As a preface, I think this merits the label hack not because it’s particularly clever or well-implemented; simply it was the fastest way for me to arrive at an optimal solution for a well-defined problem.
Splitting a class of $$k$$ students into $$n$$ disjoint meetings (‘sections’) which meet once a week on a pre-determined day of the week, and finding a mutually convenient time for each session based on the availability of each student
- Represent the availabilty of the students at each one of the $$m$$ hours in the day as a vector (length $$k$$) of truthy values
- Notice that binary numbers are basically vectors of truth values
- Perform a recursive bitwise OR on each of the $$m \choose n$$ choices of binary numbers which each represent the group’s availability at precisely one of the hours in the day
- Get the Hamming weight (number of bits set) of each result
- Order the $$m \choose n$$ choices by their associated Hamming weight
- The choices of times for the $$n$$ section meetings with maximum weight suit the greatest number of students!
I implemented this for $$n = 2$$ because that’s what I needed at the time. See here for a rubbish implementation.