Ben works quickly on his homework, but tires quickly. The first problem takes him $1$ minute to solve, and the second problem takes him $2$ minutes to solve. It takes him $N$ minutes to solve problem $N$ on his homework. If he works for an hour on his homework, compute the maximum number of problems he can solve.
1. We need to determine the maximum number of problems \( N \) that Ben can solve in 60 minutes, given that the time taken to solve the \( i \)-th problem is \( i \) minutes.
2. The total time taken to solve the first \( N \) problems is given by the sum of the first \( N \) natural numbers:
   \[
   \sum_{i=1}^N i = 1 + 2 + 3 + \cdots + N
   \]
3. The sum of the first \( N \) natural numbers can be calculated using the formula:
   \[
   \sum_{i=1}^N i = \frac{N(N+1)}{2}
   \]
4. We need to find the maximum \( N \) such that:
   \[
   \frac{N(N+1)}{2} \leq 60
   \]
5. To solve this inequality, we first multiply both sides by 2 to clear the fraction:
   \[
   N(N+1) \leq 120
   \]
6. Next, we solve the quadratic inequality \( N^2 + N - 120 \leq 0 \). This can be done by finding the roots of the corresponding quadratic equation:
   \[
   N^2 + N - 120 = 0
   \]
7. Using the quadratic formula \( N = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 1 \), and \( c = -120 \):
   \[
   N = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-120)}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 + 480}}{2} = \frac{-1 \pm \sqrt{481}}{2}
   \]
8. Approximating the square root, we get:
   \[
   \sqrt{481} \approx 21.93
   \]
9. Thus, the roots are:
   \[
   N = \frac{-1 + 21.93}{2} \approx 10.465 \quad \text{and} \quad N = \frac{-1 - 21.93}{2} \approx -11.465
   \]
10. Since \( N \) must be a positive integer, we take the integer part of the positive root:
    \[
    N \approx 10
    \]
11. To verify, we check if \( N = 10 \) satisfies the original inequality:
    \[
    \frac{10 \cdot 11}{2} = 55 \leq 60
    \]
12. If we try \( N = 11 \):
    \[
    \frac{11 \cdot 12}{2} = 66 > 60
    \]
13. Therefore, the maximum number of problems Ben can solve in 60 minutes is \( N = 10 \).

The final answer is \( \boxed{10} \)