diff --git a/Misc/Proof Techniques/parts/2 induction.tex b/Misc/Proof Techniques/parts/2 induction.tex index 7f5be5a..0528787 100644 --- a/Misc/Proof Techniques/parts/2 induction.tex +++ b/Misc/Proof Techniques/parts/2 induction.tex @@ -189,11 +189,8 @@ For example, see the proof of the statement in \ref{binomsum} on the next page. \pagebreak -\makeatletter -\@makeORMCbox{tmpbox}{Alternative Proof}{ogrape!10!white}{ogrape} -\makeatother -\begin{tmpbox} +\begin{ORMCbox}{Alternative Proof}{ogrape!10!white}{ogrape} Consider the following problem: \par How many ways are there to write a number $x$ as an ordered sum of positive integers? \par \note{ @@ -237,7 +234,7 @@ For example, see the proof of the statement in \ref{binomsum} on the next page. We've found that the number of ways to split $x$ can be written as either $\sum_{n = 1}^{x-1} \binom{x-1}{n}$ or $2^{x-1}$, and therefore $\sum_{n = 1}^{x-1} \binom{x-1}{n} = 2^{x-1}$. -\end{tmpbox} +\end{ORMCbox} \pagebreak \ No newline at end of file diff --git a/Misc/Warm-Ups/rootbound.tex b/Misc/Warm-Ups/rootbound.tex new file mode 100755 index 0000000..549f0c6 --- /dev/null +++ b/Misc/Warm-Ups/rootbound.tex @@ -0,0 +1,117 @@ +\documentclass[ + solutions, + hidewarning, + singlenumbering, + nopagenumber +]{../../resources/ormc_handout} +\usepackage{../../resources/macros} + +\def\utri#1#2{ + \draw (#1,#2) -- ++(60:1) -- ++(-60:1) -- cycle; +} +\def\dtri#1#2{ + \draw (#1,#2) -- (#1+1, #2) -- ++(-120:1) -- cycle; +} + +\def\board#1{ + \begin{center} + \begin{tikzpicture}[scale=#1] + \draw (0,0) -- (6, 0); + \foreach \y in {0,...,5} { + \foreach[expand list] \x in {-6,-5,...,\y} { + \utri{\x + 6 - 0.5 - \y * 0.5}{\y * -0.86603 - 0.86603} + } + } + \foreach \y in {5,...,0} { + \foreach[expand list] \x in {-5,-4,...,\y} { + \utri{\x + 5 - \y * 0.5}{\y * 0.86603 - 12 * 0.86603} + } + } + \draw (-3, -6 * 0.86603) -- (0, -12 * 0.86603); + \draw (9, -6 * 0.86603) -- (6, -12 * 0.86603); + \end{tikzpicture} + \end{center} +} + +\title{Rootbound} +\uptitler{\smallurl{}} +\uptitlel{Warm-ups} +\subtitle{ + Prepared by Mark on \today. \par + Based on \url{https://cjffield.com/rules/rootbound.pdf} +} + +\begin{document} + + \maketitle + + \generic{Playing the game:} + % Official size is 7 spaces per side, but variations work. + You will need a pencil, an eraser, and a printed hexagonal board. \par + Draw your pieces lightly---you may need to erase them. \par + Turns alternate. Each player draws distinct pieces (for example, circles and squares). + + \vspace{2mm} + + An example board is drawn below. \say{Spaces} are the corners of the triangular cells, \textit{not} the cells themselves. + Points on the edge of the board are also spaces. + + \board{0.3} + + During the first turn of the game, the first player must place one piece onto any empty space. + From then on, each player may place either one or two pieces onto empty spaces, respecting the restrictions below. + Players may not skip their turn unless they have no legal move. + + + \generic{Claimed regions:} + A \textit{claimed region} is an area of empty spaces that is only adjacent to pieces of a single color. \par + No regions are considered \say{claimed} until both players have had a turn. + + + \generic{Dead groups:} + A group of pieces is considered \textit{dead} if it is cut off from all other same-colored groups and is not + adjacent to a claimed region. At the end of each turn (except each player's first), erase all + dead pieces that belong to the inactive player. + + \generic{Restrictions:} + \begin{itemize}[itemsep=1mm] + \item A piece may not be placed inside a claimed region. + \item A piece may not be placed adjacent to two (or more) same-colored pieces if those pieces are also + neighbors of each other. In other words, you may not form small triangles of one color. + + \item When placing two pieces adjacent to each other in a tun, they may not form a + straight line with a third pre-existing piece of their color. + \end{itemize} + + + \generic{Ending the game:} + The game ends when both players run out of legal moves. + Each player's score is calculated by counting the number of empty spaces in regions they have