\documentclass[hyperref={colorlinks=true}]{beamer}
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% Not in French
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\usepackage{tikz,tikz-cd}
%\definecolor{GREEN}{rgb}{0.0,1.0,0.0}
\definecolor{GREEN}{RGB}{102,204,0}
\definecolor{JAUNE}{RGB}{204,204,80}
\definecolor{GREY}{RGB}{160,160,160}
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\newtheorem{definition*}{Definition}
\newtheorem{proposition*}{Proposition}

\begin{document}

\title{Wordle}
\author{David Kohel\newline
Institut de Math{\'e}matiques de Marseille}
\date{Marseille \newline January 2022}


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% TITLEPAGE
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\begin{frame}\titlepage\end{frame}
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% TABLE OF CONTENTS
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\begin{frame}\tableofcontents\end{frame}
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\section{Wordle puzzle}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{The WORDLE puzzle}

WORDLE is a puzzle in which one tries to guess a five-letter word.
A new word is chosen each day, and one has (up to) six guesses to
determine the word.
\begin{center}
\begin{tikzpicture}[box/.style={rectangle,draw=black,thick,minimum size=10mm},scale=1.1]
\foreach \x in {0,1,...,4}{
    \foreach \y in {0}
        \node[box] at (\x,\y){};
}
\end{tikzpicture}
\end{center}
Clues are given for correct letters: they are shown in
\color{GREEN}green\color{black}\ if they are in the
right position, otherwise in \color{JAUNE}yellow\color{black}.
\vspace{4mm}
\pause

So let's start with a first guess:
\begin{center}
\begin{tikzpicture}[box/.style={rectangle,draw=black,thick,minimum size=10mm},scale=1.1]
\foreach \x in {0,1,...,4}{
    \foreach \y in {0}
        \node[box] at (\x,\y){};
}
\node[box] at (0,0){{\huge\bf M}};
\node[box] at (1,0){{\huge\bf E}};
\node[box] at (2,0){{\huge\bf R}};
\node[box] at (3,0){{\huge\bf I}};
\node[box] at (4,0){{\huge\bf T}};
\end{tikzpicture}
\end{center}
{\bf Warning:} one can only guess a valid word!
\end{frame}


