
diff -r 4a22b233a5bf -r 8c007dcf31c4 cws/resit2.tex
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+++ b/cws/resit2.tex	Thu Jun 13 13:13:33 2024 +0100
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+% !TEX program = xelatex
+\documentclass{article}
+\usepackage{chessboard}
+\usepackage[LSBC4,T1]{fontenc}
+\let\clipbox\relax
+\usepackage{../styles/style}
+\usepackage{../styles/langs}
+\usepackage{disclaimer}
+\usepackage{ulem}
+%\usepackage{tipauni}
+
+
+
+\tikzset
+{%
+  pics/piece/.style n args={1}{
+    code={%
+      \fill[rounded corners]                  (-0.1,-0.1) rectangle (-0.9, -0.9);
+      \fill[left color=white,rounded corners,
+            right color=gray,
+            opacity=0.7]      (-0.1,-0.1) rectangle (-0.9, -0.9);
+      \draw[line width=0.4mm,rounded corners] (-0.1,-0.1) rectangle (-0.9, -0.9);
+      \draw[line width=0.2mm,rounded corners] (-0.2,-0.2) rectangle (-0.8, -0.8);
+      \draw[anchor=mid] (-0.5,-0.6) node {#1};
+    }},
+  pics/king/.style n args={1}{
+    code={%
+      \fill[rounded corners]                  (-0.1,-0.1) rectangle (-0.9, -0.9);
+      \fill[left color=white,rounded corners,
+            right color=gray,
+            opacity=0.7]      (-0.1,-0.1) rectangle (-0.9, -0.9);
+      \draw[line width=0.4mm,rounded corners] (-0.1,-0.1) rectangle (-0.9, -0.9);
+      \draw[line width=0.2mm,rounded corners] (-0.2,-0.2) rectangle (-0.8, -0.8);
+      \draw[anchor=mid] (-0.5,-0.6) node {#1};
+      \draw[anchor=center] (-0.5,-0.25) node {\includegraphics[scale=0.015]{crown.png}};
+    }}
+}
+
+
+\begin{document}
+
+\setchessboard{smallboard,
+               zero,
+               showmover=false,
+               boardfontencoding=LSBC4,
+               hlabelformat=\arabic{ranklabel},
+               vlabelformat=\arabic{filelabel}}
+
+\mbox{}\\[-18mm]\mbox{}
+
+\section*{Resit:\\ Implementing the Shogun Board Game\\ (Scala, 8 Marks)}
+
+\noindent
+You are asked to implement a Scala program for playing the Shogun
+board game.  The deadline for your submission is on XXX at
+16:00.  Make sure you use \texttt{scala-cli} and Scala version \textbf{3.XX}
+for the resit---the same version as during the lectures.  \medskip
+
+\IMPORTANTNONE{}
+
+\noindent
+Also note that the running time of each task will be restricted to a
+maximum of 30 seconds on my laptop: If you calculate a result once,
+try to avoid to calculate the result again.
+
+\DISCLAIMER{}
+
+\subsection*{Background}
+
+Shogun
+(\faVolumeUp\,[shōgoon]) is a game played by two players on a chess board and is somewhat
+similar to chess and checkers. A real Shogun board looks
+like in the pictures on the left.
