I'm not quite sure whether this question belongs here, because it has no definite answer. But I'll give it a shot. If any of the mods objects, then I will, of course, respectfully delete this contribution.
A friend of mine, who knows a little arithmetic and only the bare rudiments of algebra, has asked me to explain what it is exactly what mathematicians do. Also, she wants to know what a mathematical proof is.
Now I have this idea of presenting her with two sorts of mathematical 'artefacts':
A. A few non-trivial (and preferably striking or beautiful) theorems whose proofs are easy and brief enough for my friend to understand. Euclid's proof of the infinitude of the primes, or Cantor's of the uncountability of the real numbers, would fit this category. But I would like something a bit off the beaten track, that's not been done to death in thousands of `popular' books.
B. A couple of results whose proofs are not neccesarily easy, but which illustrate nicely how mathematicians deal with heuristics and with discovering new theorems. A few immediately plausible yet striking conjectures would be nice, too. Unfortunately, I can't think of any concrete examples from the top of my head right now. (Edit: the four color theorem comes to mind.)
Your suggestions will be warmly appreciated.
Best Answer
Something easily grasped by pretty much anyone are problems related to covering chessboards with dominoes. (I'm especially fond of these because learning about them was what initially got me interested in mathematics.)
A domino exactly covers two squares of chessboard. Can you always cover an $8\times 8$ chessboard with dominoes? Of course this is easy by just placing four dominoes in each row. (We always assume a sufficient supply of dominoes, in this case, $32$.)
But what if we remove two squares from opposite corners of the chessboard; can we still cover it with $31$ dominoes? In this case the answer is no, because opposite corner squares have the same colour (black, or white as shown below), which means that there are $32$ squares of one colour and $30$ of the other, while each domino covers two squares of different colours.
There are many other problems of this sort involving covering pruned chessboards with dominoes (or other so-called "polyominoes"), some of which have simple, easily explained proofs like the one above. (Though perhaps not quite that simple.)
There is a nice book of chessboard problems by John J. Watkins that includes proof of some of these kinds of problems.