\begin{frame}
\frametitle{The WORDLE puzzle}

\only<1>{
\begin{center}
\begin{tikzpicture}[box/.style={rectangle,draw=black,thick,minimum size=10mm},scale=1.1]
\foreach \x in {0,1,...,4}{
    \foreach \y in {0,1,...,5}
        \node[box] at (\x,\y){};
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREY ] at (0,5){{\huge\bf M}};
\node[box,fill=GREY ] at (1,5){{\huge\bf E}};
\node[box,fill=JAUNE] at (2,5){{\huge\bf R}};
\node[box,fill=GREY ] at (3,5){{\huge\bf I}};
\node[box,fill=GREY ] at (4,5){{\huge\bf T}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box] at (0,4){};
\node[box] at (1,4){};
\node[box] at (2,4){};
\node[box] at (3,4){};
\node[box] at (4,4){};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box] at (0,3){};
\node[box] at (1,3){};
\node[box] at (2,3){};
\node[box] at (3,3){};
\node[box] at (4,3){};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box] at (0,2){};
\node[box] at (1,2){};
\node[box] at (2,2){};
\node[box] at (3,2){};
\node[box] at (4,2){};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{tikzpicture}
\end{center}}
\only<2>{
\begin{center}
\begin{tikzpicture}[box/.style={rectangle,draw=black,thick,minimum size=10mm},scale=1.1]
\foreach \x in {0,1,...,4}{
    \foreach \y in {0,1,...,5}
        \node[box] at (\x,\y){};
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREY ] at (0,5){{\huge\bf M}};
\node[box,fill=GREY ] at (1,5){{\huge\bf E}};
\node[box,fill=JAUNE] at (2,5){{\huge\bf R}};
\node[box,fill=GREY ] at (3,5){{\huge\bf I}};
\node[box,fill=GREY ] at (4,5){{\huge\bf T}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box] at (0,4){{\huge\bf S}};
\node[box] at (1,4){{\huge\bf H}};
\node[box] at (2,4){{\huge\bf A}};
\node[box] at (3,4){{\huge\bf W}};
\node[box] at (4,4){{\huge\bf L}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{tikzpicture}
\end{center}}
\only<3>{
\begin{center}
\begin{tikzpicture}[box/.style={rectangle,draw=black,thick,minimum size=10mm},scale=1.1]
\foreach \x in {0,1,...,4}{
    \foreach \y in {0,1,...,5}
        \node[box] at (\x,\y){};
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREY ] at (0,5){{\huge\bf M}};
\node[box,fill=GREY ] at (1,5){{\huge\bf E}};
\node[box,fill=JAUNE] at (2,5){{\huge\bf R}};
\node[box,fill=GREY ] at (3,5){{\huge\bf I}};
\node[box,fill=GREY ] at (4,5){{\huge\bf T}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREEN] at (0,4){{\huge\bf S}};
\node[box,fill=GREEN] at (1,4){{\huge\bf H}};
\node[box,fill=GREEN] at (2,4){{\huge\bf A}};
\node[box,fill=GREY ] at (3,4){{\huge\bf W}};
\node[box,fill=GREY ] at (4,4){{\huge\bf L}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{tikzpicture}
\end{center}}
\only<4>{
\begin{center}
\begin{tikzpicture}[box/.style={rectangle,draw=black,thick,minimum size=10mm},scale=1.1]
\foreach \x in {0,1,...,4}{
    \foreach \y in {0,1,...,5}
        \node[box] at (\x,\y){};
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREY ] at (0,5){{\huge\bf M}};
\node[box,fill=GREY ] at (1,5){{\huge\bf E}};
\node[box,fill=JAUNE] at (2,5){{\huge\bf R}};
\node[box,fill=GREY ] at (3,5){{\huge\bf I}};
\node[box,fill=GREY ] at (4,5){{\huge\bf T}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREEN] at (0,4){{\huge\bf S}};
\node[box,fill=GREEN] at (1,4){{\huge\bf H}};
\node[box,fill=GREEN] at (2,4){{\huge\bf A}};
\node[box,fill=GREY ] at (3,4){{\huge\bf W}};
\node[box,fill=GREY ] at (4,4){{\huge\bf L}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box] at (0,3){{\huge\bf P}};
\node[box] at (1,3){{\huge\bf R}};
\node[box] at (2,3){{\huge\bf O}};
\node[box] at (3,3){{\huge\bf U}};
\node[box] at (4,3){{\huge\bf D}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{tikzpicture}
\end{center}}
\only<5>{
\begin{center}
\begin{tikzpicture}[box/.style={rectangle,draw=black,thick,minimum size=10mm},scale=1.1]
\foreach \x in {0,1,...,4}{
    \foreach \y in {0,1,...,5}
        \node[box] at (\x,\y){};
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREY ] at (0,5){{\huge\bf M}};
\node[box,fill=GREY ] at (1,5){{\huge\bf E}};
\node[box,fill=JAUNE] at (2,5){{\huge\bf R}};
\node[box,fill=GREY ] at (3,5){{\huge\bf I}};
\node[box,fill=GREY ] at (4,5){{\huge\bf T}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREEN] at (0,4){{\huge\bf S}};
\node[box,fill=GREEN] at (1,4){{\huge\bf H}};
\node[box,fill=GREEN] at (2,4){{\huge\bf A}};
\node[box,fill=GREY ] at (3,4){{\huge\bf W}};
\node[box,fill=GREY ] at (4,4){{\huge\bf L}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREY ] at (0,3){{\huge\bf P}};
\node[box,fill=JAUNE] at (1,3){{\huge\bf R}};
\node[box,fill=GREY ] at (2,3){{\huge\bf O}};
\node[box,fill=GREY ] at (3,3){{\huge\bf U}};
\node[box,fill=GREEN] at (4,3){{\huge\bf D}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{tikzpicture}
\end{center}}
\only<6>{
\begin{center}
\begin{tikzpicture}[box/.style={rectangle,draw=black,thick,minimum size=10mm},scale=1.1]
\foreach \x in {0,1,...,4}{
    \foreach \y in {0,1,...,5}
        \node[box] at (\x,\y){};
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREY ] at (0,5){{\huge\bf M}};
\node[box,fill=GREY ] at (1,5){{\huge\bf E}};
\node[box,fill=JAUNE] at (2,5){{\huge\bf R}};
\node[box,fill=GREY ] at (3,5){{\huge\bf I}};
\node[box,fill=GREY ] at (4,5){{\huge\bf T}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREEN] at (0,4){{\huge\bf S}};
\node[box,fill=GREEN] at (1,4){{\huge\bf H}};
\node[box,fill=GREEN] at (2,4){{\huge\bf A}};
\node[box,fill=GREY ] at (3,4){{\huge\bf W}};
\node[box,fill=GREY ] at (4,4){{\huge\bf L}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREY ] at (0,3){{\huge\bf P}};
\node[box,fill=JAUNE] at (1,3){{\huge\bf R}};
\node[box,fill=GREY ] at (2,3){{\huge\bf O}};
\node[box,fill=GREY ] at (3,3){{\huge\bf U}};
\node[box,fill=GREEN] at (4,3){{\huge\bf D}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box] at (0,2){{\huge\bf S}};
\node[box] at (1,2){{\huge\bf H}};
\node[box] at (2,2){{\huge\bf A}};
\node[box] at (3,2){{\huge\bf R}};
\node[box] at (4,2){{\huge\bf D}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{tikzpicture}
\end{center}}
\only<7>{
\begin{center}
\begin{tikzpicture}[box/.style={rectangle,draw=black,thick,minimum size=10mm},scale=1.1]
\foreach \x in {0,1,...,4}{
    \foreach \y in {0,1,...,5}
        \node[box] at (\x,\y){};
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREY ] at (0,5){{\huge\bf M}};
\node[box,fill=GREY ] at (1,5){{\huge\bf E}};
\node[box,fill=JAUNE] at (2,5){{\huge\bf R}};
\node[box,fill=GREY ] at (3,5){{\huge\bf I}};
\node[box,fill=GREY ] at (4,5){{\huge\bf T}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREEN] at (0,4){{\huge\bf S}};
\node[box,fill=GREEN] at (1,4){{\huge\bf H}};
\node[box,fill=GREEN] at (2,4){{\huge\bf A}};
\node[box,fill=GREY ] at (3,4){{\huge\bf W}};
\node[box,fill=GREY ] at (4,4){{\huge\bf L}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREY ] at (0,3){{\huge\bf P}};
\node[box,fill=JAUNE] at (1,3){{\huge\bf R}};
\node[box,fill=GREY ] at (2,3){{\huge\bf O}};
\node[box,fill=GREY ] at (3,3){{\huge\bf U}};
\node[box,fill=GREEN] at (4,3){{\huge\bf D}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREEN ] at (0,2){{\huge\bf S}};
\node[box,fill=GREEN ] at (1,2){{\huge\bf H}};
\node[box,fill=GREEN ] at (2,2){{\huge\bf A}};
\node[box,fill=GREEN ] at (3,2){{\huge\bf R}};
\node[box,fill=GREEN ] at (4,2){{\huge\bf D}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{tikzpicture}
\end{center}}

