Interesting but difficult board game question? I have a 3 by 3 board game. A black marble is randomly place in one of the nine squares. Distance between squares is measured as one if either diagonal or horizontal/vertically next each other, and two otherwise (so max distance of two). With an initial black marble, all squares within distance one of this black marble contains a blue one. The rest of the squares are then filled with a red marble. All 9 squares are covered. How many minimum cups (and which ones) do I need to look at in order to figure out where black marble is? 
\begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i \end{array}
So my thinking is that the center one will be blue with probability $8/9$ and black with probability $1/9$. If the black marble is one the corners (probability $4/9$), then $e$ must be blue and two of $b, d, f, h$ must be blue. This seems to be a lot of case work, but I get that the minimum is 4? Has to be symmetric?
 A: You can do it in $3$ guesses.
Look at $a$. If it’s black, you’re done. If it’s blue, the black marble is $b,d$, or $e$, and you can locate it with two more guesses. (For instance, look at $b$; if it’s black, you’re done, and if not you can look at $d$, after which you’ll know whether the black marble is $d$ or $e$.) If it’s red, the black marble is $c,f,g,h$, or $i$. Look at $i$; if it’s black, you’re done. If it’s blue, the black marble is $f$ or $h$, and one more guess will locate it. If it’s red, the black marble is $c$ or $g$, and one more guess will locate it.
Starting with $b$ works equally well and sometimes better. If $b$ is blue, the black marble is $a,c,d,e$, or $f$, and looking at $a$ will either show you the black marble, tell you that it’s $d$ or $e$, or tell you that it’s $c$ or $f$. If $b$ is red, the black marble is $g,h$, or $i$, and looking at $g$ will tell you where it is.
A: This game allows to look at $3$ fixed squares (without game tree, etc) to know where is black marble.
For example, squares $a,f,h$.
If one of them is black, then black marble is find already, and game stops.
I'll show cases, when they all are not black:
$$
\begin{array}{c}
\color{red} {\blacksquare}\;\color{gray}{\blacksquare}\;\color{gray}{\blacksquare}\\
\color{gray}{\blacksquare}\;\color{gray}{\blacksquare}\;\color{red} {\blacksquare}\\
\color{gray}{\blacksquare}\;\color{red} {\blacksquare}\;\color{gray}{\blacksquare}
\end{array}
\implies
\mbox {impossible};
$$

$$
\begin{array}{c}
\color{red} {\blacksquare}\;\color{gray}{\blacksquare}\;\color{gray}{\blacksquare}\\
\color{gray}{\blacksquare}\;\color{gray}{\blacksquare}\;\color{red} {\blacksquare}\\
\color{gray}{\blacksquare}\;\color{blue}{\blacksquare}\;\color{gray}{\blacksquare}
\end{array}
\implies
\begin{array}{c}
\color{red} {\blacksquare}\;\color{red}{\blacksquare}\;\color{red}{\blacksquare}\\
\color{blue}{\blacksquare}\;\color{blue}{\blacksquare}\;\color{red} {\blacksquare}\\
\color{black}{\blacksquare}\;\color{blue} {\blacksquare}\;\color{red}{\blacksquare}
\end{array};
$$

$$
\begin{array}{c}
\color{red} {\blacksquare}\;\color{gray}{\blacksquare}\;\color{gray}{\blacksquare}\\
\color{gray}{\blacksquare}\;\color{gray}{\blacksquare}\;\color{blue} {\blacksquare}\\
\color{gray}{\blacksquare}\;\color{red}{\blacksquare}\;\color{gray}{\blacksquare}
\end{array}
\implies
\begin{array}{c}
\color{red} {\blacksquare}\;\color{blue}{\blacksquare}\;\color{black}{\blacksquare}\\
\color{red}{\blacksquare}\;\color{blue}{\blacksquare}\;\color{blue} {\blacksquare}\\
\color{red}{\blacksquare}\;\color{red} {\blacksquare}\;\color{red}{\blacksquare}
\end{array};
$$

