This problem involves calculating the straight-line distance between points $X$ and $Y$ in a 3D arrangement of cubes. Here’s how we approach it step-by-step:

1. Visualize the Coordinates:
   Each cube has edges of length $3 \, \text{cm}$. The cubes are arranged in a $2 \times 2$ configuration, and points $X$ and $Y$ are positioned at opposite corners of the structure.

   Assign coordinates for $X$ and $Y$:

   • $X$ is at one corner of the bottom layer: $(0, 0, 0)$.
   • $Y$ is at the opposite corner of the top layer: $(3, 3, 3)$.

2. Use the Distance Formula in 3D:
   The distance formula in 3D space is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

   Substituting the coordinates of $X$ and $Y$: $d = \sqrt{(3 - 0)^2 + (3 - 0)^2 + (3 - 0)^2}$

3. Simplify the Calculation: $d = \sqrt{3^2 + 3^2 + 3^2} = \sqrt{9 + 9 + 9} = \sqrt{27} = 3\sqrt{3} \, \text{cm}.$

Final Answer:

The straight-line distance between $X$ and $Y$ is: $\boxed{3\sqrt{3} \, \text{cm}}$

Would you like a detailed diagram explanation or assistance with related concepts?

Here are 5 related questions to explore further:

1. How would the distance formula change if the cube edge length was different?
2. What is the volume of the entire structure made by these four cubes?
3. How do you calculate the surface area of this specific arrangement of cubes?
4. What is the shortest path along the edges from $X$ to $Y$?
5. How does the diagonal in a single cube relate to the diagonal across multiple cubes?

Tip: When working with 3D distances, sketching the coordinate axes and labeling points clearly can help avoid errors.