\end{frame}

\begin{frame}
\frametitle{The WORDLE puzzle}

After solving the puzzle, one can share the an emoji color grid of your wordle solution.
\begin{center}
\begin{tikzpicture}[box/.style={rectangle,draw=black,thick,minimum size=10mm},scale=1.1]

\foreach \x in {0,1,...,4}{
    \foreach \y in {2,3,...,5}
        \node[box] at (\x,\y){};
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREY ] at (0,5){};
\node[box,fill=GREY ] at (1,5){};
\node[box,fill=JAUNE] at (2,5){};
\node[box,fill=GREY ] at (3,5){};
\node[box,fill=GREY ] at (4,5){};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREEN] at (0,4){};
\node[box,fill=GREEN] at (1,4){};
\node[box,fill=GREEN] at (2,4){};
\node[box,fill=GREY ] at (3,4){};
\node[box,fill=GREY ] at (4,4){};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREY ] at (0,3){};
\node[box,fill=JAUNE] at (1,3){};
\node[box,fill=GREY ] at (2,3){};
\node[box,fill=GREY ] at (3,3){};
\node[box,fill=GREEN] at (4,3){};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[box,fill=GREEN] at (0,2){};
\node[box,fill=GREEN] at (1,2){};
\node[box,fill=GREEN] at (2,2){};
\node[box,fill=GREEN] at (3,2){};
\node[box,fill=GREEN] at (4,2){};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{tikzpicture}
\end{center}
This lets you share your game history without giving away the word of the day.
\end{frame}

