From 1a486e2d9886256424a416b6314f08f99ee8acd8 Mon Sep 17 00:00:00 2001 From: Mark Date: Wed, 22 Jan 2025 21:43:39 -0800 Subject: [PATCH] WIP ODD DICE --- src/Warm-Ups/Odd Dice/main.tex | 132 --------------------------------- src/Warm-Ups/Odd Dice/main.typ | 87 ++++++++++++++++++++++ 2 files changed, 87 insertions(+), 132 deletions(-) delete mode 100755 src/Warm-Ups/Odd Dice/main.tex create mode 100644 src/Warm-Ups/Odd Dice/main.typ diff --git a/src/Warm-Ups/Odd Dice/main.tex b/src/Warm-Ups/Odd Dice/main.tex deleted file mode 100755 index 270936d..0000000 --- a/src/Warm-Ups/Odd Dice/main.tex +++ /dev/null @@ -1,132 +0,0 @@ -\documentclass[ - nosolutions, - hidewarning, - singlenumbering, - nopagenumber -]{../../../lib/tex/ormc_handout} -\usepackage{../../../lib/tex/macros} - - - -\usepackage{tikz} -\usetikzlibrary{arrows.meta} -\usetikzlibrary{shapes.geometric} - -% We put nodes in a separate layer, so we can -% slightly overlap with paths for a perfect fit -\pgfdeclarelayer{nodes} -\pgfdeclarelayer{path} -\pgfsetlayers{main,nodes} - -% Layer settings -\tikzset{ - % Layer hack, lets us write - % later = * in scopes. - layer/.style = { - execute at begin scope={\pgfonlayer{#1}}, - execute at end scope={\endpgfonlayer} - }, - % - % Arrowhead tweak - >={Latex[ width=2mm, length=2mm ]}, - % - % Nodes - main/.style = { - draw, - circle, - fill = white, - line width = 0.35mm - } -} - -\title{Warm Up: Odd Dice} -\uptitler{\smallurl{}} -\subtitle{Prepared by Mark on \today} - - -\begin{document} - - \maketitle - - \problem{} - - We say a set of dice $\{A, B, C\}$ is \textit{nontransitive} - if, on average, $A$ beats $B$, $B$ beats $C$, and $C$ beats $A$. - In other words, we get a counterintuitive \say{rock - paper - scissors} effect. - - \vspace{2mm} - - Create a set of nontransitive six-sided dice. \par - \hint{All sides should be numbered with positive integers less than 10.} - - \begin{solution} - One possible set can be numbered as follows: - \begin{itemize} - \item Die $A$: $2, 2, 4, 4, 9, 9$ - \item Die $B$: $1, 1, 6, 6, 8, 8$ - \item Die $C$: $3, 3, 5, 5, 7, 7$ - \end{itemize} - - \vspace{4mm} - - Another solution is below: - \begin{itemize} - \item Die $A$: $3, 3, 3, 3, 3, 6$ - \item Die $B$: $2, 2, 2, 5, 5, 5$ - \item Die $C$: $1, 4, 4, 4, 4, 4$ - \end{itemize} - \end{solution} - - \vfill - - \problem{} - Now, consider the set of six-sided dice below: - \begin{itemize} - \item Die $A$: $4, 4, 4, 4, 4, 9$ - \item Die $B$: $3, 3, 3, 3, 8, 8$ - \item Die $C$: $2, 2, 2, 7, 7, 7$ - \item Die $D$: $1, 1, 6, 6, 6, 6$ - \item Die $E$: $0, 5, 5, 5, 5, 5$ - \end{itemize} - On average, which die beats each of the others? Draw a graph. \par - - \begin{solution} - \begin{center} - \begin{tikzpicture}[scale = 0.5] - \begin{scope}[layer = nodes] - \node[main] (a) at (-2, 0.2) {$a$}; - \node[main] (b) at (0, 2) {$b$}; - \node[main] (c) at (2, 0.2) {$c$}; - \node[main] (d) at (1, -2) {$d$}; - \node[main] (e) at (-1, -2) {$e$}; - \end{scope} - - \draw[->] - (a) edge (b) - (b) edge (c) - (c) edge (d) - (d) edge (e) - (e) edge (a) - - (a) edge (c) - (b) edge (d) - (c) edge (e) - (d) edge (a) - (e) edge (b) - ; - \end{tikzpicture} - \end{center} - \end{solution} - - \vfill - - Now, say we roll each die twice. What happens to the graph above? - - \begin{solution} - The direction of each edge is reversed! - \end{solution} - - \vfill - \pagebreak - -\end{document} \ No newline at end of file diff --git a/src/Warm-Ups/Odd Dice/main.typ b/src/Warm-Ups/Odd Dice/main.typ new file mode 100644 index 0000000..29c7cd5 --- /dev/null +++ b/src/Warm-Ups/Odd Dice/main.typ @@ -0,0 +1,87 @@ +#import "@local/handout:0.1.0": * + +#show: doc => handout( + doc, + quarter: link( + "https://betalupi.com/handouts", + "betalupi.com/handouts", + ), + + title: [Warm-Up: Odd Dice], + by: "Mark", +) + +#problem() +We say a set of dice ${A, B, C}$ is _nontransitive_ +if, on average, $A$ beats $B$, $B$ beats $C$, and $C$ beats $A$. +In other words, we get a counterintuitive "rock - paper - scissors" effect. + +#v(2mm) + +Create a set of nontransitive six-sided dice. \ +#hint([All sides should be numbered with positive integers less than 10.]) + +#solution([ + One possible set can be numbered as follows: + - Die $A$: $2, 2, 4, 4, 9, 9$ + - Die $B$: $1, 1, 6, 6, 8, 8$ + - Die $C$: $3, 3, 5, 5, 7, 7$ + + #v(4mm) + + Another solution is below: + - Die $A$: $3, 3, 3, 3, 3, 6$ + - Die $B$: $2, 2, 2, 5, 5, 5$ + - Die $C$: $1, 4, 4, 4, 4, 4$ + +]) + +#v(1fr) + +#problem() +Now, consider the set of six-sided dice below: +- Die $A$: $4, 4, 4, 4, 4, 9$ +- Die $B$: $3, 3, 3, 3, 8, 8$ +- Die $C$: $2, 2, 2, 7, 7, 7$ +- Die $D$: $1, 1, 6, 6, 6, 6$ +- Die $E$: $0, 5, 5, 5, 5, 5$ +On average, which die beats each of the others? Draw a diagram. + +#solution([ + /* + \begin{tikzpicture}[scale = 0.5] + \begin{scope}[layer = nodes] + \node[main] (a) at (-2, 0.2) {$a$}; + \node[main] (b) at (0, 2) {$b$}; + \node[main] (c) at (2, 0.2) {$c$}; + \node[main] (d) at (1, -2) {$d$}; + \node[main] (e) at (-1, -2) {$e$}; + \end{scope} + + \draw[->] + (a) edge (b) + (b) edge (c) + (c) edge (d) + (d) edge (e) + (e) edge (a) + + (a) edge (c) + (b) edge (d) + (c) edge (e) + (d) edge (a) + (e) edge (b) + ; + \end{tikzpicture} + */ +]) + +#v(1fr) + +#problem() +Now, say we roll each die twice. What happens to the graph fromE the previous problem? + +#solution([ + The direction of each edge is reversed! +]) + +#v(1fr)