[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.
Best Answer
"At any party, there are at least two people with the same number of friends there".
This typically won't be obvious to someone who isn't a mathematician. However, it's quite easy for many people to see this by picturing everyone at the party wearing a t-shirt displaying how many of their friends are present. The number of possible t-shirts is one less than the number of attendees, so there must be a double somewhere.