Amanda has the list of even numbers $2, 4, 6, \dots 100$ and Billy has the list of odd numbers $1, 3, 5, \dots 99$. Carlos creates a list by adding the square of each number in Amanda's list to the square of the corresponding number in Billy's list. Daisy creates a list by taking twice the product of corresponding numbers in Amanda's list and Billy's list. What is the positive difference between the sum of the numbers in Carlos's list and the sum of the numbers in Daisy's list?

[i]2016 CCA Math Bonanza Individual #3[/i]
1. **Identify the sequences and their properties:**
   - Amanda's list of even numbers: \(2, 4, 6, \ldots, 100\)
   - Billy's list of odd numbers: \(1, 3, 5, \ldots, 99\)

2. **Define the sequences mathematically:**
   - Amanda's list: \(a_n = 2n\) for \(n = 1, 2, \ldots, 50\)
   - Billy's list: \(b_n = 2n - 1\) for \(n = 1, 2, \ldots, 50\)

3. **Construct Carlos's list:**
   - Carlos's list is formed by adding the squares of corresponding numbers from Amanda's and Billy's lists:
     \[
     c_n = (2n - 1)^2 + (2n)^2
     \]
   - Simplify \(c_n\):
     \[
     c_n = (2n - 1)^2 + (2n)^2 = (4n^2 - 4n + 1) + 4n^2 = 8n^2 - 4n + 1
     \]

4. **Construct Daisy's list:**
   - Daisy's list is formed by taking twice the product of corresponding numbers from Amanda's and Billy's lists:
     \[
     d_n = 2 \cdot (2n - 1) \cdot (2n)
     \]
   - Simplify \(d_n\):
     \[
     d_n = 2 \cdot (2n - 1) \cdot (2n) = 4n(2n - 1) = 8n^2 - 4n
     \]

5. **Find the difference between corresponding terms in Carlos's and Daisy's lists:**
   - The difference for each term is:
     \[
     c_n - d_n = (8n^2 - 4n + 1) - (8n^2 - 4n) = 1
     \]

6. **Sum the differences:**
   - Since there are 50 terms in each list, the total difference is:
     \[
     \sum_{n=1}^{50} (c_n - d_n) = \sum_{n=1}^{50} 1 = 50
     \]

Conclusion:
\[
\boxed{50}
\]