You will probably get better answers if you ask this in a community dedicated to teaching/professors. Posting on general asklemmy seems like you’re going to get flamed a bit.

I gave my students a take home exam over spring break. (This is normal where I teach)

That is rough. Nothing you can do about it this time, but, in the future, I wouldn’t recommend giving work over break even if others are doing so. Breaks are there for a reason.

It came down to a factor of three in the solution. That factor inexplicably appeared with no justification on many of their exams.

It’s hard to say without seeing exactly what you mean, but this sounds a little flimsy. You want to be pretty sure before you accuse someone of cheating. You can always just mark the answer as wrong if they didn’t prove to you that they understand it.

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?

If I strongly suspected cheating, I would probably do something like that. Just be aware that the environment is different from a paper exam, so you need to be lenient. They are not used to standing in front of a board and working while someone watches. Also, a problem on a take-home exam could be worked on for hours, whereas you presumably expect them to do it quickly. You may need to give them the solution they wrote and see whether they can explain it to you. Or, give them most of the solution, but have them fill in some missing details that they should know if they actually did the problem.

Also, as others have said, there was no cheating unless you were very clear on what resources were allowed and not allowed on the exam.

FWIW, I do strongly disagree with the folks who are saying that any take-home exam should be open-everything. The argument that you will be able to do it in your career doesn’t hold water. School isn’t the workplace. Students are working on simple problems to build up skills that they can use to solve more complicated problems later on. If people want workplace rules about collaboration in the classroom, then the problems need to be scaled up accordingly. In many schools, that does happen later in the curriculum with things like senior projects or some project-based upper-level courses. But, teaching that way from the start wouldn’t give students the time and support they need to gradually improve, so allowed resources need to be scaled back accordingly to account for the deliberate oversimplification of the problems.

On a more personal note, sorry that you have to deal with this. Everyone can appreciate that the situation is tough for the students, but a lot of people don’t realize that dealing with cheating is also very stressful and disheartening for teachers.