-button sidebar-button">
<span class="inline-circle"></span>
<span class="question-name">BAHAHAHAHAHAHAH!</span>
</button>
<button data-checkpoint-name="cp4" class="checkpoint-sidebar-button sidebar-button">
<span class="inline-big material-icons">flag</span>
<span><strong>Checkpoint 4</strong></span>
</button>
<hr class="mo-mobile">
<button data-question-name="analytic" class="question-sidebar-button sidebar-button">
<span class="inline-circle"></span>
<span class="question-name">isomorphism?????</span>
</button>
<button data-question-name="derivative" class="question-sidebar-button sidebar-button">
<span class="inline-circle"></span>
<span class="question-name">“cannot do” - a prof</span>
</button>
<button data-checkpoint-name="cp5" class="checkpoint-sidebar-button sidebar-button">
<span class="inline-big material-icons">flag</span>
<span><strong>Checkpoint 5</strong></span>
</button>
<hr class="mo-mobile">
<button data-question-name="checkers" class="question-sidebar-button sidebar-button">
<span class="inline-circle"></span>
<span class="question-name">Now the fun begins</span>
</button>
<button data-question-name="fibonacci" class="question-sidebar-button sidebar-button">
<span class="inline-circle"></span>
<span class="question-name">Fibonacci (Nice)</span>
</button>
<button data-checkpoint-name="cp6" class="checkpoint-sidebar-button sidebar-button">
<span class="inline-big material-icons">flag</span>
<span><strong>Checkpoint 6</strong></span>
</button>
<hr class="mo-mobile">
<button data-target-elem="info-page" class="sidebar-button">
<span class="material-icons inline-big">
info
</span>
<span><strong>Information & Privacy</strong></span>
</button>
</div>
</div>
<div id="checkpoint-area-template" class="main-page checkpoint-page content-page">
<h1>Checkpoint</h1>
<p>This checkpoint will check your answers to questions <span class="checkpoint-question-list"></span>.</p>
<button class="check-answers-button">Check Answers</button>
<div class="checkpoint-output-box vert-center">
<div class="checkpoint-success" style="display: none;">
<span class="checkpoint-success-ign" style="display: none">
All checked answers are correct. Checkpoint code:
<br>
<div class="checkpoint-code-output monospaced">
</div>
</span>
<span class="checkpoint-success-noign" style="display: none">
All checked answers are correct.
<br>
To get a checkpoint code,
please enter your nickname/IGN on the first page.
</span>
</div>
<div class="checkpoint-fail" style="display: none;">
Sorry, at least one of your answers are incorrect.
<span class="checkpoint-fail-more-info">
Please also note that your answer to question <span
class="inline-circle checkpoint-fail-question"></span>
gave an undefined value for the input:
<div class="checkpoint-fail-inputs"></div>
Please make sure you are not dividing by zero, or putting too large parameters into factorials
or exponentials.
</span>
</div>
<div class="checkpoint-error" style="display: none;">
There was a problem evaluating your answer to question <span
class="inline-circle checkpoint-error-question"></span>.
<hr>
<div class="checkpoint-error-output monospaced">
<span class="checkpoint-error-box"></span>
<br>
<span class="checkpoint-error-msg"></span>
</div>
</div>
</div>
</div>
<div id="loading-screen" class="vert-center hidden">
<div>
Loading math libraries and questions... please wait...
</div>
</div>
<div id="question-area" class="vert-center invisible" tabindex="-1">
<div id="welcome-page" class="main-page content-page">
<h1>Welcome</h1>
<p>
Well, would you look at that? Your favourite maths test franchise has
been completely revamped! It's pretty, isn't it? Not only that, but
I am now using a custom-made maths library with some neat custom features.
</p>
<p>
<strong>Please read the tutorial</strong> as the new math library works
slightly differently from before.
</p>
<p>
Well, what are you waiting for? Go ahead and enter your nickname/IGN and
have fun(TM)! (If you do not want checkpoint codes, you do not need to enter your IGN.
Your answers will be checked nonetheless).
</p>
<input id="ign-input" class="mono" type="text" placeholder="Please enter your nickname/IGN...">
</div>
<div id="info-page" class="main-page">
<h1>Privacy</h1>
<p>
This website is purely client-side and does not store identifiable user data on any server. The
developer has made no efforts to send any user-generated data to any server. This can be verified by
viewing the source code of this project.
