For context, I'm trying to get better at writing proofs, and, to that end, want to get better at recognizing proofs that are wrong in subtle ways. Recognizing incorrect proofs of true statements is especially difficult.

In textbooks and papers, I usually see polished proofs that are correct but abbreviated to varying degrees. The way I read a proof like this is quite different from how I read a perhaps-proof that I'm skeptical of, and I frequently skip over stuff I partially understand if I understand the theorem in question. I'd like to be able to read proofs skeptically in a time-efficient way.

I was wondering whether there were any books or references out there consisting of proof puzzles (by analogy with a chess puzzle), presenting the reader with a proof and asking them to assess its validity (correctness of the argument) and soundness (truth of the premises). Ideally such a book would not require too much background.

The solution verification tag on this site is close, I suppose, but it's sort of a fire hose and you get some clues, such as the vote total, as to whether the proof attempt in the question succeeded or not.

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    $\begingroup$ I'm not aware of anything like this, but I think this would be a great resource! $\endgroup$ Jan 8, 2023 at 22:13
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    $\begingroup$ I'm also not aware of anything like this. At the high school summer program PROMYS we had "PODASIP" exercises: Prove or Disprove and Salvage if Possible. Sometimes the statements were true, sometimes completely false, sometimes false but similar to a true statement. I thought the "salvage if possible" was particularly educational and I've never seen it anywhere else. $\endgroup$ Jan 8, 2023 at 23:18
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    $\begingroup$ One problem is that writing "productively gappy" proofs is really hard. It would take a lot of work - at least for me - to produce a text of this sort. $\endgroup$ Jan 8, 2023 at 23:39
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    $\begingroup$ This is not exactly what you want but still interesting: amazon.com/Mathematical-Fallacies-Flaws-Flimflam-Spectrum/dp/… $\endgroup$ Jan 9, 2023 at 0:43
  • $\begingroup$ Unfortunately it looks like it is out of print, but you might be able to access it using interlibrary loan. $\endgroup$ Jan 9, 2023 at 0:46


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