Solution 9f4206b2-f3fd-4352-bfc2-7b6d67eb54e5

Points: 4 pts

Lemma 1: We can reflect $ABCD$ multiple times to make the ray $PQ$ straight.

Whenever the path meets a side, reflect the square across that side
instead of reflecting the path. The equal-angle rule makes the unfolded
path a straight ray through the reflected squares.

\begin{tikzpicture}[x=0.62cm,y=0.62cm,line cap=round,line join=round]
  % The original square, its reflection across BC, and further copies.
  \ReflectedGrid{3}{2}
  % Keep the original square unshaded with four matching black edges.
  \draw[fill=\SolutionBackground,draw=black,line width=0.85pt]
    (0,0) rectangle (4,4);

  \draw[red!85!black,line width=1.05pt,
        -{Stealth[length=2.4mm,width=1.6mm]}]
    (0,{8/5})--(8.65,{18/25*8.65+8/5});

  \ReflectedVertices{3}{2}
  \MarkRayStart
\end{tikzpicture}

Lemma 2: The ray satisfies $18x-25y=-40$.

Using rise over run,
\[
\text{slope}
=\frac{\text{rise}}{\text{run}}
=\frac{4-\frac85}{\frac{10}{3}}
=\frac{\frac{12}{5}}{\frac{10}{3}}
=\frac{18}{25}.
\]
Since the ray starts at $y=\frac85$, its equation is
\[
y=\frac{18}{25}x+\frac85
\quad\Longrightarrow\quad
18x-25y=-40.
\]
At a reflected vertex, $x$ and $y$ are multiples of $4$, so we seek
integer solutions of this Diophantine equation.

Lemma 3: The first vertex reached is $(20,16)$.

Using modulo $2$ and $5$, we find:
\[
\begin{aligned}
3x&\equiv0\pmod{5} &&\Rightarrow 5\mid x\\
y&\equiv0\pmod{2} &&\Rightarrow 2\mid y
\end{aligned}
\]
Substituting $(x,y)=(5i,2j)$, where $i$ and $j$ are integers, gives
\[
90i-50j=-40
\quad\Longrightarrow\quad
9i-5j=-4.
\]
We can rearrange this as
\[
9(i+1)=5(j+1).
\]
Since the ray travels to the right, we have $i\geq0$. As $5$ and $9$
are coprime, the smallest such solution satisfies
\[
i+1=5,\qquad j+1=9.
\]
Therefore,
\[
(i,j)=(4,8)
\quad\Longrightarrow\quad
(x,y)=(20,16).
\]
Both coordinates are multiples of $4$, so $(20,16)$ is the first
vertex of the reflected grid that the ray hits.

Lemma 4: The point $(20,16)$ is labeled $B$.

\begin{tikzpicture}[x=0.44cm,y=0.44cm,line cap=round,line join=round]
  \ReflectedGrid{5}{4}

  % The unfolded ray terminates at its first grid vertex.
  \draw[red!85!black,line width=1.15pt] (0,{8/5})--(20,16);
  \draw[red!85!black,line width=1.15pt,
        -{Stealth[length=2.3mm,width=1.6mm]}]
    (10.2,{18/25*10.2+8/5})--(11.6,{18/25*11.6+8/5});

  \ReflectedVertices{5}{4}
  \MarkRayStart

  \draw[red!85!black,line width=0.85pt]
    (20,16) circle[radius=2.7pt];
  \node[above left,xshift=-2pt,yshift=3pt,inner sep=3pt,font=\small]
    at (20,16) {$(20,16)$};
\end{tikzpicture}

The point is five squares to the right and four squares up from $A$,
so its label is $B$.

Final answer:
B

Notation definitions (retained for meaning):
\newcommand{\ReflectedGrid}[2]{  \pgfmathtruncatemacro{\gridwidth}{4*#1}  \pgfmathtruncatemacro{\gridheight}{4*#2}  \fill[problemblue!12!white] (0,0) rectangle (4,4);
  \draw[bodytext!32,line width=0.4pt,step=4]
    (0,0) grid (\gridwidth,\gridheight);
  \draw[bodytext,line width=0.85pt] (0,0) rectangle (4,4);

  \draw[bodytext,line width=0.6pt,
        -{Stealth[length=1.9mm,width=1.3mm]}]
    (0,0)--({\gridwidth+0.95},0)
    node[right,inner sep=2pt] {$x$};
  \draw[bodytext,line width=0.6pt,
        -{Stealth[length=1.9mm,width=1.3mm]}]
    (0,0)--(0,{\gridheight+0.95})
    node[above,inner sep=2pt] {$y$};

  \foreach \i in {0,...,#1}{    \pgfmathtruncatemacro{\tickvalue}{4*\i}    \draw[bodytext,line width=0.45pt]
      (\tickvalue,0)--(\tickvalue,-0.13);
    \node[below,inner sep=2.5pt,font=\footnotesize]
      at (\tickvalue,-0.13) {$\tickvalue$};
  }  \foreach \j in {1,...,#2}{    \pgfmathtruncatemacro{\tickvalue}{4*\j}    \draw[bodytext,line width=0.45pt]
      (0,\tickvalue)--(-0.13,\tickvalue);
    \node[left,inner sep=2.5pt,font=\footnotesize]
      at (-0.13,\tickvalue) {$\tickvalue$};
  }}
\newcommand{\ReflectedVertices}[2]{  \foreach \i in {0,...,#1}{    \foreach \j in {0,...,#2}{      \pgfmathtruncatemacro{\vx}{4*\i}      \pgfmathtruncatemacro{\vy}{4*\j}      \pgfmathtruncatemacro{\colparity}{mod(\i,2)}      \pgfmathtruncatemacro{\rowparity}{mod(\j,2)}      \ifnum\rowparity=0
        \ifnum\colparity=0\def\vertexlabel{A}\else\def\vertexlabel{B}\fi
      \else
        \ifnum\colparity=0\def\vertexlabel{D}\else\def\vertexlabel{C}\fi
      \fi
      \fill[bodytext] (\vx,\vy) circle[radius=1.05pt];
            \ifnum\i=1
        \ifnum\j=1
          \node[below right,inner sep=1.5pt,font=\small]
            at (\vx,\vy) {$\vertexlabel$};
        \else
          \node[above right,inner sep=1.5pt,font=\small]
            at (\vx,\vy) {$\vertexlabel$};
        \fi
      \else
        \node[above right,inner sep=1.5pt,font=\small]
          at (\vx,\vy) {$\vertexlabel$};
      \fi
    }  }}
\newcommand{\MarkRayStart}{  \fill[bodytext] (0,{8/5}) circle[radius=1.3pt];
  \fill[bodytext] ({10/3},4) circle[radius=1.3pt];
  \node[left,inner sep=4pt] at (0,{8/5}) {$P$};
  \node[below,yshift=-4pt,inner sep=1pt]
    at ({10/3},4) {$Q$};
}