<p>
However, this site is hosted on Cloudflare Pages, and hence is on Cloudflare's network. Thus, usage
of this site may be subject to Cloudflare's privacy policy, available
<a href="https://www.cloudflare.com/en-gb/privacypolicy/" target="_blank">here</a>. In
this case, you would be categorised as an End User, and a Website Visitor should you access
Cloudflare's website for the purpose of viewing the privacy policy.
</p>
<p>
All required libraries and files are in this website's repository, and thus no requests
are made to any server other than those CloudFlare's CDN.
</p>
<p>
Your IGN, previous answers, and information about completed checkpoints are saved between sessions
using your browser's local storage. This data is not accessible by the developer.
</p>
</p>
<h1>This Website is Open Source</h1>
<p>
Source Code and License:
<a href="https://github.com/CAG2Mark/MathsTest2024" target="_blank">GitHub</a>,
<a href="https://github.com/CAG2Mark/MathsTest2024/blob/main/LICENSE.md">AGPL v3 License</a>.
</p>
<h1>Open Source Licenses</h1>
<p>
KaTeX: <a href="https://github.com/KaTeX/KaTeX/blob/main/LICENSE" target="_blank">MIT License</a>.
</p>
<p>
wasm-math-evaluator:
<a href="https://github.com/CAG2Mark/wasm-math-evaluator/" target="_blank">GitHub</a>,
<a href="https://github.com/CAG2Mark/wasm-math-evaluator/blob/main/LICENSE.md" target="_blank">GPL
v2 License</a>.
</p>
<p>
Inter Font: <a href="fonts/inter/license.txt" target="_blank">OFL License</a>.
</p>
<p>
JetBrains Mono Font: <a href="fonts/JetBrains-Mono-LICENSE-OFL.txt" target="_blank">OFL License</a>.
</p>
<p>
Material icons: <a href="fonts/material-icons-LICENSE-APACHE.txt" target="_blank">Apache
License</a>.
</p>
</div>
<div id="tutorial-page" class="main-page content-page">
<h1>Tutorial</h1>
<p>
When asked for an answer, you will need to input a mathematical expression.
This could be a number or a function, depending on the question. <strong>Your answer
will be checked to a higher precision than most calculators can handle, so
please input your answers exactly</strong> (or, if you must, to 25 significant figures. :P)
</p>
<p>
If you are sure your simplified expression is exactly the correct answer, then you
can enter it. However, if the answer is something like $2\pi$, do not enter $6.28$
or any other approximation. Just directly enter <span class="mono">2pi</span>.
</p>
<p>
For example, if the question is asking for the value of $2+2$, you can input <span
class="mono">4</span>,
<span class="mono">2+2</span>, <span class="mono">1+3</span> or similar:
</p>
<div class="math-input" id="tut1"></div>
<p>
As you may have noticed, the input box doubles as a calculator. You can use
many standard mathematical functions in your input. All available functions are
<span class="mono">sin, cos, tan, arcsin, arccos, arctan, sinh, cosh, tanh,
arsinh, arcosh, artanh, exp, ln, sqrt, sgn, floor, ceil, abs, nCr, nPr, atan2</span>.
The available operators are <span class="mono">+, -, *, /, %, ^, !</span>.
</p>
<p>
For example, if the answer is $\sqrt{\left(\binom{10}{5} \mod 4\right) + (\sin 5)^2}$, you could
type
<span class="mono">sqrt(nCr(10, 5) % 4 + (sin5)^2</span>. Note that brackets
can be left open.
</p>
<div class="math-input" id="tut2"></div>
<p>
Sometimes, you will need to input a function or expression of some variables as the answer.
For example, if you are asked to find $\sin(a + b)$, you could type
<span class="mono">sin a cos b + sin b cos a</span> or <span class="mono">sin(a+b)</span>.
</p>
<div class="math-input" id="tut3"></div>
<p>
You can also define variables within your answer. For example, if you need to compute
$\sin\frac{x^2+4}{|\sin x|} + \sqrt{\frac{x^2+4}{|\sin x|}} + (\frac{x^2+4}{|\sin x|} )!$, you can
type <span class="mono">let r = sin((x^2+4)/abs sin x); sin r + sqrt r + r!</span> instead of
typing out the whole thing.
