I was writing a 3-question examination today (undergrad) with 1200 other students when our professor comes in after ~2/3 of the exam and changes a question to make it solvable. This was a 1.5 hour exam where each question was designed to take 30 minutes so unless you did the other two questions knowing that question was impossible to solve and waited for an announcement on instructions of how to solve, you would not be able to finish. When I walked out of the exam, you could tell that everyone was mad that this changed question could have impacted their overall mark by 15-20%. What should I do to help out myself and my fellow classmates who were screwed over by this change? Has anyone ever had a similar situation?
This does happen sometimes, despite a professor's best efforts to check the exam beforehand. Professors are humans and make mistakes.
You can write a polite email to the professor (or whoever is in charge of grading the exam, if different), letting them know that you feel this had a disproportionate negative impact on your score.
That's about all you can do. It is ultimately up to the professor (or grading committee, etc) to decide what to do about this issue, if anything. They might:
Do nothing, reasoning that although the correction was unfortunate, it affected all students equally.
Give credit to students who made an appropriate attempt to solve the impossible version of the problem.
Adjust the "curve" or other statistical correction of the exam score to take this into account.
Discard the question's score, and reweight the scores on the other questions.
Discard the entire exam and hold a new one.
Discard the entire exam and reweight other exams in the course to compensate.
In principle, if you don't agree with the professor's decision, you may be able to appeal to some higher authority. This would depend on your university's regulations, and my guess is that it would be unlikely to succeed, if the professor did anything halfway reasonable. I'd consider that any of the above options would satisfy that.
Yes, these things happen. No one is perfect, not even a professor. But what you need is a fair resolution. One would be to just cancel the exam and adjust grading rubric accordingly. Another, not quite as good, would be to reschedule another exam.
But you need to find a, hopefully polite, way to let the professor know that some people spent a lot of time on an impossible question and others did not. Even giving everyone full marks on that question isn't fair due to the frustration that some experienced.
If the professor is focused on teaching and not just on grading, then it should be possible to work out a solution.
With 1200 people it is hard to form a delegation to meet with the professor, but that would be a logical step.
But if this just happened, it may be that the professor will announce a suitable accommodation at the next meeting. If not, you might bring it up with the TAs for the course. I wouldn't escalate it to any formal complaint, however, until you have more evidence about how the professor intends to deal with it.
My advice to the professor, however, is that if you give a diagnostic that you know is invalid, you need to drop it entirely. It can't be finessed.
Consider not doing anything.
The issue isn't time sensitive. Its not like the grades can't be changed after the fact. It's very reasonable to believe the professor is going to analyze the grades that came out of the exam and find a solution.
The professor will have information you don't have. While you know your exam was affected, and you can estimate how it affected 1199 other people, the professor will be making decisions with all 1200 graded exams in front of them.
Now if the professor hands you back the graded papers and doesn't do anything to resolve the issues, that's a good time to start making noise.
One of my professors had a blanket rule that was applied to handle situations like these - You must solve the question to the best of your ability. If the missing piece of information can be simply substituted by a variable, say 'x', your answer must be in terms of 'x'. If you think that a question is not solvable, you should prove so in your answer. If you are successfully able to do it, you get full points for that question. Based on the difficulty of said proof, they also awarded bonus points, thus turning a potentially problematic situation on its head.
This worked wonders. The students were thrilled when they were able to successfully do this, and the professor had achieved a higher goal than what a simple exam would do. In fact there were unconfirmed rumours of the professor 'making a mistake' on purpose every once in a while. You can suggest that your professor adopt a similar policy in the future.
As for what you can do now, your options are limited. People make mistakes. You can contact the professor via a polite email and make your concerns known. You can also ask what strategy they would apply, to make it fairer. Whatever they do, it's probably not going to be 100% fair anyway. If they do nothing, or their strategy is blatantly unfair, that would be the time for you to take your complaint further if need be. But any half decent strategy is probably going to get the support of any grading committee(s) and/or department heads.
Other answers respond to the questions raised in the main body of your message, I'd like to comment on the broader question (asked in the title), What should I do if my professor changes the question mid-exam?, with particular reference to
unless you did the other two questions knowing [the other] question was impossible to solve and waited for an announcement on instructions of how to solve, you would not be able to finish.
Exam strategy can help here: I recommend considering the entire examination script before writing. If you're able to identify an issue with any question, then immediately raise it with invigilators. (They should promptly raise such issues with the professor.) This maximises the window during which a professor can respond to the issue. Divide the remaining time between questions, with the goal of maximising your score. If you were able to identify an issue with a question, then that question should be delayed, because you might receive additional information during the exam. Returning to the question:
What should I do if my professor changes the question mid-exam?
Be prepared: Anticipate this possibility and adopt an exam strategy that optimises your advantage.
Response to comments
You're not answering the question. You're giving advice about [what] one can do before a change [of an exam script] to reduce the impact, not what to do in response to a change.
The question of How to preparation for when a professor changes the question mid-exam would be well answered by this. What to do after the fact is the OP's question. Unless you have a time-machine allowing the OP to follow your advice "preemptively", this does not answer the question.
After an event such as the OP's, one must reflect and consider how to improve themselves. My answer explains how the OP should improve themselves for a similar such event in the future. I consider this to be a crucial part of the OP's response.