The solutions of the first two puzzlesand maybe the third as wellmake good use of the Intermediate Value Theorem, which says if you go continuously from one real number to another, you must pass through all the real numbers in between. The most famous application is perhaps the Ham Sandwich Theorem, which says, given any ham-and-cheese sandwich, no matter how sloppily made, there is a planar cut dividing the ham, cheese, and bread, each into two equal-size portions.
The two solved problems are perhaps a bit easier than the Ham Sandwich Theorem but still tricky and rewarding enough to be worth your attention and effort.
All readers are encouraged to submit prospective puzzles for future columns to email@example.com.
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I think the truth of 2 and 3 may depend on how you define (closed) curve, so please give your precise definitions. For instance, there are unbounded, space filling curves for which 2 would not hold, and one can make such a curve 'closed' by self-crossing. So, maybe 2 should be restricted to simple (Jordan) curves?
A continuous tangent should be sufficient. Are weaker conditions sufficient ?
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