</p>
<div class="math-input" id="tut4"></div>
</div>
<div data-question-name="easy" class="main-page question-page content-page">
<h1>
Simple Trigonometry
</h1>
<p>
What is $180 - 42 - 42$?
</p>
<div class="math-input" data-answer-name="ans"></div>
</div>
<div data-question-name="unixtime" class="main-page question-page content-page">
<h1>Detroit: Become Creepy</h1>
<p>
At what Unix time (in seconds) did CAG2 change his MC username to CAG2? (Hint: NameMC shows it in
your local timezone.)
</p>
<div class="math-input" data-answer-name="ans"></div>
</div>
<div data-question-name="profile" class="main-page question-page content-page">
<h1>Garlic Phone Animation</h1>
<p>
Suppose CAG2's profile picture spins clockwise at a rate of $2918572878278522^\circ$ per second.
</p>
<p>
After $2817274$ seconds, how many degrees would his profile picture be rotated clockwise from its
original position? Give your answer as a positive integer that is as small as possible.
</p>
<p>
For example, if you think the answer is $362^\circ$, your answer should evaluate to <span
class="mono">2</span>.
</p>
<div class="math-input" data-answer-name="ans"></div>
</div>
<div data-question-name="empirical" class="main-page question-page content-page">
<h1>Empirically Empirical</h1>
<p>
Suppose the $\mu$-ness of $\mathcal{R}$, denoted $\sigma(\mathcal{R})$ is related to its $v$-ness
$\mu_v$, $\upsilon$-ness $\rho_v$, $i$-ness $v_\upsilon$ and highness $\upsilon_v$, and $D$-ness
$\mathbb{D}$. They are empirically linked by the equation:
\[
\sigma(\mathcal{R}) = \frac{\arctan(0.4412852^{\upsilon_v
v_\upsilon^{1.002412}})}{\mathbb{D}\mu_{\upsilon} \rho_{v}}
\]
with the formulae $\mu_v = \frac{1}{\mu_\upsilon}$, $\rho_v = \frac{1}{\sqrt[0.441285]{\rho_v}}$. It
is
assumed that
\[
-\frac{\pi}{2} < \sigma(\mathcal{R})\mathbb{D}\mu_{\upsilon} \rho_{v} < \frac{\pi}{2} \] Given
that
the
$\mu$-ness of $\mathcal{R}$ is $0.69$, $i$-ness is $0.4$, $D$-ness is $0.6$ and $v$-ness is $0.4$,
what is the highness of $\mathcal{R}$? (Please note that $v$ and $\upsilon$ are different
characters; one is slightly wider than the other. Therefore, it is advised to use a magnifying glass
and ruler for this question.)
</p>
<p>
(Hint: Any typo you think you see is probably intentional.)
</p>
<div class="math-input" data-answer-name="ans"></div>
</div>
<div data-question-name="leapyear" class="main-page question-page content-page">
<h1>Leap of Faith</h1>
<p>
Suppose we randomly pick a day between the first day of 2000 and the last day of
9999 inclusive. Assume all days have an equal probability of being picked. What is the probability we
pick leap day, i.e. the 29th of February?
</p>
<p>
Note: Most people get this wrong because the actual condition to be a leap year is
not well-known.
</p>
<div class="math-input" data-answer-name="ans"></div>
</div>
<div data-question-name="passant" class="main-page question-page content-page">
<h1>Google En Passant</h1>
<p>
In chess, a pawn can move ahead two squares on its first move. If it does so, and lands adjacent to
an
enemy pawn, it may be captured by that enemy pawn. This is known as <i>en passant</i>.</p>
<p>
For this question, define two occurences of en passant to be <i>different</i> if any of the
following
are true:
<ul>
<li>The pawn being captured is different. Two pawns are different if they originated from different
files
or are different colours.</li>
<li>The capturing pawn is different.
</li>
<li>The pawn was captured from a different file. For example, suppose the black pawn on the $c$ file
was captured en passant. Two different ways for this to happen is if the the white pawn was on
the $b$ file or the $d$ file.
