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 number of completely painted sides of this randomly selected cube.

To find the expected value of the number of completely painted sides of a randomly selected small cube from a larger cube made up of \( 29,791 \) small cubes, we start by determining the dimensions of the larger cube.

Given that the total number of small cubes is \( 29,791 \), we can write:
\[
n^3 = 29791
\]
Taking the cube root of both sides:
\[
n = \sqrt[3]{29791}
\]
Calculating the cube root of 29791:
\[
n = 31
\]
Thus, the larger cube has dimensions \( 31 \times 31 \times 31 \).

### Step 1: Number of Small Cubes by Category

1. **Corner Cubes**: Each corner of the cube has 3 faces painted. There are 8 corners.
   \[
   \text{Number of corner cubes} = 8
   \]

2. **Edge Cubes (excluding corners)**: Each edge of the cube has \( 31 - 2 = 29 \) cubes excluding the corners, with 2 faces painted. There are 12 edges.
   \[
   \text{Number of edge cubes} = 12 \times 29 = 348
   \]

3. **Face Cubes (excluding edges and corners)**: Each face of the cube has \( 31 \times 31 = 961 \) cubes. Excluding the edges and corners, each face has:
   \[
   (31 - 2)^2 = 29^2 = 841 \text{ cubes with 1 face painted}
   \]
   There are 6 faces.
   \[
   \text{Number of face cubes} = 6 \times 841 = 5046
   \]

4. **Internal Cubes**: These cubes are completely inside and have no faces painted. The inner cube is \( 29 \times 29 \times 29 \).
   \[
   \text{Number of internal cubes} = 29^3 = 24389
   \]

### Step 2: Probability of Selecting Each Type of Cube

To find the expected value of the number of painted sides, we need the probabilities:
- Probability of selecting a corner cube:
  \[
  P(\text{corner}) = \frac{8}{29791}
  \]

- Probability of selecting an edge cube:
  \[
  P(\text{edge}) = \frac{348}{29791}
  \]

- Probability of selecting a face cube:
  \[
  P(\text{face}) = \frac{5046}{29791}
  \]

- Probability of selecting an internal cube:
  \[
  P(\text{internal}) = \frac{24389}{29791}
  \]

### Step 3: Expected Value Calculation

Each type of cube contributes to the expected number of painted faces as follows:
- Corner cube: 3 painted faces
- Edge cube: 2 painted faces
- Face cube: 1 painted face
- Internal cube: 0 painted faces

The expected value \( E \) is:
\[
E = (3 \times P(\text{corner})) + (2 \times P(\text{edge})) + (1 \times P(\text{face})) + (0 \times P(\text{internal}))
\]

Substituting the probabilities:
\[
E = 3 \left( \frac{8}{29791} \right) + 2 \left( \frac{348}{29791} \right) + 1 \left( \frac{5046}{29791} \right)
\]
\[
E = \frac{24}{29791} + \frac{696}{29791} + \frac{5046}{29791}
\]
\[
E = \frac{24 + 696 + 5046}{29791}
\]
\[
E = \frac{5766}{29791}
\]

Simplifying the fraction:
\[
E \approx 0.1937
\]

Therefore, the expected value of the number of completely painted sides of a randomly selected small cube is:
\[
\boxed{0.1937}
\]

Would you like to ask for details or have any questions about the problem? Here are some related questions to explore:

1. What is the probability of selecting a cube with exactly one face painted?
2. How many small cubes make up the internal part of the larger cube?
3. What is the total surface area painted on the larger cube?
4. How many cubes are completely surrounded by other cubes?
5. What is the number of edge cubes on each face of the larger cube?
6. How would the problem change if the larger cube had \( 27 \times 27 \times 27 \) dimensions?
7. Can you generalize the expected number of painted sides for an \( n \times n \times n \) cube?
8. How does the expected value change if you paint only some faces of the cube?

**Tip:** When solving probability problems, ensure you account for all categories and correctly compute the probabilities based on the total number of elements.

Explore the expected value of the number of completely painted sides of a randomly selected small cube from a larger cube made up of 29,791 small cubes. Learn how to calculate probabilities for corner, edge, face, and internal cubes to find the expected number of painted sides. Suitable for high school students studying probability and expected value.

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