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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.

Mike Pierce
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Anthony Quas
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    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. – JRN May 13 '20 at 07:15
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    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. –  May 13 '20 at 12:00
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    Nice question. I am struggling too with students that think it is fine to get answers from the internet. The answers in the github link are quite poor (for example, questions 3, 5 and 6.I didn't read further), so your student will "pay the price" of relying on weak sources sooner or later. I don't know if it is comforting or alarming though. – Taladris May 13 '20 at 15:31
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    @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. –  May 13 '20 at 16:58
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    Note that github allows users to fake dates. – user253751 May 14 '20 at 13:04
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    The way a lot of these sites work is to pay current students to post questions and solutions with credit for the site. They mobilize the cheating students to serve as their spies. – James S. Cook May 14 '20 at 21:08
  • Was the mistake that they thought the intervals for $a$ were instead symmetric about $0$ and then said the obvious? Because I was reading quickly and that's what I would have made. – Vandermonde May 16 '20 at 03:02
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    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. – Anthony Quas May 16 '20 at 06:37
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    In my recently acquired experience with Chegg (deep sigh...), some of the answerers in their "Chegg Study" service are good, but the majority are simply students trying to make a quick buck out of their even more cluless peers. Note that Chegg "experts" are not employed by Chegg -- just like Ebay sellers aren't employed by Ebay. And although Chegg claims that they have some kind of quality control in place, and at the very least there's a rating feature, it doesn't do much good. I've seen my exam questions on Chegg with an "expert" answer carefully retyped from Symbolab. – zipirovich May 25 '20 at 18:01
  • In my experience teaching discrete math, most Chegg answers have been total nonsense, with a few good ones. The nonsense answers are how I found that this was happening. – Kevin Carlson Jun 18 '20 at 14:50

1 Answers1

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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]

Nick C
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