+
+
+\begin{center}
+\begin{tabular}{@{}ccc@{}}
+\raisebox{2mm}{\includegraphics[scale=0.1]{shogun2.jpeg}}
+&
+\raisebox{2mm}{\includegraphics[scale=0.14]{shogun.jpeg}}
+&
+\begin{tikzpicture}[scale=0.45,every node/.style={scale=0.45}]
+% chessboard
+\draw[very thick,gray] (0,0) rectangle (8,8);
+\foreach\x in {0,...,7}\foreach\y in {7,...,0}
+{
+  \pgfmathsetmacro\blend{Mod(\x+\y,2)==0?75:25}
+  \fill[gray!\blend] (\x,\y) rectangle ++ (1,1);
+}
+% black pieces
+\foreach\x/\y/\e in {1/1/1,2/1/3,3/1/2,4/1/3,6/1/3,7/1/1,8/1/2}
+  \pic[fill=white] at (\x,\y) {piece={\e}};
+% white pieces
+\foreach\x/\y/\e in {1/8/4,2/8/2,3/8/4,5/8/4,6/8/2,7/8/3,8/8/1}
+  \pic[fill=red] at (\x,\y)     {piece={\e}};
+\pic[fill=white] at (5.0,1.0) {king={1}};
+\pic[fill=red]   at (4.0,8.0) {king={2}};
+% numbers
+\foreach\x in {1,...,8}
+{\draw (\x - 0.5, -0.4) node {\x};
+}
+\foreach\y in {1,...,8}
+{\draw (-0.4, \y - 0.6, -0.4) node {\y};
+}
+\end{tikzpicture}
+\end{tabular}
+\end{center}
+
+
+\noindent
+In what follows we shall use board illustrations as shown on the right.  As
+can be seen there are two colours in Shogun for the pieces, red and white. Each
+player has 8 pieces, one of which is a king (the piece with the crown)
+and seven are pawns. At the beginning the pieces are lined up as shown
+above.  What sets Shogun apart from chess and checkers is that each
+piece has, what I call, a kind of \textit{energy}---which for pawns is
+a number between 1 and 4, and for kings between 1 and 2. The energy
+determines how far a piece has to move. In the physical version of
+Shogun, the pieces and the board have magnets that can change the
+energy of a piece from move to move---so a piece on one field can have
+energy 2 and on a different field the same piece might have energy
+3. There are some further constraints on legal moves, which are
+explained below.  The point of the resit is to implement functions
+about moving pieces on the Shogun board.\medskip\medskip
+
+%and testing for when a
+%checkmate occurs---i.e.~the king is attacked and cannot move
+%anymore to an ``unattacked'' field (to simplify matters for
+%the resit we leave out the case where the checkmate can be averted by capturing
+%the attacking piece).\medskip
+
+\noindent
+Like in chess, in Shogun the players take turns of moving and
+possibly capturing opposing pieces.
+There are the following rules on how pieces can move:
+
+\begin{itemize}
+\item The energy of a piece determines how far, that is how many
+  fields, a piece has to move (remember pawns have an energy between 1 --
+  4, kings have an energy of only 1 -- 2). The energy of a piece might
+  change when the piece moves to new field.
+\item Pieces can move in straight lines (up, down, left, right), or in
+  L-shape moves, meaning a move can make a single
+  90$^{\circ}$-turn. S-shape moves with more than one turn are not
+  allowed. Also in a single move a piece cannot go forward and then
+  go backward---for example with energy 3 you cannot move 2 fields up and
+  then 1 field down. A piece can never move diagonally.
+\item A piece cannot jump over another piece and cannot stack up on top of your own pieces.
+  But you can capture an opponent's piece if you move to an occupied field. A captured
+  piece is removed from the board.
+\end{itemize}
+
+\noindent
+Like in chess, checkmate is determined when the king of a player cannot
+move anymore to a field that is not attacked, or a player cannot
+capture or block the attacking piece, or the king is the only
+piece left for a player. A non-trivial board that is checkmate is the following:
+
+\begin{center}
+\begin{tikzpicture}[scale=0.5,every node/.style={scale=0.5}]
+% chessboard
+\draw[very thick,gray] (0,0) rectangle (8,8);
+\foreach\x in {0,...,7}\foreach\y in {7,...,0}
+{
+  \pgfmathsetmacro\blend{Mod(\x+\y,2)==0?75:25}
+  \fill[gray!\blend] (\x,\y) rectangle ++ (1,1);
+}
+% redpieces
+\pic[fill=red]   at (4,2) {king={2}};
+\pic[fill=red]   at (6,1) {piece={3}};
+\pic[fill=red]   at (4,4) {piece={4}};
+\pic[fill=red]   at (5,3) {piece={4}};
+% white pieces
+\pic[fill=white] at (7,1) {king={2}};
+\pic[fill=white] at (8,5) {piece={2}};
+\pic[fill=white] at (4,1) {piece={2}};
+% numbers
+\foreach\x in {1,...,8}
