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What are examples of well received mathematical papers in which the author provides detail on how a surprising solution to a problem has been found.

I am especially looking for papers that also document the dead ends of investigation, i.e. ideas that seemed promising but lead nowhere, and where the motivation and inspiration that lead to the right ideas came from.

By "surprising solution" I mean solutions that feel right at first reading and it isn't clear why they haven't been found earlier.

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    $\begingroup$ A long time ago, a referee asked for more information on how we had gone about finding the solution to the problem. We added half a page on that. Then we got a new report saying "the authors carry on like they solved the Riemann Hypothesis". $\endgroup$
    – Brendan McKay
    Commented Nov 21, 2018 at 10:15
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    $\begingroup$ @BrendanMcKay its too sad that it seems to be imperative in mathematics to keep up the impression mathematicians don't experience emotions when doing research; no wonder that mathematics is judged as dry and the most undesirable thing to do. $\endgroup$
    – Manfred Weis
    Commented Nov 21, 2018 at 10:26
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    $\begingroup$ I'm surprised JR Stallings' paper "How Not To Prove The Poincaré Conjecture" has not yet been mentioned. $\endgroup$
    – Sam Hopkins
    Commented Nov 21, 2018 at 16:15
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    $\begingroup$ Not a paper, but Villani's "Birth of a Theorem" does that kind of thing in detail. $\endgroup$
    – Piyush Grover
    Commented Nov 21, 2018 at 17:17
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    $\begingroup$ Dirac in Recollections of an exciting era (1977) recounts how he came up with his matrices while trying to “take the square root of a sum of four squares”. $\endgroup$
    – Francois Ziegler
    Commented Nov 21, 2018 at 18:40

9 Answers 9

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Richard P. Stanley's How the Upper Bound Conjecture was proved ends with two morals:

  1. The shortest path may not be the best.
  2. Even if you don’t arrive at your destination, the journey can still be worthwhile.
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    $\begingroup$ These kind of papers surely deserve more awareness and should be recommended to novices to mathematical research. $\endgroup$
    – Manfred Weis
    Commented Nov 21, 2018 at 8:17
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    $\begingroup$ @DavidRicherby Looking at your edit, I should probably mention that the picture is mostly my fault. For a more detailed explanation, see here. $\endgroup$
    – Martin Sleziak
    Commented Nov 21, 2018 at 21:33
  • $\begingroup$ @MartinSleziak Ah. Makes sense, now. I didn't look at the edit history before removing it. $\endgroup$
    – David Richerby
    Commented Nov 21, 2018 at 21:35
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The paper

Rawnsley, John; Schmid, Wilfried; Wolf, Joseph A., Singular unitary representations and indefinite harmonic theory, J. Funct. Anal. 51, 1-114 (1983). ZBL0511.22005.

contains an unusual “Historical Note” (pp. 102–107). E.g.:

For various reasons one expects to get $\mu_n$ by... That does not work directly because... In 1975, S & W tried... At that point it became clear that an intrinsic higher $L_2$ cohomology theory was needed... In 1977, R & W looked... They did not see how to... This was the point at which S & W had been stopped... During the following academic year B succeeded in... but the method did not extend past... R & W made some progress in... These results were not published formally because... During the summer of 1979, S & W discussed the apparent disparity and clarified... then carried out a computation... then looked at the case... Thus the original S & W problem was settled... At the end of the summer of 1980, S & W saw that... could be simplified... The present version was completed in... There are two important parallel developments which we understood only after... (etc.)

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The prime example is Euler's papers. This style is out of fashion in 20th century. Polya in Mathematics and Plausible reasoning discusses this question at length and even reproduces completely (in English) one of Euler's papers (on partitions).

Of the 20th century examples I can mention

MR1555091 Malmquist, J. Sur les fonctions a un nombre fini de branches définies par les équations différentielles du premier ordre. Acta Math. 36 (1913), no. 1, 297–343.

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The first example that came to mind was

MR0270881 (42 #5764) van der Waerden, B. L. How the proof of Baudet's conjecture was found. 1971 Studies in Pure Mathematics (Presented to Richard Rado) pp. 251–260 Academic Press, London.

There, van der Waerden describes some of the history as well as his proof of his well-known theorem.

Another example:

MR2245898 (2007j:05091) Seymour, Paul. How the proof of the strong perfect graph conjecture was found. Gaz. Math. No. 109 (2006), 69–83.

From Wilson's review in Mathematical Reviews: "In this interesting and revealing paper, Seymour describes in graphic terms their assaults on the problem, the difficulties they came across, and the means they used to overcome these difficulties."

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Ryan Williams provides a Casual Tour Around a Circuit Complexity Bound (the bound in question being that NEXP lacks nonuniform polysized ACC circuits) which may fit the bill, although I believe that Williams's goal is to give a motivated exposition rather than a 100% historically accurate account of how he came up with his proof.

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A nice example is the article "The method of undetermined generalization and specialization, illustrated with Fred Galvin's amazing proof of the Dinitz Conjecture" by Doron Zeilberger in Amer. Math. Monthly 103, no. 3, 233-239, 1996 (see also here for a freely accessible version).

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Another type of example. Textbooks on non-Euclidean geometry may often begin with a chapter on historical failed attempts to prove the Parallel Postulate.

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I enjoy The genesis of the Macdonald polynomial statistics, complete with journal entries, and detailed descriptions of the experimental method.

This paper describes how the researchers came up with a nice formula for the combinatorial (aka modified) Macdonald polynomials.

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Famous example: Wiles’ Modular elliptic curves and Fermat’s Last Theorem, Ann. Math. 141 (1995) 443-551, has such a “research report” filling pp. 449-454:

The following is an account of the origins of this... I began working on these problems in the late summer of 1986... I hoped rather naively that... After several months... I made the first real breakthrough... For a long time the ring-theoretic version of the problem, although more natural, did not look any simpler... The turning point... came in the spring of 1991... Not long afterwards I realized... This I had been trying to do for a long time but without success... Then, in August, 1991, I learned of a new construction... By the Fall of 1992, I believed I had achieved this... For several months I tried... Then unexpectedly in May 1993... I made a crucial and surprising breakthrough... Believing now that the proof was complete, I sketched the whole theory in three lectures in Cambridge... However... I made, almost unconsciously, a critical switch... frustrated in the efforts to repair the Euler system argument, I began to work with Taylor... The attempt to use $p = 2$ reached an impasse at the end of August... I decided in September to take one last look... In doing this I came suddenly to a marvelous revelation... I had unexpectedly found the missing key to my old abandoned approach... we spent the next few days making sure of the details...

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