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# Puzzled: Variations on the Ham Sandwich Theorum

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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.

1. A pair of intrepid computer programmers spend a weekend hiking the Cascade Range in Washington. On Saturday morning they begin an ascent of Mt. Bakerall 10,781 feet of itreaching the summit by nightfall. They spend the night there and start down the mountain the following morning, reaching the bottom at dusk on Tuesday.
Prove that at some precise time of day, these programmers were at exactly the same altitude on Sunday as they were on Saturday.
2. Prove that Lake Champlain can be inscribed in a square. More precisely, show that, given any closed curve in the plane, there is a square containing the curve all four sides of which touch the curve. A corner counts for both incident sides.
3. Be the first person ever to prove (or disprove) that every closed curve in the plane contains the corners of some square.

### Author

Peter Winkler (puzzled@cacm.acm.org) is Professor of Mathematics and of Computer Science and Albert Bradley Third Century Professor in the Sciences at Dartmouth College, Hanover, NH.

### Footnotes

All readers are encouraged to submit prospective puzzles for future columns to puzzled@cacm.acm.org.

DOI: http://doi.acm.org/10.1145/1735223.1735250

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##### Valentin Goranko

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?

##### Kenneth Hertz

A continuous tangent should be sufficient. Are weaker conditions sufficient ?

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