The problem asks us to determine which color(s) can be assigned to the gray square, such that no two neighboring squares share the same color. Neighboring squares are defined as those sharing either a side or a vertex. Here's the analysis:

Step 1: Analyze the colors of neighboring squares

The gray square has the following neighboring squares:

• Top-right corner square: Colored $A$
• Top-left corner square: Colored $C$
• Bottom-left corner square: Colored $B$
• Bottom-right corner square: Colored $D$

Step 2: Rule out existing colors

Since the gray square cannot share a color with any of its neighbors, it cannot be $A, B, C,$ or $D$. Thus, the only possible color options for the gray square are colors that have not yet been used in the grid.

Step 3: Possible colors

Given that $A, B, C,$ and $D$ are the only available colors, and all four are already used for neighboring squares, there is no valid color left for the gray square.

Answer:

E) It is impossible to know.

Let me know if you'd like further clarification! Below are some related questions to deepen understanding.

Related Questions:

1. What would happen if additional colors were introduced to the problem?
2. How can the coloring constraints be verified for larger grids?
3. What is the general algorithm to determine if a grid is colorable under similar rules?
4. If vertex neighbors were not considered, how would the solution change?
5. How do graph-coloring principles apply to this problem?

Tip:

In problems involving coloring and constraints, a systematic approach to eliminate invalid options often simplifies the solution process.