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Suppose you're meeting with programmers who have taken some professional programming courses (/ self thought) but didn't study a university level math.

In order to show them the beauty of TCS, I'd like to gather some nice results/open questions coming from TCS which can easily be explained.

A good candidate for this purpose (IMHO) will be showing that the halting problem is not decidable. Another will be showing a lower bound on the running time of comparison based sorting (although that's a bit pushing it from what I expect them to understand).

I can also use the ideas from Explain P = NP problem to 10 year old, assuming some of them are unfamiliar with it.

So, questions has to be:

(0. Beautiful)