It's a fun set of optimisations: to get the path-clearing effect do you run the pipes on less optimal routes (greater capital costs), or do you using varying grades of insulation (less effective under paths to get the clearing effect, more effective on the short-cuts between paths), and so on. I hope Idaho uses these scenarios in their maths lessons, in place of the hoary "the phone company needs to put a line across a property that includes a river where it costs three times as much per metre..." that I teethed with.

Well the shortest distance between two points is a straight line. Pipe and sidewalk routing have similar objectives and constraints.

Not all sidewalks are heated. It’s more like they paved the top of the service vaults.

I don’t recall this problem specifically but the university employs a lot of students and alums.

Probably less than you lose in energy from heating the paths all year round.

As I explained they aren’t heated year round.