+{\draw (\x - 0.5, -0.4) node {\x};
+}
+\foreach\y in {1,...,8}
+{\draw (-0.4, \y - 0.6, -0.4) node {\y};
+}
+\end{tikzpicture}
+\end{center}
+
+\noindent
+The reason for the checkmate is that the white king on field (7, 1) is
+attacked by the red pawn on \mbox{(5, 3)}. There is nowhere for the
+white king to go, and no white pawn can be moved into the way of this
+red pawn and white can also not capture it. When determining a possible
+move, you need to be careful with pieces that might be in the
+way. Consider the following position:
+
+\begin{equation}\label{moves}
+\begin{tikzpicture}[scale=0.5,every node/.style={scale=0.5}]
+% chessboard
+\draw[very thick,gray] (0,0) rectangle (8,8);
+\foreach\x in {0,...,7}\foreach\y in {7,...,0}
+{
+  \pgfmathsetmacro\blend{Mod(\x+\y,2)==0?75:25}
+  \fill[gray!\blend] (\x,\y) rectangle ++ (1,1);
+}
+% redpieces
+\fill[blue!50] (0,2) rectangle ++ (1,1);
+\fill[blue!50] (1,1) rectangle ++ (1,1);
+\fill[blue!50] (0,4) rectangle ++ (1,1);
+\fill[blue!50] (1,5) rectangle ++ (1,1);
+\fill[blue!50] (2,6) rectangle ++ (1,1);
+%%\fill[blue!50] (3,7) rectangle ++ (1,1);
+\fill[blue!50] (4,6) rectangle ++ (1,1);
+\fill[blue!50] (5,5) rectangle ++ (1,1);
+\fill[blue!50] (6,4) rectangle ++ (1,1);
+\fill[blue!50] (6,2) rectangle ++ (1,1);
+\fill[blue!50] (7,3) rectangle ++ (1,1);
+\fill[blue!50] (4,0) rectangle ++ (1,1);
+\fill[blue!50] (2,0) rectangle ++ (1,1);
+\pic[fill=red]   at (4,4) {piece={4}};
+\pic[fill=red]   at (4,8) {piece={4}};
+\pic[fill=white] at (2,5) {piece={3}};
+\pic[fill=white] at (4,3) {piece={2}};
+\pic[fill=white] at (6,3) {piece={1}};
+\pic[fill=white] at (8,4) {piece={1}};
+% numbers
+\foreach\x in {1,...,8}
+{\draw (\x - 0.5, -0.4) node {\x};
+}
+\foreach\y in {1,...,8}
+{\draw (-0.4, \y - 0.6, -0.4) node {\y};
+}
+\end{tikzpicture}
+\end{equation}
+
+\noindent
+The red piece in the centre on field (4, 4) can move to all the blue fields.
+In particular it can move to (2, 6), because it can move 2 fields up
+and 2 fields to the left---it cannot reach this field by moving two
+fields to the left and then two up, because jumping over the white
+piece at (2, 5) is not allowed. Similarly, the field at (6, 2) is
+unreachable for the red piece because of the two white pieces at (4,
+3) and (6, 3) are in the way and no S-shape move is allowed in
+Shogun. The red piece on (4, 4) cannot move to the field (4, 8) at the
+top, because a red piece is already there; but it can move to (8, 4)
+and capture the white piece there. The moral is we always have to
+explore all possible ways in order to determine whether a piece can be
+moved to a field or not: in general there might be several ways and some of
+them might be blocked.
+
+
+\subsection*{Hints}
+
+Useful functions about pieces and boards are defined at the beginning
+of the template file. The function \texttt{.map} applies a function to
+each element of a list or set; \texttt{.flatMap} works like
+\texttt{map} followed by a \texttt{.flatten}---this is useful if a
+function returns a set of sets, which need to be ``unioned up''.  Sets
+can be partitioned according to a predicate with the function
+\texttt{.partition}.  For example
+
+\begin{lstlisting}
+val (even, odd) = Set(1,2,3,4,5).partition(_ % 2 == 0)
+// --> even = Set(2,4)
+//     odd  = Set(1,3,5)
+\end{lstlisting}
+
+\noindent
+The function \texttt{.toList} transforms a set into a list. The function
+\texttt{.count} counts elements according to a predicate. For example
+
+\begin{lstlisting}
+Set(1,2,3,4,5).count(_ % 2 == 0)
+// --> 2
+\end{lstlisting}
+
+%% \newpage
+
+\subsection*{Tasks}
+
+You are asked to implement how pieces can move on a Shogun board.  Let
+us first fix the basic datastructures for the implementation.  A
+\emph{position} (or field) is a pair of integers, like $(3, 2)$. The
+board's dimension is always 8 $\times$ 8.  A \emph{colour} is either
+red (\texttt{Red}) or white (\texttt{Wht}).  A \emph{piece} is either
+a pawn or a king, and has a position, a colour and an energy (an
+integer).  In the template file there are functions \texttt{incx},
+\texttt{decx}, \texttt{incy} and \texttt{decy} for incrementing and
+decrementing the x- and y-coordinates of positions of pieces.
