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I'm a layman in mathematics, so please excuse me in advance for anything in this question that may be inappropriate :D. Well: Four years ago, I was reading (and working to solve the puzzles on) Winkler's Mathematical Puzzles: A Connoisseur's Collection, and in the section "Unsolved Puzzles", there was the problem "Squaring the Lake" at p. 143: "Prove that every simple closed curve in the plane contains four points forming the vertices of a square." Obviously, I was not able to prove it (the max I could get, after much effort, was to prove that it was true for triangles!!!), so I went to "Comments and Sources" at p. 148, where he mentioned a web-site, a journal paper and a book on the subject, and he also made the comment that "It's a little embarrassing that mathematicians cannot [...]" (bolds is mine. "[...]" is equivalently to "solve the puzzle", I don't want to make a full quote because if I quote too much I may infringe some copyright...).

Well, much time later I came across http://www.ams.org/journals/notices/201404/rnoti-p346.pdf and according to https://en.wikipedia.org/wiki/Inscribed_square_problem, the problem is still open. So, I ask, why this (innocent-looking :D) problem is actually so hard? (I probably won't understand why, but you guys don't need to give an "answer to layman", it can be a technical answer!)

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    $\begingroup$ The problem is deceptively hard because a "Jordan curve" can be very badly behaved and look quite different from what intuitively one might think of as a "closed curve." $\endgroup$
    – Sam Hopkins
    Commented Jul 31, 2015 at 21:13

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Let me elaborate on Sam Hopkins' comment.

The main reason that makes this and other problems on continuous curves so hard is that a "simple closed curve" or "Jordan curve", i.e. a non-self-intersecting continuous loop in the plane, can be a horrible object, for instance a nowhere differentiable curve such as the Koch snowflake and other fractal curves. There are also Jordan curves of positive area (first constructed by Osgood in 1903).

In fact, as it is also explained in the Wikipedia article that you linked, the problem is actually solved for "well-behaved" curves, such as convex curves or piecewise analytic curves, i.e. for objects that are close to our intuitive notion of "continuous closed loop".

A possible strategy to solve the problem in the general case is to try to approximate your Jordan curve by using well-behaved curves, for which we know that the conjecture is true, and then pass to the limit. The technical difficulty with this approach is that a limit of squares is not necessarily a square, but it can be a single point (i.e., a square "of side length 0"), so it is not clear whether it is always possible to find an approximating sequences of curves that works.

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    $\begingroup$ Thank you for answering! I have an additional (informal) question to make here on comments section: Is it possible to transform such problems on general Jordan curves, so of an object with origin in visual geometry, to problems of pure abstract algebra? If possible for this specific problem, would this be more likely to help in the search for a solution or would make it more difficult? $\endgroup$
    – N. G. M.
    Commented Jul 31, 2015 at 21:52
  • $\begingroup$ Another: Has it been already proved for some specific fractal (of the famous ones, with more badly behaviour)? $\endgroup$
    – N. G. M.
    Commented Jul 31, 2015 at 21:52
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    $\begingroup$ I'm not an expert on the topic. However, regarding your first question, I would be very surprised if one could "detopologize" the problem and make it into a pure algebraic question; I presume that the answer is no. $\endgroup$
    – Francesco Polizzi
    Commented Jul 31, 2015 at 22:04
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    $\begingroup$ Concerning your second question, as explained in Wikipedia's article the problem is solved for curves with central symmetry (Nielsen and Wright, 1995). So it is true for every centrally symmetric fractal (for instance, for the Koch snowflake). $\endgroup$
    – Francesco Polizzi
    Commented Jul 31, 2015 at 22:07

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