$$
\begin{array}{c}
\color{red} {\blacksquare}\;\color{gray}{\blacksquare}\;\color{gray}{\blacksquare}\\
\color{gray}{\blacksquare}\;\color{gray}{\blacksquare}\;\color{blue} {\blacksquare}\\
\color{gray}{\blacksquare}\;\color{blue}{\blacksquare}\;\color{gray}{\blacksquare}
\end{array}
\implies
\begin{array}{c}
\color{red} {\blacksquare}\;\color{red}{\blacksquare}\;\color{red}{\blacksquare}\\
\color{red}{\blacksquare}\;\color{blue}{\blacksquare}\;\color{blue} {\blacksquare}\\
\color{red}{\blacksquare}\;\color{blue} {\blacksquare}\;\color{black}{\blacksquare}
\end{array};
$$

$$
\begin{array}{c}
\color{blue} {\blacksquare}\;\color{gray}{\blacksquare}\;\color{gray}{\blacksquare}\\
\color{gray}{\blacksquare}\;\color{gray}{\blacksquare}\;\color{red} {\blacksquare}\\
\color{gray}{\blacksquare}\;\color{red}{\blacksquare}\;\color{gray}{\blacksquare}
\end{array}
\implies
\mbox{impossible};
$$

$$
\begin{array}{c}
\color{blue} {\blacksquare}\;\color{gray}{\blacksquare}\;\color{gray}{\blacksquare}\\
\color{gray}{\blacksquare}\;\color{gray}{\blacksquare}\;\color{red} {\blacksquare}\\
\color{gray}{\blacksquare}\;\color{blue}{\blacksquare}\;\color{gray}{\blacksquare}
\end{array}
\implies
\begin{array}{c}
\color{blue} {\blacksquare}\;\color{blue}{\blacksquare}\;\color{red}{\blacksquare}\\
\color{black}{\blacksquare}\;\color{blue}{\blacksquare}\;\color{red} {\blacksquare}\\
\color{blue} {\blacksquare}\;\color{blue} {\blacksquare}\;\color{red}{\blacksquare}
\end{array};
$$

$$
\begin{array}{c}
\color{blue}{\blacksquare}\;\color{gray}{\blacksquare}\;\color{gray}{\blacksquare}\\
\color{gray}{\blacksquare}\;\color{gray}{\blacksquare}\;\color{blue} {\blacksquare}\\
\color{gray}{\blacksquare}\;\color{red} {\blacksquare}\;\color{gray}{\blacksquare}
\end{array}
\implies
\begin{array}{c}
\color{blue} {\blacksquare}\;\color{black}{\blacksquare}\;\color{blue}{\blacksquare}\\
\color{blue}{\blacksquare}\;\color{blue}{\blacksquare}\;\color{blue} {\blacksquare}\\
\color{red} {\blacksquare}\;\color{red} {\blacksquare}\;\color{red}{\blacksquare}
\end{array};
$$

$$
\begin{array}{c}
\color{blue}{\blacksquare}\;\color{gray}{\blacksquare}\;\color{gray}{\blacksquare}\\
\color{gray}{\blacksquare}\;\color{gray}{\blacksquare}\;\color{blue} {\blacksquare}\\
\color{gray}{\blacksquare}\;\color{blue}{\blacksquare}\;\color{gray}{\blacksquare}
\end{array}
\implies
\begin{array}{c}
\color{blue}{\blacksquare}\;\color{blue}{\blacksquare}\;\color{blue}{\blacksquare}\\
\color{blue}{\blacksquare}\;\color{black}{\blacksquare}\;\color{blue} {\blacksquare}\\
\color{blue}{\blacksquare}\;\color{blue} {\blacksquare}\;\color{blue}{\blacksquare}
\end{array}.
$$
So, strategy can be so easy.