+
+A \emph{board} consists of a set of pieces. We always assume that we
+start with a consistent board and every move only generates another
+consistent board. In this way we do not need to check, for example,
+whether pieces are stacked on top of each other or located outside the
+board, or have an energy outside the permitted range. There are
+functions \texttt{-} and \texttt{+} for removing, respectively adding,
+single pieces to a board.  The function \texttt{occupied} takes a
+position and a board as arguments, and returns an \texttt{Option} of a
+piece when this position is occupied, otherwise \texttt{None}. The
+function \texttt{occupied\_by} returns the colour of a potential piece
+on that position. The function \texttt{is\_occupied} returns a boolean
+for whether a position is occupied or not; \texttt{print\_board} is a
+rough function that prints out a board on the console. This function
+is meant for testing purposes.
+
+
+
+\begin{itemize}
+\item[(1)] You need to calculate all possible moves for a piece on a Shogun board. In order to
+  make sure no piece moves forwards and backwards at the same time,
+  and also exclude all S-shape moves, the data-structure \texttt{Move}
+  is introduced. A \texttt{Move} encodes all simple moves (up, down, left,
+  right) and L-shape moves (first right, then up and so on). This is defined
+  as follows:
+
+{\small\begin{lstlisting}
+abstract class Move
+case object U extends Move    // up
+case object D extends Move    // down
+case object R extends Move    // right
+case object L extends Move    // left
+case object RU extends Move   // ...
+case object LU extends Move
+case object RD extends Move
+case object LD extends Move
+case object UR extends Move
+case object UL extends Move
+case object DR extends Move
+case object DL extends Move
+\end{lstlisting}}
+
+You need to implement an \texttt{eval} function that takes a piece
+\texttt{pc}, a move \texttt{m}, an energy \texttt{en} and a board
+\texttt{b} as arguments. The idea is to recursively calculate all
+fields that can be reached by the move \texttt{m} (there might be more than
+one). The energy acts as a counter and decreases in each recursive
+call until 0 is reached (the final field). The function \texttt{eval} for a piece \texttt{pc}
+should behave as follows:
+
+\begin{itemize}
+\item If the position of a piece is outside the board, then no field can be reached (represented by
+  the empty set \texttt{Set()}).
+\item If the energy is 0 and the position of the piece is \textit{not} occupied, then the field can be reached
+  and the set \texttt{Set(pc)} is returned whereby \texttt{pc} is the piece given as argument.
+\item If the energy is 0 and the position of the piece \textit{is} occupied, but occupied by a piece
+  of the opposite colour, then also the set \texttt{Set(pc)} is returned. Otherwise the empty set
+    \texttt{Set()} is returned.
+\item In case the energy is > 0 and the position of the piece
+  \texttt{pc} is occupied, then this move is blocked and the set
+  \texttt{Set()} is returned.
+\item In all other cases we have to analyse the move
+    \texttt{m}. First, the simple moves (that is \texttt{U}, \texttt{D},
+    \texttt{L} and \texttt{R}) we only have to increment / decrement the
+    x- or y-position of the piece, decrease the energy and call eval
+    recursively with the updated arguments. For example for \texttt{U} (up)
+    you need to increase the y-coordinate:
+
+  \begin{center}
+  \texttt{U} $\quad\Rightarrow\quad$ new arguments: \texttt{incy(pc)}, \texttt{U}, energy - 1, same board
+  \end{center}
+
+  The move \texttt{U} here acts like a ``mode'', meaning if you move
+   up, you can only move up; the mode never changes for simple moves. Similarly for the other simple moves: if
+   you move right, you can only move right and so on. In this way it is
+   prevented to go first to the right, and then change direction in order to go
+   left (same with up and down).
+
+  For the L-shape moves (\texttt{RU}, \texttt{LU}, \texttt{RD} and so on) you need to calculate two
+  sets of reachable fields. Say we analyse \texttt{RU}, then we first have to calculate all fields
+  reachable by moving to the right; then we have to calculate all moves by changing the mode to \texttt{U}.