</li>
</ul>
</p>
<p>
Assume a standard $8 \times 8$ chess board and standard chess rules. Assume no pawn promotions
occur. How many <i>different</i> ways are there for en passant to theoretically occur? (Don't forget
to account for both players capturing.)
</p>
<div class="math-input" data-answer-name="ans"></div>
</div>
<div data-question-name="quadratic" class="main-page question-page content-page">
<h1>Cos why not.</h1>
<p>
Consider the quadratic function:
\[
f(x) = x^2 \sin^2 a + 2x \cos a - 1
\]
What is the smallest positive value of $a$ such that $f(x)$ has a root at $x = 1$?
</p>
<div class="math-input" data-answer-name="ans"></div>
</div>
<div data-question-name="panda" class="main-page question-page content-page">
<h1>ren pamdy</h1>
<p>
Suppose there are $27$ red pandas in a zoo and there are $3$ zookeepers who have been assigned to
take care of these red pandas.
</p>
<p>
The zoo policy states each zookeeper has to take care of at least $3$ red pandas. How many ways are
there to assign each red panda to a zookeeper so that this requirement is met?
</p>
<p>
NOTE: This question originally involved only $20$ red pandas, but it was possible to brute force
the answer. So I've increased it to $27$. If you did the question properly it should take no effort
to update your answer :)
</p>
<p>
(Hint: $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap
C|$.)
</p>
<div class="math-input" data-answer-name="ans"></div>
</div>
<div data-question-name="rotation" class="main-page question-page content-page">
<h1>spinspinspinspin</h1>
<p>
Consider the matrix for $(a, b) \neq (0, 0)$:
\[
A = \begin{bmatrix}
a & -b\\
b & a
\end{bmatrix}.
\]
First, note that $A$ is a rotation matrix multiplied by some positive constant, i.e.
\[
\begin{bmatrix}
a & -b\\
b & a
\end{bmatrix}
=
c \begin{bmatrix}
\cos \theta & -\sin \theta\\
\sin \theta & \cos \theta
\end{bmatrix}
\]
for some $\theta$ and $c > 0$. Please give the value of that $c$ below in terms of $a$ and $b$.
</p>
<div class="math-input" data-answer-name="constant"></div>
<p>
Now, give a closed form expession for $A^n$ for integers $n$. Note $A^n$ is also a $2 \times 2$
matrix.
You may want to use the function <span class="mono">atan2(y, x)</span>.
</p>
<p>
Top left term of $A^n$:
</p>
<div class="math-input" data-answer-name="ii"></div>
<p>
Top right term of $A^n$:
</p>
<div class="math-input" data-answer-name="ij"></div>
<p>
Bottom left term of $A^n$:
</p>
<div class="math-input" data-answer-name="ji"></div>
<p>
Bottom right term of $A^n$:
</p>
<div class="math-input" data-answer-name="jj"></div>
</div>
<div data-question-name="recursion" class="main-page question-page content-page">
<h1>HHHHHHHHHHHHHHH!</h1>
<p>
For $a, b \in \mathbb R$, consider the function $H$ defined on the non-negative integers $n, k \geq
0$:
\[
H(n,k) = \begin{cases}
1 & n = 0, k = 0\\
1 + aH(k-1, 0) + bH(0, k-1) & n = 0, k \neq 0\\
1 + aH(n-1, 0) + bH(0, n-1) & n \neq 0, k = 0\\
1 + aH(n-1, k) + bH(n, k-1) & \text{otherwise}.
\end{cases}
\]
Find a closed form expression for $H(n, k)$ in terms of $n, k, a, b$. You may assume $a + b \neq 1$.
</p>
<p>
(Hint: Try solving it for $a = 1, b = 1$ first.)
</p>
<div class="math-input" data-answer-name="ans"></div>
</div>
<div data-question-name="hastring" class="main-page question-page content-page">
<h1>HAHAHAHAHAHAHAH!</h1>
<p>
A $HA$-string is a string of uppercase characters that can be permuted into a sequence of zero or
more $HA$, optionally followed by an $H$. For example, $AHAHAHAH$, $AAAHHHH$ and $H$ are all
$HA$-strings, as they can be permuted into $HAHAHAHA$, $HAHAHAH$ and $H$ respectively. However,
$HAHAAHAA$ is not a $HA$-string. The empty string is a $HA$-string.
