As a research mathematician, I am often asked by non-mathematicians "what's there left to do in mathematics?" As a number theorist, my go-to example of an unsolved problem is the Twin Prime Conjecture, as it is interesting to me and relatively easy to explain. Very often, I am met with the response "well, there are infinitely many integers, so why wouldn't there be infinitely many twin primes?"
This question gets at a deeper issue with non-mathematicians; namely a lack of understanding of the standards mathematicians have for proof and truth. We mathematicians readily understand that a mathematical statement is not considered true unless we can give a rigorous proof of it. Often though, it seems that a non-mathematician will consider a statement true/proved if there's no obvious reason for it not to be true.
To get at this, my goal lately has been to try to come up with statements or questions which seem like they "should" be true or answerable, but in fact are provably false or have an unexpected answer. An example might be "Can you find two positive cubes that add to a cube?" This is, of course, the simplest case of Fermat's Last Theorem, and has the "unexpected" answer that this is impossible. Thus my question:
What are some mathematical statements/questions which are relatively easy to explain, seem like they should be true/answerable, but are in fact provably false/have an unexpected answer?
I'd love to see ideas from the community about this. I'd especially like number theoretic examples, but all ideas are welcome. As well, if there is another post like this one already on MSE, I'd appreciate a reference to it (moderators may then decide whether to leave this question open).
