Nothing.
Having any homework for the holidays is already enough. Of course most of them would just want to have that gone ASAP.

If you don't want students to work together and learn from each other don't give home assignments. It's not like they won't be able to work together irl

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.

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Did they cheat? You were lazy & let them take an exam at home. Sounds like you should’ve expected them to use any resources available. Just because something is normal, doesn’t mean it’s right.

I’m sure your students love you…

That's a terrible thing to be considered normal. Those students were on break, on vacation. I didn't do work for my job when I was on vacation! I hate cheating, but I hate that you made them take an exam on break even more.

Edit: "Class, I see some of you did not understand the way I require work to be shown. For this reason, I will reteach my requirements. Those of you who did not understand will be given an opportunity to retake the exam."

Hold an in class quiz with essentially the same problem but with different values. The students that actually worked through the problem should be able to do it again with the changes. Those who didn't understand and just put down what their peers got will struggle with a quiz. Bonus points if you can restructure the problem in a way to elucidate which specific aspects you think the students were skipping over with help from their peers. Feel free to have specific requirements assigned point values in the problem statement.

Don't call them into your office and put them on the spot. That will make this adversarial. Your job is to teach them how to solve problems and communicate their methods in a clear fashion. You should reevaluate your problem writing and grading policies if just looking up answers can earn a passing grade. If you give a quiz, be up front with them that you have concerns about some students skipping the work and copying answers. Reiterate that the point of the exam was to make sure they can solve problems, the correct answer is merely a byproduct.

I will add speculation that there is a difference between what your students think you expect from an answer and what your expectations actually are. Mismatches in expectations are immensely frustrating for both parties. So don't leave your students guessing. Give them specific examples of work of different quality and what aspects earn full points and what things might lead to point deductions. Some of the best professors I had would publish all the prior year exams with their solutions. That gave everyone the opportunity to mimic the workflow and match the level of detail expected. That also elliminates the concern of students finding the answers online or from prior year students for exams as the teacher will have had to avoid reused questions entirely.

I wouldn't do anything. Your job is to teach, not to discipline. Your students can choose to do or not do whatever work you set them; it's their education and their choice. Ultimately cheating only affects them and their learning.

Also, seconding the fact that if you give people a graded take home exam that implies open book (including the internet and each other)

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.

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 appreciate the feedback. Even the negative feedback. You guys really think I'm some kind of asshole 🤣. I typed this from my phone in bed, so now that I'm at a keyboard, let me explain fully.

When I said I gave the exam over spring break, I didn't mean it began at the beginning of spring break and ended at the end of it. That time was available for them to work on it. When I give exams, I give them a little over a week. From Tuesday to Thursday the following week. In this case, it began the Tuesday before spring break and ended the Thursday after. The reason I did this is because, like many of you, I remember papers being due immediately before or right at the end of breaks. By saying I gave it over spring break, I meant I gave them plenty of time.

I am very clear what is and is not permitted for an exam in my syllabus. They get an equation sheet, the allotted time, and they can work with a partner. Nothing else. Except for AI in which case they must screenshot everything. This is mostly for my curiosity. It still doesn't work for physics.

When I say normal at my institution, I mean to give a take home exam. I wasn't deviating from the norm by doing this, and it is the way I typically do it. As we have all experienced, you may have a day when you have 3 exams. Maybe that happens to only a few students. It disproportionately effects them. Giving this time, they can work it into their schedules.

So what did I see that constitutes cheating? It's very clear to me that the students used solutions from Chegg and/or other sites. If you've done this sort of thing with code, you know that folks will change the names of the variables, but not the structure or logic. It reads exactly the same. That was the case here. A few students were so (hilariously) guilty of cheating, they actually rewrote the solution to a similar, but different problem. Those problems had a different number of parts!

This is not my first time doing this. I've done this at several other universities. In those cases, I didn't have the issue of cheating, so I don't have a very explicit cheating policy in my syllabus. I'm taking the advice that some have given and giving them credit for what they've done. I will however be telling them on Tuesday (a conversation I am NOT looking forward to having) that I know many of them cheated, that I have evidence of it, and that I will refer them to the honor council should it happen again.

The part that sucks the most is I trust students. Having done this before, I've found that if you trust students, respect them, they in turn respect your expectations. Given how blatant this cheating is, it feels like a betrayal. Thanks again to everyone who replied, it has given me plenty to think on.