+  That means there are two recursive calls to \texttt{eval}:
+
+  \begin{center}
+    \begin{tabular}{@{}lll@{}}
+    \texttt{RU} & $\Rightarrow$ & new args for call 1: \texttt{incx(pc)}, \texttt{RU}, energy - 1, same board\\
+                &               & new args for call 2: \texttt{pc}, \texttt{U}, same energy, same board
+    \end{tabular}
+    \end{center}
+
+  In each case we receive some new piece(s) on reachable fields and therefore we return the set
+  containing all these fields. Similarly in the other cases.
+\end{itemize}
+
+For example in the left board below, \texttt{eval} is called with the white
+piece in the centre and the move \texttt{RU} generates then a set of
+new pieces corresponding to the blue fields. The difference with the
+right board is that \texttt{eval} is called with a red piece and therefore the
+field (4, 8) is not reachable anymore because it is already occupied by
+another red piece. But (7, 5) becomes reachable because it is occupied
+by a piece of the opposite colour.
+
+\begin{center}
+\begin{tabular}{cc}
+\begin{tikzpicture}[scale=0.45,every node/.style={scale=0.45}]
+% chessboard
+\draw[very thick,gray] (0,0) rectangle (8,8);
+\foreach\x in {0,...,7}\foreach\y in {7,...,0}
+{
+  \pgfmathsetmacro\blend{Mod(\x+\y,2)==0?75:25}
+  \fill[gray!\blend] (\x,\y) rectangle ++ (1,1);
+}
+\fill[blue!50] (5,5) rectangle ++ (1,1);
+\fill[blue!50] (3,7) rectangle ++ (1,1);
+\fill[blue!50] (4,6) rectangle ++ (1,1);
+%\fill[blue!50] (6,4) rectangle ++ (1,1);
+\fill[blue!50] (7,3) rectangle ++ (1,1);
+
+% black pieces
+\foreach\x/\y/\e in {2/1/3,3/1/2,4/1/3,6/1/3,7/1/1,7/5/2}
+  \pic[fill=white] at (\x,\y) {piece={\e}};
+% white pieces
+\foreach\x/\y/\e in {1/8/4,2/8/2,3/8/4,5/8/4,6/8/2,7/8/3,8/8/1}
+  \pic[fill=red] at (\x,\y)     {piece={\e}};
+\pic[fill=white] at (5.0,1.0) {king={1}};
+\pic[fill=red]   at (4.0,8.0) {king={2}};
+
+\pic[fill=white] at (4,4) {piece={4}};
+% numbers
+\foreach\x in {1,...,8}
+{\draw (\x - 0.5, -0.4) node {\x};
+}
+\foreach\y in {1,...,8}
+{\draw (-0.4, \y - 0.6, -0.4) node {\y};
+}
+\end{tikzpicture}
+&
+\begin{tikzpicture}[scale=0.45,every node/.style={scale=0.45}]
+% chessboard
+\draw[very thick,gray] (0,0) rectangle (8,8);
+\foreach\x in {0,...,7}\foreach\y in {7,...,0}
+{
+  \pgfmathsetmacro\blend{Mod(\x+\y,2)==0?75:25}
+  \fill[gray!\blend] (\x,\y) rectangle ++ (1,1);
+}
+\fill[blue!50] (5,5) rectangle ++ (1,1);
+\fill[blue!50] (4,6) rectangle ++ (1,1);
+\fill[blue!50] (6,4) rectangle ++ (1,1);
+\fill[blue!50] (7,3) rectangle ++ (1,1);
+
+
+% black pieces
+\foreach\x/\y/\e in {1/1/1,2/1/3,3/1/2,4/1/3,6/1/3,7/1/1,7/5/2}
+  \pic[fill=white] at (\x,\y) {piece={\e}};
+% white pieces
+\foreach\x/\y/\e in {1/8/4,2/8/2,3/8/4,5/8/4,6/8/2,7/8/3}
+  \pic[fill=red] at (\x,\y)     {piece={\e}};
+\pic[fill=white] at (5.0,1.0) {king={1}};
+\pic[fill=red]   at (4.0,8.0) {king={2}};
+
+\pic[fill=red] at (4,4) {piece={4}};
+% numbers
+\foreach\x in {1,...,8}
+{\draw (\x - 0.5, -0.4) node {\x};
+}
+\foreach\y in {1,...,8}
+{\draw (-0.4, \y - 0.6, -0.4) node {\y};
+}
+\end{tikzpicture}
+\\[-5mm]
+\end{tabular}
+\end{center}\hfill[3 Marks]
+
+\item[(2)] Implement an \texttt{all\_moves} function that calculates for a
+  piece and a board, \textit{all} possible onward positions. For this
+  you have to call \texttt{eval} for all possible moves \texttt{m} (that is \texttt{U},
+  \texttt{D}, \ldots, \texttt{DL}). An example for all moves for the red piece on (4, 4) is
+  shown in \eqref{moves} on page \pageref{moves}. Be careful about possible modifications
+  you need to apply to the board  before you call the \texttt{eval} function.
