On January $20$, $2018$, Sally notices that her $7$ children have ages which sum to a perfect square: their ages are $1$, $3$, $5$, $7$, $9$, $11$, and $13$, with $1+3+5+7+9+11+13=49$. Let $N$ be the age of the youngest child the next year the sum of the $7$ children's ages is a perfect square on January $20$th, and let $P$ be that perfect square. Find $N+P$.

[i]2018 CCA Math Bonanza Lightning Round #2.3[/i]
1. We start by noting that the sum of the ages of Sally's 7 children on January 20, 2018, is given by:
   \[
   1 + 3 + 5 + 7 + 9 + 11 + 13 = 49
   \]
   This sum is a perfect square, specifically \(49 = 7^2\).

2. Let \(x\) be the number of years after 2018. The sum of the children's ages \(x\) years later will be:
   \[
   49 + 7x
   \]
   We need to find the smallest positive integer \(x\) such that \(49 + 7x\) is a perfect square.

3. Let \(49 + 7x = k^2\) for some integer \(k\). Rearranging, we get:
   \[
   7x = k^2 - 49
   \]
   \[
   x = \frac{k^2 - 49}{7}
   \]
   For \(x\) to be an integer, \(k^2 - 49\) must be divisible by 7. Since \(49 = 7^2\), we can rewrite the equation as:
   \[
   k^2 \equiv 49 \pmod{7}
   \]
   \[
   k^2 \equiv 0 \pmod{7}
   \]
   This implies that \(k\) must be a multiple of 7. Let \(k = 7m\) for some integer \(m\).

4. Substituting \(k = 7m\) into the equation \(49 + 7x = k^2\), we get:
   \[
   49 + 7x = (7m)^2
   \]
   \[
   49 + 7x = 49m^2
   \]
   \[
   7x = 49m^2 - 49
   \]
   \[
   x = 7(m^2 - 1)
   \]

5. We need the smallest positive integer \(x\), so we start with the smallest integer \(m > 1\). The smallest \(m\) is 2:
   \[
   x = 7(2^2 - 1) = 7(4 - 1) = 7 \times 3 = 21
   \]

6. Therefore, \(x = 21\). The sum of the children's ages 21 years later is:
   \[
   49 + 7 \times 21 = 49 + 147 = 196
   \]
   This is a perfect square, specifically \(196 = 14^2\).

7. The youngest child, who was 1 year old in 2018, will be:
   \[
   1 + 21 = 22 \text{ years old}
   \]

8. Let \(N\) be the age of the youngest child and \(P\) be the perfect square. We have:
   \[
   N = 22, \quad P = 196
   \]

9. Therefore, \(N + P\) is:
   \[
   22 + 196 = 218
   \]

The final answer is \(\boxed{218}\)