sweep line

The fewest meeting rooms you need equals the highest number of meetings active simultaneously. That reframing is the entire interview; two O(n log n) solutions drop out of it, and the tie-handling catches careless candidates.

Merging intervals is the entry point to the whole interval family. The trick is sorting by start so overlaps end up next to each other, then collapsing them in one linear sweep. Here is the pattern and the sweep-line generalization behind it.