To solve this problem, we need to analyze the scenario carefully:

Problem Setup:

1. Vertices of the Cube: The coordinates of the cube's vertices are given:

   • $A(0, 0, 0), B(3, 0, 0), C(3, 3, 0), D(0, 3, 0)$ (bottom face),
   • $E(0, 0, 3), F(3, 0, 3), G(3, 3, 3), H(0, 3, 3)$ (top face).

2. Midpoints of Edges: There are 12 edges in the cube, each with a midpoint:

   • Bottom face edges: $M_1(1.5, 0, 0), M_2(3, 1.5, 0), M_3(1.5, 3, 0), M_4(0, 1.5, 0)$,
   • Top face edges: $M_5(1.5, 0, 3), M_6(3, 1.5, 3), M_7(1.5, 3, 3), M_8(0, 1.5, 3)$,
   • Vertical edges: $M_9(0, 0, 1.5), M_{10}(3, 0, 1.5), M_{11}(3, 3, 1.5), M_{12}(0, 3, 1.5)$.

3. Shortest Path: The particle moves from $A(0, 0, 0)$ to $G(3, 3, 3)$ using the shortest path, either through vertices or midpoints. Excluding $A$ and $G$, the coordinates visited contribute to $N$, the sum of the digits of the $x$-, $y$-, and $z$-coordinates.

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Step-by-Step Solution:

1. Path Selection: The shortest path from $A(0, 0, 0)$ to $G(3, 3, 3)$ involves increasing each coordinate $x, y, z$ from $0$ to $3$. Along the way, the particle can use either vertices or midpoints.

   A valid shortest path includes exactly three steps:

   • $x$ increases from $0 \to 3$,
   • $y$ increases from $0 \to 3$,
   • $z$ increases from $0 \to 3$.

2. Possible Intermediate Points: The intermediate points may include:

   • Vertices such as $B(3, 0, 0), C(3, 3, 0), F(3, 0, 3), H(0, 3, 3)$,
   • Midpoints such as $M_1(1.5, 0, 0), M_2(3, 1.5, 0), M_{10}(3, 0, 1.5)$.

3. Representative Path: Let’s choose a specific shortest path that alternates between midpoints and vertices: $A(0, 0, 0) \to M_1(1.5, 0, 0) \to B(3, 0, 0) \to M_{10}(3, 0, 1.5) \to F(3, 0, 3) \to M_6(3, 1.5, 3) \to C(3, 3, 3) \to G(3, 3, 3).$

4. Excluding $A$ and $G$: The coordinates visited are:

   • $M_1(1.5, 0, 0), B(3, 0, 0), M_{10}(3, 0, 1.5), F(3, 0, 3), M_6(3, 1.5, 3), C(3, 3, 3)$.

5. Sum of Digits Calculation: For each point, calculate the sum of the digits in the $x, y, z$-coordinates:

   • $M_1(1.5, 0, 0): 1 + 5 + 0 + 0 = 6$,
   • $B(3, 0, 0): 3 + 0 + 0 = 3$,
   • $M_{10}(3, 0, 1.5): 3 + 0 + 1 + 5 = 9$,
   • $F(3, 0, 3): 3 + 0 + 3 = 6$,
   • $M_6(3, 1.5, 3): 3 + 1 + 5 + 3 = 12$,
   • $C(3, 3, 3): 3 + 3 + 3 = 9$.

   Excluding $A(0, 0, 0)$ and $G(3, 3, 3)$, the total sum is: $6 + 3 + 9 + 6 + 12 + 9 = 45.$

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Final Answer:

The sum of the digits $N$ is: $\boxed{45}.$

Would you like further clarifications or have any additional questions? 😊

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Related Questions:

1. What are the coordinates of all midpoints of a cube with arbitrary side length?
2. How many shortest paths exist between two opposite vertices of a cube using only vertices?
3. How can the problem change if diagonal paths through the interior of the cube are allowed?
4. What is the total distance traveled by the particle in the chosen path?
5. How would the answer change if we included starting and ending vertices in the sum?

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Tip:

For problems involving geometry in three dimensions, visualizing the scenario with a diagram or sketch often simplifies understanding and solution planning.