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{{unsolved|mathematics|Does every [[Jordan curve]] have an inscribed square?}}
[[Image:Inscribed square.svg|thumb|right|Example: The black dashed curve goes through all corners of several blue squares.]]
The '''inscribed square problem''', also known as the '''square peg problem''' or the '''Toeplitz' conjecture''', is an unsolved question in [[geometry]]: ''Does every [[Jordan curve|plane simple closed curve]] contain all four vertices of some [[Square (geometry)|square]]?'' This is true if the curve is [[convex set|convex]] or piecewise [[Smooth function|smooth]] and in other special cases. The problem was proposed by [[Otto Toeplitz]] in 1911.<ref>{{citation | last= Toeplitz | first = O. | authorlink = Otto Toeplitz | title = Über einige Aufgaben der Analysis situs | journal = Verhandlungen der Schweizerischen Naturforschenden Gesellschaft | volume = 94 | date = 1911 | page = 197 | language = de }}</ref> Some early positive results were obtained by [[Arnold Emch]]<ref name="Emch">{{citation |last=Emch |first=Arnold | authorlink=Arnold Emch |year=1916 |title=On some properties of the medians of closed continuous curves formed by analytic arcs |journal=American Journal of Mathematics |doi=10.2307/2370541 |mr=1506274 |volume=38 |issue=1 |pages=6–18|jstor=2370541 }}</ref> and [[Lev Schnirelmann]].<ref name="Schnirelmann 1944">{{citation |last=Šnirel'man |first=L. G. |author-link=Lev Schnirelmann |year=1944 |title=On certain geometrical properties of closed curves |journal=Akademiya Nauk SSSR I Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk |mr=0012531 |volume=10 |pages=34–44}}</ref>
== Problem statement ==
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