\begin{frame}
  \frametitle{WORDLE}

  \vspace{16mm}
  \begin{center}{\large\bf But where's the math?}\end{center}
  \vspace{16mm}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Five-letter word graphs}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
  \frametitle{Word graphs}

  % Normally we avoid manually forcing a newline, but here we force
  % a few line breaks with \\.

  Let $V$ be the set of five-letter words. We define a word graph\\
  on the vertices $V$ by addding an edge $e = \{w_1,w_2\}$ for each\\
  pair $(w_1,w_2)$ of distinct words in $V \times V$, if the two words
  have disjoint sets of letters.
  \vspace{8mm}

  In order to test as many letters as possible, with as few guesses\\
  as possible, we look for adjacent pairs, triples, quadruples, and
  quintuplets in the graph.
\end{frame}

\begin{frame}
  \frametitle{Word graphs}

  \begin{definition*}
    A graph is {\em complete} on $V$ if there is an edge $e = \{v_1,v_2\}$
    for every pair $(v_1,v_2)$ of distinct vertices in $V \times V$.
  \end{definition*}
  \vspace{4mm}

  \begin{proposition*}
    A set of $n$ words $W = \{w_1,\dots,w_n\}$ is pairwise disjoint if
    and only if the subgraph of the word graph on the vertex set $W$
    is complete.
  \end{proposition*}
  \vspace{8mm}

  Do there exist complete subgraphs of the five-letter word graph?
\end{frame}

\begin{frame}[fragile]
\frametitle{Word graphs}

\noindent{\bf Yes!}
\begin{center}
\begin{tikzpicture}[main/.style = {draw, circle}]
% Magma:
% [ [Round(200*Cos(Pi(RR)*n/20)),Round(200*Sin(Pi(RR)*n/20))] : n in [2..40 by 8] ];
% [ 0, 200 ], [ +/-190, 62 ], [ +/-118, -162 ]
% BRAWN FLECK GHOST JUMPY VIVID
\node[main] (GHOST) at (0.00,2.00) {{\bf\tiny GHOST}};
\node[main] (BRAWN) at (1.90,0.62) {{\bf\tiny BRAWN}};
\node[main] (FLECK) at (1.18,-1.62) {{\bf\tiny FLECK}};
\node[main] (JUMPY) at (-1.18,-1.62) {{\bf\tiny JUMPY}};
\node[main] (VIVID) at (-1.90,0.62) {{\bf\tiny VIVID}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw (GHOST) -- (BRAWN);
\draw (GHOST) -- (FLECK);
\draw (GHOST) -- (JUMPY);
\draw (GHOST) -- (VIVID);
\draw (BRAWN) -- (FLECK);
\draw (BRAWN) -- (JUMPY);
\draw (BRAWN) -- (VIVID);
\draw (FLECK) -- (JUMPY);
\draw (FLECK) -- (VIVID);
\draw (JUMPY) -- (VIVID);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% [ [Round(200*Cos(Pi(RR)*n/20))-380,Round(200*Sin(Pi(RR)*n/20))] : n in [2..40 by 8] ];
% [ -570, 62 ], [ -380, 200 ], [ -190, 62 ], [ -262, -162 ], [ -498, -162 ]
% BOTCH FLUNG JERKY SWAMP VIVID
\node[main] (JERKY) at (-3.80,2.00) {{\bf\tiny JERKY}};
\node[main] (FLUNG) at (-5.70,0.62) {{\bf\tiny FLUNG}};
\node[main] (SWAMP) at (-2.62,-1.62) {{\bf\tiny SWAMP}};
\node[main] (BOTCH) at (-4.98,-1.62) {{\bf\tiny BOTCH}};
%\node[main] (5) at (-1.90,0.62) {{\bf\tiny VIVID}};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw (VIVID) -- (JERKY);
\draw (VIVID) -- (FLUNG);
\draw (VIVID) -- (SWAMP);
\draw (VIVID) -- (BOTCH);
\draw (JERKY) -- (FLUNG);
\draw (JERKY) -- (SWAMP);
\draw (JERKY) -- (BOTCH);
\draw (FLUNG) -- (SWAMP);
\draw (FLUNG) -- (BOTCH);
\draw (SWAMP) -- (BOTCH);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Labelled:
%\draw (SWAMP) to [out=330, in=240] node[midway, above] {{\tiny 5}} (FLECK);
% Unlabelled:
\draw (JERKY) to [out=36, in=144] (GHOST);
\draw (SWAMP) to [out=324, in=252] (FLECK);
\draw (BOTCH) to [out=288, in=216] (JUMPY);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{tikzpicture}
\end{center}
This subgraph contains two complete subgraphs on five vertices.
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Wordle quintuplets}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{WORDLE quintuplets}