claimed. \par + The player with the highest score wins. + + \vfill + + \begin{ORMCbox}{Notes}{ogrape!10!white}{ogrape} + \begin{itemize} + \item + Each player needs to make multiple groups before the first player's second turn. \par + Otherwise, all their pieces will be removed from the game, resulting in an early loss. \par + Do not make your first placements next to each other! + + \item + Live groups are easiest to secure near the corners or along the edges. Play there first. \par + \end{itemize} + \end{ORMCbox} + + + + + \pagebreak + + \null\vfill + \board{1.4} + \vfill\null + + \pagebreak + +\end{document} \ No newline at end of file diff --git a/Misc/Warm-Ups/roots.tex b/Misc/Warm-Ups/roots.tex deleted file mode 100755 index fcbe939..0000000 --- a/Misc/Warm-Ups/roots.tex +++ /dev/null @@ -1,103 +0,0 @@ -\documentclass[ - solutions, - hidewarning, - singlenumbering, - nopagenumber -]{../../resources/ormc_handout} -\usepackage{../../resources/macros} - -\def\utri#1#2{ - \draw (#1,#2) -- ++(60:1) -- ++(-60:1) -- cycle; -} -\def\dtri#1#2{ - \draw (#1,#2) -- (#1+1, #2) -- ++(-120:1) -- cycle; -} - -\def\board#1{ - \begin{center} - \begin{tikzpicture}[scale=#1] - \draw (0,0) -- (6, 0); - \foreach \y in {0,...,5} { - \foreach[expand list] \x in {-6,-5,...,\y} { - \utri{\x + 6 - 0.5 - \y * 0.5}{\y * -0.86603 - 0.86603} - } - } - \foreach \y in {5,...,0} { - \foreach[expand list] \x in {-5,-4,...,\y} { - \utri{\x + 5 - \y * 0.5}{\y * 0.86603 - 12 * 0.86603} - } - } - \draw (-3, -6 * 0.86603) -- (0, -12 * 0.86603); - \draw (9, -6 * 0.86603) -- (6, -12 * 0.86603); - \end{tikzpicture} - \end{center} -} - -\title{Rootbound} -\uptitler{\smallurl{}} -\uptitlel{Warm-ups} -\subtitle{Prepared by Mark on \today.} - -\begin{document} - - \maketitle - - \generic{Playing the game:} - % Official size is 7 spaces per side, but variations work. - You will need a pencil, an eraser, and a printed hexagonal board. \par - Draw your pieces lightly---you may need to erase them. - - - Turns alternate. - On the first player's first turn, they must place one piece onto any empty space. - From then on, players may place either one or two pieces onto empty spaces, respecting the restrictions below. - Players may not skip their turn unless they have no legal move. - - - \generic{Claimed regions:} - A \textit{claimed region} is an area of empty spaces that is adjacent to only pieces of a single color. \par - No regions are Claimed Regions until both players have had a turn. - - - \generic{Dead groups:} - A group of pieces is considered \textit{dead} if it is cut off from all other same-colored groups and is not - adjacent to a claimed region. At the end of each turn (except each player's first), remove all - dead pieces that belong to the inactive player. - - \generic{Restrictions:} - \begin{itemize} - \item A piece may not be placed in a claimed region. - \item A piece may not be placed adjacent to two or more same-colored pieces if those pieces are also - neighbors of each other (i.e, if they do not form any small triangles of one color). - - \item Additionally, when placing two pieces adjacent to each other in a turn, they may not form a - straight line with a third pre-existing piece of their color. Rapid growth cannot go in a straight - line. - \end{itemize} - - - \generic{Ending the game:} - The game ends when both players run out of legal moves. - Each player's score is calculated by counting the number of empty spaces in regions they have claimed. - The player with the highest score wins. - -%It is important to note that players will need to make multiple groups by the second player’s -%first turn or the first player’s second. Otherwise, their groups will be considered dead and -%removed from the game, resulting in an early loss. Do not make your first placements next to -%each other. - -%Live groups are easiest to secure near the corners or along the edges. Play there first. But do -%not get too greedy by trying to secure too much space. It is very easy for the enemy to invade -%and ruin your plans. - - - \vfill - \pagebreak - - \null\vfill - \board{1.4} - \vfill\null - - \pagebreak - -\end{document} \ No newline at end of file