+  Also for this task, ignore the fact that a king cannot move onto an attacked field.\\
+  \mbox{}\hfill[1 Mark]
+
+\item[(3)] Implement a function \texttt{attacked} that takes a colour and a board
+  and calculates all pieces of the opposite side that are attacked. For example
+  below in the left board are all the attacked pieces by red, and on the right all for white:
+
+
+\begin{center}
+\begin{tabular}{cc}
+\begin{tikzpicture}[scale=0.45,every node/.style={scale=0.45}]
+% chessboard
+\draw[very thick,gray] (0,0) rectangle (8,8);
+\foreach\x in {0,...,7}\foreach\y in {7,...,0}
+{
+  \pgfmathsetmacro\blend{Mod(\x+\y,2)==0?75:25}
+  \fill[gray!\blend] (\x,\y) rectangle ++ (1,1);
+}
+\fill[blue!50] (7,3) rectangle ++ (1,1);
+\fill[blue!50] (6,0) rectangle ++ (1,1);
+
+
+% red pieces
+\foreach\x/\y/\e in {6/1/3,4/4/4,5/3/4,6/5/3}
+  \pic[fill=red] at (\x,\y) {piece={\e}};
+% white pieces
+\foreach\x/\y/\e in {8/4/1,4/1/2,8/7/3,6/7/2}
+  \pic[fill=white] at (\x,\y)     {piece={\e}};
+
+\pic[fill=red] at (4,2) {king={2}};
+\pic[fill=white] at (7,1) {king={2}};
+
+% numbers
+\foreach\x in {1,...,8}
+{\draw (\x - 0.5, -0.4) node {\x};
+}
+\foreach\y in {1,...,8}
+{\draw (-0.4, \y - 0.6, -0.4) node {\y};
+}
+\end{tikzpicture}
+&
+\begin{tikzpicture}[scale=0.45,every node/.style={scale=0.45}]
+% chessboard
+\draw[very thick,gray] (0,0) rectangle (8,8);
+\foreach\x in {0,...,7}\foreach\y in {7,...,0}
+{
+  \pgfmathsetmacro\blend{Mod(\x+\y,2)==0?75:25}
+  \fill[gray!\blend] (\x,\y) rectangle ++ (1,1);
+}
+\fill[blue!50] (5,0) rectangle ++ (1,1);
+\fill[blue!50] (5,4) rectangle ++ (1,1);
+
+% red pieces
+\foreach\x/\y/\e in {6/1/3,4/4/4,5/3/4,6/5/3}
+  \pic[fill=red] at (\x,\y) {piece={\e}};
+% white pieces
+\foreach\x/\y/\e in {8/4/1,4/1/2,8/7/3,6/7/2}
+  \pic[fill=white] at (\x,\y)     {piece={\e}};
+
+\pic[fill=red] at (4,2) {king={2}};
+\pic[fill=white] at (7,1) {king={2}};
+
+% numbers
+\foreach\x in {1,...,8}
+{\draw (\x - 0.5, -0.4) node {\x};
+}
+\foreach\y in {1,...,8}
+{\draw (-0.4, \y - 0.6, -0.4) node {\y};
+}
+\end{tikzpicture}
+\\[-5mm]
+\end{tabular}
+\end{center}\mbox{}\hfill[1 Mark]
+
+\item[(4)] Implement a function \texttt{attackedN} that takes a piece and a board
+  and calculates the number of times this pieces is attacked by pieces of the opposite colour.
+  For example the piece on field (8, 4) above is attacked by 3 red pieces, and
+  the piece on (6, 1) by 1 white piece. In this number also include kings even
+  if they cannot move to this field because the would be in ``check''.
