Post Made Community Wiki by S. Carnahan

The Gale-Sharpley stable marriage theorem, http://en.wikipedia.org/wiki/Stable_marriage_problem .

The algorithm and its proof are very much accessible to school students. Despite its innocuous look, the algorithm is not easy at all to invent.

On a similar note, Hall's theorem: http://en.wikipedia.org/wiki/Hall%27s_marriage_theorem#Graph_theory . This looks like a recreational puzzle but actually is closer to university mathematics than everything done in high school.

Here is another combinatorial exercise which, properly presented, does not even look like mathematics: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=279550#p279550 . The thing I don't like about it is that the standard "gotcha" proof (explained in the usual, informal way) requires a bit too much concentration to understand - some students might fail at it and take it as an example that mathematical proofs are something one either believes or not, rather than something one can check. Of course, one can formalize the proof, but this requires quite an amount of time in a high school class.

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The Gale-Sharpley stable marriage theorem, http://en.wikipedia.org/wiki/Stable_marriage_problem .

The algorithm and its proof are very much accessible to school students. Despite its innocuous look, the algorithm is not easy at all to invent.

On a similar note, Hall's theorem: http://en.wikipedia.org/wiki/Hall%27s_marriage_theorem#Graph_theory . This looks like a recreational puzzle but actually is closer to university mathematics than everything done in high school.