I would parade the cheaters through town naked while ringing a bell and saying "shame" over and over again.

Or just give them a 0 for the assignment if I had evidence of cheating.

Not being able to solve a problem in class that they could solve at home is not evidence of cheating. Neither is not showing your work on hard problems, especially in the take home format where students could not only use other resources, but other sheets of paper, if they wanted.

If showing all your work is required for answers, then I would have clearly stated that prior to giving the students any work and remind them before all tests to do so.

If you are sending take home tests over a vacation, you also need to, as a teacher, clearly define what is and isn't cheating if it's not defined in your syllabus.

As the teacher it's your job to set the requirements and boundaries clearly, and not be reactionary when you've failed to do so.

It's unclear from your description if you gave proper guidelines on all of this, but it does seem like you didn't set up the requirement of "show your work, or I will accuse you of cheating without any evidence," so I would prepare to get much deserved backlash from this.

Getting the problem wrong on the board isn't evidence of cheating, but it might be evidence that you need to cover that subject in more depth for the students. Learning is the point after all, not test scores and your pride.