In Star Trek II, the suggestion is that the cadet knows it IS the no-win scanario by virtue that no one has passed it ... except one. Kirk's commendation for "original thinking" was because he did pass the test. If he revealed to anyone at Starfleet he hacked into the program he would have been booted out for cheating.
So, it is my perception that the Cadets know its not passable, but because one person did pass means it IS passable and that's their challenge. The fact Savek didn't know how he passed is testament that what he did (reprogramming AND solution) was not known to Cadets.