Engineering student:
When I was in university all egineersengineers had to take a spacialspatial awareness test. Maybe you can adapt this into a roleplaying puzzle.
- give them an odd view of a keyhole and make them select the correct key.
Specific to Mechanical Engineering:
Anything that has to due with simple machines (pullies, levers, gears) should work. Maybe a rube goldberg machine or something similiar.
Psychology student:
This one might be a stretch, but how about something involving the Prisoners Dilemma. http://serendip.brynmawr.edu/bb/pd.html