+  \mbox{}\hfill[1 Mark]
+
+\item[(5)] Implement a function \texttt{protectedN} that takes a piece and a board
+  and calculates the number of times this pieces is protected by pieces of the same colour.
+  For example the piece on field (8, 4) above is protected by 1 white pieces (the one on (8, 7)),
+  and the piece on (5, 3) is protected by three red pieces ((6, 1), (4, 2), and (6, 5)).
+  Similarly to \texttt{attackedN}, include in the calculated number here also the king provided it
+   can reach the given piece.
+  \\
+  \mbox{}\hfill[1 Mark]
+
+  \item[(6)] Implement a function \texttt{legal\_moves} that behaves like \texttt{all\_moves} from (2) for
+    pawns, but for kings, in addition, makes sure that they do not move to an attacked field.
+    For example in the board below on the left, there are three possible fields the white king can
+    reach, but all of them are attacked by red pieces. In the board on the right where the
+    white king has an energy of 1, there is only one legal move, namely to move to field (8, 1).
+    The field (7, 2) is reachable, but is attacked; similarly capturing the red piece on field (6, 1) is
+    not possible because it is protected by at least another red piece.
+    %
+    \begin{center}
+    \begin{tabular}{cc}
+    \begin{tikzpicture}[scale=0.45,every node/.style={scale=0.45}]
+  % chessboard
+  \draw[very thick,gray] (0,0) rectangle (8,8);
+  \foreach\x in {0,...,7}\foreach\y in {7,...,0}
+  {
+    \pgfmathsetmacro\blend{Mod(\x+\y,2)==0?75:25}
+    \fill[gray!\blend] (\x,\y) rectangle ++ (1,1);
+  }
+  \fill[blue!50] (5,1) rectangle ++ (1,1);
+  \fill[blue!50] (6,2) rectangle ++ (1,1);
+  \fill[blue!50] (7,1) rectangle ++ (1,1);
+
+
+  % red pieces
+  \foreach\x/\y/\e in {6/1/3,4/4/4,5/3/4,6/5/3}
+    \pic[fill=red] at (\x,\y) {piece={\e}};
+  % white pieces
+  \foreach\x/\y/\e in {8/4/1,4/1/2,8/7/3,6/7/2}
+    \pic[fill=white] at (\x,\y)     {piece={\e}};
+
+  \pic[fill=red] at (4,2) {king={2}};
+  \pic[fill=white] at (7,1) {king={2}};
+
+  % numbers
+  \foreach\x in {1,...,8}
+  {\draw (\x - 0.5, -0.4) node {\x};
+  }
+  \foreach\y in {1,...,8}
+  {\draw (-0.4, \y - 0.6, -0.4) node {\y};
+  }
+  \end{tikzpicture}  &
+   \begin{tikzpicture}[scale=0.45,every node/.style={scale=0.45}]
+  % chessboard
+  \draw[very thick,gray] (0,0) rectangle (8,8);
+  \foreach\x in {0,...,7}\foreach\y in {7,...,0}
+  {
+    \pgfmathsetmacro\blend{Mod(\x+\y,2)==0?75:25}
+    \fill[gray!\blend] (\x,\y) rectangle ++ (1,1);
+  }
+  \fill[blue!50] (5,0) rectangle ++ (1,1);
+  \fill[blue!50] (6,1) rectangle ++ (1,1);
+  \fill[blue!50] (7,0) rectangle ++ (1,1);
+
+
+  % red pieces
+  \foreach\x/\y/\e in {6/1/3,4/4/4,5/3/3,6/5/3}
+    \pic[fill=red] at (\x,\y) {piece={\e}};
+  % white pieces
+  \foreach\x/\y/\e in {8/4/1,4/1/2,8/7/3,6/7/2}
+    \pic[fill=white] at (\x,\y)     {piece={\e}};
+
+  \pic[fill=red] at (4,2) {king={2}};
+  \pic[fill=white] at (7,1) {king={1}};
+
+  % numbers
+  \foreach\x in {1,...,8}
+  {\draw (\x - 0.5, -0.4) node {\x};
+  }
+  \foreach\y in {1,...,8}
+  {\draw (-0.4, \y - 0.6, -0.4) node {\y};
+  }
+  \end{tikzpicture}
+  \end{tabular}
+  \end{center}
+  \mbox{}\hfill[1 Mark]
+
+\end{itemize}
+
+\end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:




