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I recently taught an introduction to real analysis. I assigned a (covid-induced) take-home final, which included the question:

Define the set S by $$ \bigcup_{n=1}^\infty \left\{ \frac{a}{2^n}\colon 0\le a\le 4^n, a\in\mathbb{Z} \right\}. $$ Identify $\bar S$, showing that $\bar S$ is what you claim. [ Hint: for any real number $x$ and any positive integer $k$, there exists $a \in \mathbb Z$ such that $kx \in [a,a+1)$, so that $|x−\frac a k|<\frac 1 k$.]

Shockingly (to me), I received 9 essentially identical solutions that all contained the same serious, and somewhat subtle error. I first suspected a "homework help" site, and indeed found that the question had been posted and answered on chegg (a colleague has subscribed to that web site, and showed me the solution that was posted there, which was identical to the incorrect solution).

However, that's not the end of the story! Another student, when challenged said she "had never heard of chegg", and claimed she had found it in some materials she had been studying "before the exam". She then gave me a link to a github repository (see question 19) that appears to be dated 3 years ago that contained almost exactly my question (that I thought I was the first to invent!) -- with the same incorrect solution as appears on chegg. Given the quite distinctive formatting of the solutions and the distinctive error that they all contained, I am convinced this is not a case of two people independently making the same mistake.

My question: How were the "cheggspert" (i.e. the person employed by chegg to solve students' exam questions) and my student able to locate the github repository containing this question?

Needless to say, I harbour the hope that understanding how this kind of cheating takes place will give me ways to prevent it in the future.

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    $\begingroup$ One way to prevent this type of problem in the future (used in mathematics contests) is to use the current year as part of the problem. So a problem containing "2020" as a constant is most likely not in any repository containing problems from previous years. $\endgroup$ – Joel Reyes Noche May 13 at 7:15
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    $\begingroup$ Chegg has two different services, Chegg Q&A and Chegg Tutors. The only reason you were able to find your question on Chegg by googling is that it was in Chegg Q&A, and that was also why n-1 of your cheaters found it. If students pay for the Chegg Tutors service, then I don't think the results are publicly visible. $\endgroup$ – Ben Crowell May 13 at 12:00
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    $\begingroup$ @Taladris: My experience is that Chegg Q&A answers are usually of very high quality. I think they have some facility that lets users rate answers as good or bad. Presumably this works best for things like freshman physics, where tens of thousands of students are trying to solve the same homework problems. I've heard a lot of my colleagues say that they don't care if students cheat on the internet because the answers they get are bad, but this has not been the case for the classes I teach. $\endgroup$ – Ben Crowell May 13 at 16:58
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    $\begingroup$ Note that github allows users to fake dates. $\endgroup$ – user253751 May 14 at 13:04
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    $\begingroup$ No it was worse than that. They fixed a denominator, $k$, and then found an $a$ [ the numerator $a$ depends on $k$. ] They then found an $n$ such that $4^n$ exceeds $a$. They then changed $k$ to be $2^n$ without realizing that that changes the $a$ also... It gets a bit jarring when you see it for the 9th time. $\endgroup$ – Anthony Quas May 16 at 6:37
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How were the "cheggspert" (i.e. the person employed by chegg to solve students' exam questions) and my student able to locate the github repository containing this question?

Here's a possible timeline:

  1. A student posts the question to Chegg.
  2. The Cheggster, who probably keeps a trove of solutions to real analysis problems, digs up the solution and sells it to the student.
  3. Student shares it with other students.
  4. One of the students searches the internet for the text of the answer from Chegg and stumbles on the Github repo, points to that as the source.

This may work if the solution is easier to search for than the text of the problem. [i.e. More typed words in the solution]

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  • $\begingroup$ That’s a smart idea! $\endgroup$ – Anthony Quas Jun 5 at 3:04

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