Is it unethical to accidentally benefit from online material in a take-home exam? Many years ago, as an undergraduate, I was taking a physics class which had a peculiar exam arrangement: on Tuesday, the professor gave us a take-home portion of the exam which would be submitted on Thursday, but on Thursday, there would also be an in-class portion of the exam with a different set of problems.
In my case, I remember working out the solution to a difficult problem on the take-home exam throughout Tuesday and Wednesday. I had a solution, but I was unsure whether it was correct. Some equations didn't seem to make sense. At the same time, I was also preparing myself for the exam on Thursday. As usual, I read the textbook, but I also was in the habit of consulting other websites to help me whenever the textbook wasn't enough.
While doing this, I accidentally stumbled upon a Wikipedia article which explained the solution of a similar problem to the one I was working on in the take-home exam. It was not the same problem, but the general method used was sufficiently similar so that I immediately realized that my approach to the problem on the take-home exam was probably correct.
Perhaps the most ethical thing to do at the time was to stop consulting the Wikipedia article immediately. I have to confess that I didn't do this exactly: I couldn't help but glance at it a few more times. Coming across that article made me confident in my approach to the take-home problem, and it drove me to scrutinize my previously written solution more deeply. This led me to discover a simple algebraic mistake which I had made in my own solution, after which my solution made sense and was surely correct. In the end, I never copied anything directly from the Wikipedia article, but I cannot deny that reading it helped me realize my own mistakes.
On Thursday I handed in my work on the take-home portion, and did well in the in-class portion as well. I never told the professor about the incident. Internally, I reasoned to myself that when I came across the Wikipedia article I was not trying to cheat on the take-home, but simply preparing for the in-class exam.
Years afterwards, however, I started to have doubts over what I did was completely ethical. I did not exactly cheat - in the sense of copying solutions from the Internet - but I was indeed &quot;inspired&quot; to become confident in my existing solution because of something I found on the Internet.
Was what I did dishonest? Unethical? Did it rise to the level of cheating that should have been disclosed to the professor? Or was it just a fortunate accident, the product of an odd exam arrangement?
(Note that the professor did not have a clear policy on consulting materials other than the official textbook from Tuesday-Thursday during the take home portion of the exam. While it was generally considered cheating to search out solutions on the Internet for specific problems on a take-home exam, I did not consider it unethical to look up online resources when studying a course and preparing for an in-class exam. In this case, I was doing the latter and accidentally stumbled upon material which was more similar to the former. The fact that there was a take home exam occuring right before an in-class exam resulted in a gray area.)
I suggest that you don't lose a lot of sleep over something that happened several years ago. While you may have crossed a line, your description suggests it wasn't a bright line. Assuming your description is honest, you started out (probably) appropriately looking for alternate explanations of the material, not solutions to the problem at hand. But, rather than the internet it might just as well have been a different textbook.
But if it bothers you, then, assuming you are an academic, make sure that your own students get more explicit advice about what is and is not permitted.
It would have been better, of course, if you had pointed out the situation to the professor. But let the past be the past. Make the future better than the past if you are able.