Unless you explicitly stated in the exam that they had to show their work with their answer or fail, even if the answer was correct, then I say pass them. It's on you to be specific as to what you want to see on the exam. Maybe they worked it out on scratch paper and didn't turn that part in with the exam?

For the students that can’t solve it in person, explain why cheating is bad and that they’re only hurting themselves, and give them a 0 on the exam. That’s probably enough for them to get the picture.

Also maybe focus on them a bit more, since they’re struggling. If someone cheated because they can’t do the assignment, then they’re struggling. So maybe offer them extra credit to make up for the exam if they come in after school to study with you where you can answer their questions.

One of the questions was particulary difficult.

If its the only question that the kids "cheated" on, probably just make that question weigh less compared to the rest of the questions. Bonus question or something.

I mean, unless the point of the test is to weed out kids who shouldn't be taking the course...

I've seen various ways to handle it. My favorite is to grade the rest of the exam extremely harshly, where even a minor mistake could get full points taken off. Unfortunately, it's extremely difficult to prove cheating. I've had students copy word-for-word from Google and that still wasn't enough evidence. I don't think even having the students solve it in the board could convince the higher ups to do anything

Probably treating as “forgetting to write down their steps” before jumping to cheating; presumably you’ve been teaching them to show their work and they forgot to do that here.

Give them the chance to show you how they solved it on the board, if they can then great! Give ‘em the points and send em on their way. If not, then give them a zero for that part of the test and move on.

Seems like people are missing the part where you said “this is normal where I teach” and just judging you for a take home exam. Anywhere I’ve been to school has an office for handling academic dishonesty. I’d consult with them, even if only to protect yourself.

Pass them. Grades aren't proper representation of intellect or ability anyways, and failing students will only hurt their lively hood and chances of success later in life.

I'd ask them to come in and redo the problem on paper at the same time, if many have the answer without any calculations, the moment you got the first one in your office, the others would be aware of what's going on.

I think getting them to show their work is appropriate and for any that can’t replicate their work explain to them the downfalls of cheating. The other comments here justifying likely haven’t ever been in an academic setting. Relying on cheating is setting yourself up for failure if you intend to continue studying at a tertiary level.

I don’t think a punishment is necessary for cheaters just a lecture. Let them know people can and have had their degrees rescinded years after the fact when their cheating was detected with newer methods.

Edit: downvotes for suggesting that cheating is bad lmao. Like I said cheating at uni is easily detected these days. Fuck the getting caught, you’re paying however much to get an education, you may as well actually learn.