A cube made of 29,791 small cubes get s all of it's sides painted, Let S be the set of all cubes enclosed in the 29.791 small cubes structure that are made up of at least one small cube. A random element in S will be drawn. Find the expected value of number of completely painted sides of this randomly selected cube.

To solve this problem, we first need to understand the structure of the cube and how many small cubes get their sides painted. Let’s break this down step by step:

### Step 1: Determine the size of the cube
We know that the cube is made up of 29,791 small cubes. If the cube has \( n^3 \) small cubes, we can find \( n \) by solving the equation:
\[
n^3 = 29,791
\]
Taking the cube root of both sides:
\[
n = \sqrt[3]{29,791} = 31
\]
So, the cube has dimensions \( 31 \times 31 \times 31 \), meaning it consists of 31 small cubes along each edge.

### Step 2: Categories of cubes based on their position
Now, we classify the small cubes into different categories based on how many sides of each cube are completely painted:

- **Corner cubes**: A corner cube has 3 painted sides. There are 8 corners in a cube.
  
- **Edge cubes (excluding corners)**: An edge cube that is not at the corner has 2 painted sides. Each edge of the large cube contains 31 cubes, but the corner cubes are excluded, leaving \( 31 - 2 = 29 \) cubes per edge. Since there are 12 edges, the total number of edge cubes (excluding corners) is:
  \[
  12 \times 29 = 348
  \]

- **Face cubes (excluding edges)**: A cube that lies on a face but is not on the edge has 1 painted side. Each face of the large cube contains \( 31 \times 31 = 961 \) cubes, but we must exclude the edges. The edge of a face consists of 31 cubes per side, so the number of edge cubes per face is \( 4 \times 31 - 4 = 120 \) (4 edges, subtracting corner duplicates). Thus, the number of face cubes per face (excluding edges) is:
  \[
  961 - 120 = 841
  \]
  Since there are 6 faces, the total number of face cubes (excluding edges) is:
  \[
  6 \times 841 = 5,046
  \]

- **Interior cubes**: These cubes are completely inside the large cube and have no painted sides. The number of interior cubes forms a smaller cube of side length \( 29 \) (since the outer layer of cubes has been removed), so the number of interior cubes is:
  \[
  29 \times 29 \times 29 = 24,389
  \]

### Step 3: Calculate the expected value
The expected value of the number of painted sides for a randomly selected cube is a weighted average. The weights are the number of cubes in each category, and the values are the number of painted sides for cubes in that category. We calculate the expected value as follows:

\[
E(X) = \frac{\sum (\text{number of painted sides} \times \text{number of cubes})}{\text{total number of cubes}}
\]

Substitute the values:

\[
E(X) = \frac{(3 \times 8) + (2 \times 348) + (1 \times 5,046) + (0 \times 24,389)}{29,791}
\]

Simplify:

\[
E(X) = \frac{24 + 696 + 5,046 + 0}{29,791} = \frac{5,766}{29,791}
\]

Perform the division:

\[
E(X) \approx 0.1935
\]

### Final Answer:
The expected value of the number of completely painted sides of a randomly selected cube is approximately \( 0.1935 \).

---

Do you need further clarification on any part of the solution? 

Here are five related questions to expand the topic:
1. How would the expected value change if the cube had dimensions \( 21 \times 21 \times 21 \)?
2. What would the result be if we instead wanted the expected value of non-painted sides?
3. How many cubes have exactly one painted side in a \( 51 \times 51 \times 51 \) cube?
4. How does the total surface area of the cube change with increasing cube dimensions?
5. Can this method be applied to other 3D shapes, like rectangular prisms?

**Tip:** Understanding the symmetry of a cube and categorizing positions based on corners, edges, and faces can simplify many problems involving geometry and surface area.

A cube made of 29,791 small cubes gets all of its sides painted. Let S be the set of all cubes enclosed in the 29,791 small cubes structure that are made up of at least one small cube. A random element in S will be drawn. Find the expected value of the number of completely painted sides of this randomly selected cube.

Find the expected value of completely painted sides for a random cube from a 31x31x31 cube structure made of 29,791 small cubes. This problem involves geometric classification and probability concepts. Suitable for high school students, Grades 10-12.

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