\noindent{\bf Examples.}
There exist numerous WORDLE quintuplets, even if we assume that four
out of five words consist of five distinct letters.

\begin{center}
\begin{tabular}{ccccc|l}
  BLITZ & FROND & JUMPY & WHACK & GEESE & $\lnot\ \{$Q,V$\}$\\
  BLITZ & FJORD & SHUCK & WAXEN & PYGMY & $\lnot\ \{$Q,V$\}$\\
  %BIGHT & CRANK & FOWLS & JUMPY & VEXED & $\lnot\ \{$Q,Z$\}$\\
  BLOCK & DWARF & JUMPS & ZINGY & TEETH & $\lnot\ \{$Q,V,X$\}$\\
  %BLOND & FRITZ & JACKY & WHUMP & GEESE & $\lnot\ \{$Q,V,X$\}$\\
  BOTCH & FLUNG & JERKY & SWAMP & VIVID & $\lnot\ \{$Q,X,Z$\}$\\
  BRAWN & FLECK & GHOST & JUMPY & VIVID & $\lnot\ \{$Q,X,Z$\}$\\
  %BRONX & FETCH & GAWKS & JUMPY & LIVID & $\lnot\ \{$Q,Z$\}$\\
  CURVY & JAMBS & PHLOX & TWINK & EDGED & $\lnot\ \{$Q,Z$\}$\\
  DOWRY & FLUNG & JAMBS & THICK & PEEVE & $\lnot\ \{$Q,X,Z$\}$\\
  %FJORD & GUNKY & SWEPT & ZILCH & MAMBA & $\lnot\ \{$Q,V,X$\}$\\\
  %FJORD & HACKS & PLUMB & VYING & TWEET & $\lnot\ \{$Q,X,Z$\}$\\
  & & $\vdots$ &
\end{tabular}
\end{center}
\end{frame}

\begin{frame}
  \frametitle{Edge-colored graphs}

  In order to take into account known characters and their positions,
  we define the complete graph on $V$ with {\it edge coloring}:
  $$
  E \longrightarrow \NN
  %  \{w_1,w_2\} \arrow[r] & |w_1 \cap w_2|
  %\begin{tikzcd}
  %  E \arrow[r] & \NN \\
  %  \{w_1,w_2\} \arrow[r] & |w_1 \cap w_2|
  %\end{tikzcd}
  $$
  taking an edge $\{w_1,w_2\}$ to the cardinality $|w_1\cap w_2|$ of
  their common letters.
  \vspace{4mm}

  The previous graph is the subgraph on $V$ of $5$-colored edges.
  \vspace{4mm}

  How can we use the structure of this graph to solve WORDLE?
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Variants}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{WORDLE variants}

WORDLE only appeared online to the public in October 2021.\\
As of November 1, it had 90 players, and by two months later\\
had grown to 300,000 players. This spawned numerous spinoffs.

\vspace{4mm}
N.B. The word VIRAL is a five-letter word.
\vspace{8mm}

So, what about other languages ... like {\it marseillais} ?
\begin{center}
  {\large\url{motchus.fr}}
\end{center}
\vspace{8mm}
\begin{center}
  {\large\bf Questions?}
\end{center}

\end{frame}

\end{document}
