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[This is just the kind of vague community-wiki question that I would almost certainly turn my nose up at if it were asked by someone else, so I apologise in advance, but these sorts of questions do come up on MO with some regularity now so I thought I'd try my luck]

I have just been asked by the Royal Society of Arts to come along to a lunchtime seminar on "ingenuity". As you can probably guess from the location, this is not a mathematical event. In the email to me with the invitation, it says they're inviting me "...as I suppose that some mathematical proofs exhibit ingenuity in their methods." :-)

The email actually defines ingenuity for me: it says it's "ideas that solve a problem in an unusually neat, clever, or surprising way.". My instinct now would usually be to collect a bunch of cute low-level mathematical results with snappy neat clever and/or surprising proofs, e.g. by scouring my memory for such things, over the next few weeks, and then to casually drop some of them into the conversation.

My instinct now, however, is to ask here first, and go back to the old method if this one fails.

Question: What are some mathematical results with surprising and/or unusually neat proofs?

Now let's see whether this question (a) bombs, (b) gets closed, (c) gets filled with rubbish, (d) gets filled with mostly rubbish but a couple of gems, which I can use to amuse, amaze and impress my lunchtime arty companions and get all the credit myself.

This is Community Wiki of course, and I won't be offended if the general consensus is that these adjectives apply to the vast majority of results and the question gets closed. I'm not so sure they do though---sometimes the proof is "grind it out". Although I don't think I'll be telling the Royal Society of Arts people this, I always felt that Mazur's descent to prove his finiteness results for modular curves was pretty surprising (in that he had enough data to pull the descent off). But I'm sure there are some really neat low-level answers to this.

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    $\begingroup$ I'm in a mellow mood and shan't vote to close, but I think that as penance for asking the question you should do a report on the seminar for us. Slightly more seriously, I think it'll be quite an amazing result that a) has a surprising proof, b) is explainable to non-mathematicians, and c) the fact that the proof is surprising is explainable to non-mathematicians. $\endgroup$
    – Andrew Stacey
    Commented Sep 28, 2010 at 16:56
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    $\begingroup$ +1 for 'I suppose that some mathematical proofs exhibit ingenuity'. Two cultures, anyone? $\endgroup$
    – HJRW
    Commented Sep 28, 2010 at 17:31
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    $\begingroup$ Check out mathoverflow.net/questions/8846/proofs-without-words $\endgroup$
    – Martin Brandenburg
    Commented Sep 28, 2010 at 17:36
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    $\begingroup$ Kevin: doesn't do it for me. Why would I be interested in the sum of the first 1000 numbers? Sounds like the sort of thing my kids would do before they knew any better. $\endgroup$
    – Andrew Stacey
    Commented Sep 28, 2010 at 19:33
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    $\begingroup$ I think, by the way, that "I need some examples for a talk" is a completely legitimate excuse for a big-list community wiki question. $\endgroup$
    – gowers
    Commented Sep 28, 2010 at 20:03

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I like very much the proof of fundamental theorem of algebra (using the winding number around the origin), but it will probably take too long to explain...

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A recent reference is "Street fighting mathematics" by Sanjoy Mahajan.

AT http://www.amazon.com/Street-Fighting-Mathematics-Educated-Guessing-Opportunistic/dp/026251429X

I have browse and read some part. It look as if it almost manages to render NavierStokes equation edible for an hard die finitist. It uses mainly dimensionality but is full of ingenuity.

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a. Euler's computation of Zeta(2), (at first with only very weak, handwaving, or no convergence arguments), as redacted in Polya or here. Obviously amazingly ingenious, and interesting for artists, connecting numbers and circles. Requires unerstanding that a polynomial is equal to the product of its first degree factors, possibly it is too well known, however. It also allows one to then revisit the convergence argument and show what mathematicians actually worry about, after the "flash of ingenuity"

b. A lovely agument I heard a long time ago on sci.math relating to Sagan's book "Contact" in which a message is encoded in the bits of Pi. Someone asked whether a deity could arrange for Pi to be a different real number. Opinions went back and forth relating to spacetime, etc. Then someone asked, in light of any of the familiar series expansions, "If Pi were different, which natural number would be missing or duplicated, and how might that be?" Leads to a discussion of the Peano axioms. Everyone goes home wondering. :-)

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Alon Amit's comment to Kevin Buzzard's answer (on the equivalence between the "upon STOP flip over the top card" game vs. the "upon STOP flip over the last card") reminded me of the Ants on a Meter Stick problem (which has been a favorite of mine since I first read it in the Harvey Mudd Fun Facts site):

One hundred ants are dropped on a meter stick. Each ant is traveling either to the left or the right with constant speed 1 meter per minute. When two ants meet, they bounce off each other and reverse direction. When an ant reaches an end of the stick, it falls off.

At some point all the ants will have fallen off. The time at which this happens will depend on the initial configuration of the ants.

Question: over ALL possible initial configurations, what is the longest amount of time that you would need to wait to guarantee that the stick has no more ants?

The insight lies in the fact that this is equivalent to the "Ghost Ants on a Meter Stick" problem, where the ants pass right through each other instead of bouncing off each other.

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I don't know if this meets your requirement for being elementary, but the Eilenberg Swindle is very easy, very clever, and widely applicable.

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    $\begingroup$ This obviously doesn't meet the requirement for being elementary. And widely applicable? Only if you are a mathematician; this talk is for artists!!! $\endgroup$
    – Sheikraisinrollbank
    Commented Nov 30, 2010 at 15:50
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    $\begingroup$ @Sheikraisinrollbank: The general scheme of the swindle could surely be adapted to a simpler problem, no? $\endgroup$
    – Harry Gindi
    Commented Dec 1, 2010 at 2:52
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