Fit the most tasks into an hour across machine slots

Your team runs a batch cluster of n machines with IDs 0 to n-1. Each machine has m execution slots, so it can run up to m tasks at the same time. A slot runs one task at a time; when a task ends, the slot can start the next one immediately.

You get a list durations of task run times in whole minutes. A task must run on a single slot from start to finish: it cannot be split, paused, or moved. All slots are free at minute 0.

Implement:

def max_tasks(n: int, m: int, durations: list[int], limit: int = 60) -> int:
    ...

Return the largest number of tasks that can finish by minute limit (60 by default). You choose which tasks to run and where.

Constraints

• 0 <= n, m, and n * m <= 30
• 0 <= len(durations) <= 40
• 1 <= durations[i] <= 10^4 (a task longer than limit can never finish)

Example 1

n = 1, m = 1, durations = [10, 20, 30, 40]
-> 3

One slot runs 10, 20 and 30 back to back, ending exactly at minute 60.

Example 2

n = 2, m = 1, durations = [3, 3, 4, 5, 5], limit = 10
-> 5

Machine 0 runs 5 then 5. Machine 1 runs 3, 3 and 4. Placing each shortest task on the least-busy machine finishes only 4, so the order of placement matters.

Be ready to explain whether your method is always optimal.

Reconstructed from the reported question; details may differ from what interviewers ask.