</p>
<p>
How many $HA$-strings of length $n$ are there for $n \geq 0$? You can type <span
class="mono">floor(x)</span> to
represent $\lfloor x \rfloor$ and <span class="mono">ceil(x)</span> to
represent $\lceil x \rceil$.
</p>
<div class="math-input" data-answer-name="ans"></div>
</div>
<div data-question-name="hagoteem" class="main-page question-page content-page">
<h1>BAHAHAHAHAHAHAH!</h1>
<p>
A $HA$-$GOTEEM$-string is a string of uppercase characters such that its Levenshtein distance from
some $HA$-string is $1$. In other words, you can delete, insert, or modify exactly one character
from any $HA$-$GOTEEM$-string such that the resulting string is a $HA$-string.
</p>
<p>
For example: $GAH$, $HA$ and $AAAGHHH$ are all $HA$-$GOTEEM$ strings.
</p>
<p>
How many $HA$-$GOTEEM$-strings of length $n$ are there for $n \geq 0$?
</p>
<p>
You may assume that $\binom{a}{b} = 0$ for $a, b \in \mathbb Z$ if $a < b$ or $b < 0$. You
may also want to define constants using <span class="mono">let</span> (see the tutorial page).
</p>
<div class="math-input" data-answer-name="ans"></div>
</div>
<div data-question-name="analytic" class="main-page question-page content-page">
<h1>isomorphism?????</h1>
<p>
Suppose that for some fixed $C \in \mathbb R$, an (analytic) function $f(x)$ on $\mathbb{R}$ is characterised by
\[
f(f(x)) - f(x) - x = C
\]
Find the $f(x)$ satisfying this constraint that has the largest $f'(0)$.
</p>
<div class="math-input" data-answer-name="function"></div>
<p>
Also, how many such functions are there?
</p>
<div class="math-input" data-answer-name="howmany"></div>
</div>
<div data-question-name="derivative" class="main-page question-page content-page">
<h1>“cannot do” - a prof</h1>
<p>
Let $f : \mathbb{R} \to \mathbb{R}$ be differentiable at $x_0$. Let:
\[
\begin{align*}
f_0(x, h) &= h\\
f_n(x, h) &= f(x + f_{n-1}(x, h)) - f(x)
\end{align*}
\]
</p>
<p>
Define the <i>$n$-th level derivative</i> of $f$ to be:
\[
\frac{d_n}{dx}f(x) = \lim_{h \to 0} \frac{f_n(x, h)}{h}
\]
Express $\frac{d_n}{dx}f(x_0)$ in terms of $f'(x_0)$ for $n \geq 0$. You may use $F$ to denote
$f'(x_0)$.
</p>
<p>
(You may be tempted to use L'hopital's rule. However, it requires that $f$ is differentiable on an
open interval containing $x_0$, except possibly at $x_0$ itself, which is not a given condition.)
</p>
<div class="math-input" data-answer-name="ans"></div>
</div>
<div data-question-name="checkers" class="main-page question-page content-page">
<h1>Now the fun begins</h1>
<p>
Suppose you want to arrange red and green checkers on an $8 \times 8$ red and green checkboard in a
certain way.
</p>
<p>
You can place red checkers on green tiles and green checkers on red tiles, but no green checker may
be adjacent to a red checker.
</p>
<p>
How many valid ways are there to arrange the red and green checkers? Note that an empty board also
counts as a valid arrangement.
</p>
<p>
Note: There is no general formula.
</p>
<div class="math-input" data-answer-name="ans"></div>
</div>
<div data-question-name="fibonacci" class="main-page question-page content-page">
<h1>Fibonacci (Nice)</h1>
<p>
Recall the Fibonacci numbers $F_n := 1, 1, 2, 3, 5, 8, 13, \cdots$.
</p>
<p>
Suppose two Fibonacci numbers $x_1, x_2$ are independently drawn at random with uniform probability.
What is the probability that $x_1 + x_2$ shares a factor with $69$ other than $1$?
</p>
<p>
If you do not like the notion of drawing a random integer that could be arbitrarily large, find or
bound the probability with the restriction $x_1, x_2 < N$ for any $N$ and take $N \to \infty$.
</p>
<div class="math-input" data-answer-name="ans"></div>
</div>
</div>
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