Turns out what Hogwarts really needs is a course on cryptography
Contains read-between-the-lines spoilers
for The Half-Blood Prince
.) This page on problems of trust in the Harry Potter Universe
is even better:
Time Turner: This allows a wizard to be in two places at once. In the Prisoner of Azkaban, Harry and Hermione use this to frustrate the plans of the Ministry of Magic, while retaining a cast-iron alibi.
Interestingly, Hermione's possession of the Time Turner had previously been authorized by the Ministry of Magic - presumably by a separate department. Clearly the wizarding world has failed to embrace Joined-Up-Government.
Quidditch:: The opportunities for cheating at Quidditch seem endless. For example, there are several instances in the book where key artefacts (brooms, bludgers) are jinxed by watching stakeholders.
(In contrast, some of the cheating that goes on around the Triwizard Tournament seems touchingly naive.)
Both this one and the last one via GeekPress