A [i]pucelana[/i] sequence is an increasing sequence of $16$ consecutive odd numbers whose sum is a perfect cube. How many pucelana sequences are there with $3$-digit numbers only?
1. **Define the sequence and sum**:
   Let \( x \) be the first number of the pucelana sequence. The sequence consists of 16 consecutive odd numbers starting from \( x \). Therefore, the sequence is:
   \[
   x, x+2, x+4, \ldots, x+30
   \]
   The sum of these 16 numbers can be written as:
   \[
   S = x + (x+2) + (x+4) + \cdots + (x+30)
   \]

2. **Sum of an arithmetic series**:
   The sum of an arithmetic series with the first term \( a \), common difference \( d \), and \( n \) terms is given by:
   \[
   S = \frac{n}{2} \left( 2a + (n-1)d \right)
   \]
   Here, \( a = x \), \( d = 2 \), and \( n = 16 \). Substituting these values, we get:
   \[
   S = \frac{16}{2} \left( 2x + (16-1) \cdot 2 \right) = 8 \left( 2x + 30 \right) = 16x + 240
   \]

3. **Sum is a perfect cube**:
   We are given that the sum \( S \) is a perfect cube. Therefore, we can write:
   \[
   16x + 240 = k^3
   \]
   for some integer \( k \).

4. **Simplify the equation**:
   Since \( 16x + 240 \) must be divisible by 8, we can write:
   \[
   16x + 240 = 8(2x + 30) = k^3
   \]
   Let \( k = 2p \), then:
   \[
   8(2x + 30) = (2p)^3 = 8p^3
   \]
   Dividing both sides by 8, we get:
   \[
   2x + 30 = p^3
   \]

5. **Solve for \( x \)**:
   \[
   2x = p^3 - 30 \implies x = \frac{p^3 - 30}{2}
   \]
   Since \( x \) must be an integer, \( p^3 - 30 \) must be even. Therefore, \( p^3 \) must be even, implying \( p \) must be even. Let \( p = 2q \), then:
   \[
   2x + 30 = (2q)^3 = 8q^3
   \]
   \[
   2x = 8q^3 - 30 \implies x = 4q^3 - 15
   \]

6. **Find the range for \( q \)**:
   Since \( x \) must be a 3-digit number, we have:
   \[
   100 \leq x \leq 999
   \]
   Substituting \( x = 4q^3 - 15 \), we get:
   \[
   100 \leq 4q^3 - 15 \leq 999
   \]
   Adding 15 to all parts:
   \[
   115 \leq 4q^3 \leq 1014
   \]
   Dividing by 4:
   \[
   28.75 \leq q^3 \leq 253.5
   \]
   Taking the cube root:
   \[
   \sqrt[3]{28.75} \leq q \leq \sqrt[3]{253.5}
   \]
   Approximating the cube roots:
   \[
   3 \leq q \leq 6
   \]

7. **Check for integer values of \( q \)**:
   The integer values of \( q \) in this range are \( q = 3, 4, 5, 6 \). However, \( q \) must be even to ensure \( x \) is an integer. Therefore, \( q \) can be \( 4 \) or \( 6 \).

8. **Count the valid sequences**:
   Thus, there are two valid values for \( q \), which correspond to two pucelana sequences.

The final answer is \(\boxed{2}\)