black and white figure

book.tex

@@ -3344,25 +3344,26 @@ If you can color the remaining vertices in the graph with the nine
 colors, then you have also solved the corresponding game of Sudoku.
 Figure~\ref{fig:sudoku-graph} shows an initial Sudoku game board and
 the corresponding graph with colored vertices.  We map the Sudoku
-number 1 to blue, 2 to yellow, and 3 to red.  We only show edges for a
+number 1 to black, 2 to white, and 3 to gray.  We only show edges for a
 sampling of the vertices (the colored ones) because showing edges for
 all of the vertices would make the graph unreadable.
 
 \begin{figure}[tbp]
 \includegraphics[width=0.45\textwidth]{figs/sudoku}
-\includegraphics[width=0.5\textwidth]{figs/sudoku-graph}
+\includegraphics[width=0.5\textwidth]{figs/sudoku-graph-bw}
 \caption{A Sudoku game board and the corresponding colored graph.}
 \label{fig:sudoku-graph}
 \end{figure}
 
-It turns out that some techniques for playing Sudoku correspond to
-heuristics used in graph coloring algorithms.  For example, one of the
-basic techniques for Sudoku is called Pencil Marks. The idea is to use
-a process of elimination to determine what numbers are no longer
-available for a square and write down those numbers in the square
-(writing very small). For example, if the number $1$ is assigned to a
-square, then write the pencil mark $1$ in all the squares in the same
-row, column, and region.
+Some techniques for playing Sudoku correspond to heuristics used in
+graph coloring algorithms.  For example, one of the basic techniques
+for Sudoku is called Pencil Marks. The idea is to use a process of
+elimination to determine what numbers are no longer available for a
+square and write down those numbers in the square (writing very
+small). For example, if the number $1$ is assigned to a square, then
+write the pencil mark $1$ in all the squares in the same row, column,
+and region to indicate that $1$ is no longer an option for those other
+squares.
 %
 The Pencil Marks technique corresponds to the notion of
 \emph{saturation}\index{subject}{saturation} due to \cite{Brelaz:1979eu}.  The
@@ -3380,10 +3381,10 @@ Using the Pencil Marks technique leads to a simple strategy for
 filling in numbers: if there is a square with only one possible number
 left, then choose that number! But what if there are no squares with
 only one possibility left? One brute-force approach is to try them
-all: choose the first one and if it ultimately leads to a solution,
+all: choose the first one and if that ultimately leads to a solution,
 great.  If not, backtrack and choose the next possibility.  One good
 thing about Pencil Marks is that it reduces the degree of branching in
-the search tree. Nevertheless, backtracking can be horribly time
+the search tree. Nevertheless, backtracking can be terribly time
 consuming. One way to reduce the amount of backtracking is to use the
 most-constrained-first heuristic. That is, when choosing a square,
 always choose one with the fewest possibilities left (the vertex with