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

If this is normal, that just means a lot teachers have no respect for personal time.

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.

So? Are you saying a lot of them cooperated on it? Did they copy work from a separate source? Where is the problem?

You assigned graded work during a vacation, which I would assume means you can use any material you have access too, including teamwork and the entire internet. Does it not?

I intend to have the students I suspect of cheating come to my office to solve the problem on the board.

And if they fail, what does this prove? That they can't reproduce an answer constructed over (potentially) many days of work with references on hand, in a few minutes of high-stress with their teacher breathing down their neck?

What would you do?

Not send graded work home with students if you don't expect them to cooperate. Procter an exam if you want them to use only their brains.

In fact, you should procter an exam during your vacation, because they didn't get one either.