I gave my students a take home exam over spring break. (This is normal where I teach) One of the questions was particulary difficult. It came down to a factor of three in the solution. That factor inexplicably appeared with no justification on many of their exams. I intend to have the students I suspect of cheating come to my office to solve the problem on the board. What would you do?
Edit: I gave them the Tuesday before spring break until the Thursday after. I didn't want it to be right before or right after.
When I say normal I mean giving take home exams.
Kids cheat when they're not engaged with the material enough to learn it properly, or when the consequences for not cheating are too much for them to bear.
You gave them an assignment to do when you weren't actually teaching them, which means there's no way they can be properly engaged with the material. And you threatened their spring break with sitting in a room alone doing homework if they didn't get it done fast enough. You created a perfect breeding ground for cheating. Try creating an environment where kids don't feel that they need to cheat.
When I was in university I never heard of anyone cheating, because we were all treated like adults and we were engaging in material we liked. Try inspiring your students and treating them like adults. That means respecting their free time. If you don't give them respect as people, you won't get any respect as an authority.
See edit please
Did you tell them they were only expected to work on the assignment during the school term?