To solve the problem, we need to determine the number of natural numbers \( n \) such that the characteristic of \( n^2 \) to the base 8 is 2.
### Step-by-Step Solution:
1. **Understanding Characteristic:**
The characteristic of a logarithm is the integer part of the logarithm. For example, if \( \log_b(a) = c \), then the characteristic is the integer part of \( c \).
2. **Setting Up the Inequality:**
Given that the characteristic of \( n^2 \) to the base 8 is 2, we can express this as:
\[
2 \leq \log_8(n^2) < 3
\]
3. **Using the Definition of Logarithm:**
We can convert the logarithmic inequality into an exponential form:
- From \( \log_8(n^2) \geq 2 \):
\[
n^2 \geq 8^2 = 64
\]
- From \( \log_8(n^2) < 3 \):
\[
n^2 < 8^3 = 512
\]
4. **Combining the Inequalities:**
We now have:
\[
64 \leq n^2 < 512
\]
5. **Taking Square Roots:**
Taking the square root of the entire inequality gives:
\[
\sqrt{64} \leq n < \sqrt{512}
\]
This simplifies to:
\[
8 \leq n < 16\sqrt{2}
\]
6. **Calculating \( 16\sqrt{2} \):**
We know that \( \sqrt{2} \approx 1.414 \), thus:
\[
16\sqrt{2} \approx 16 \times 1.414 \approx 22.624
\]
Therefore, we can write:
\[
8 \leq n < 22.624
\]
7. **Finding Integer Values:**
The possible integer values of \( n \) are from 8 to 22. The integers in this range are:
\[
8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22
\]
This gives us a total of:
\[
22 - 8 + 1 = 15
\]
8. **Conclusion:**
Thus, the number of possible values of \( n \) is \( \boxed{15} \).
