Neil writes: << Many years ago i saw an abstract in the Oberwolfach Vortragsbuch entitled "Squares in Lake Michigan", which for a long time I thought proved theorems such as "any Jordan curve - or distorted circle in the plane - contains 4 points which are at the vertices of a square". But I never saw anything more about this, and began to doubt my memory. . . .

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When I was about 12, I recall reading the same statement in a popular math book -- it was either Mathematical Snapshots, by Hugo Steinhaus, or Through the Mathescope, by C. Stanley Ogilvy. If anyone has access to the right one of these, maybe it contains a reference. Then again, I seem to recall hearing years later (but also years ago) that this theorem had never been properly proved. --Dan