One problem which I think is mentioned in Guy's book is the integer block problem: does there exist a cuboid (aka "brick") where the width, height, breadth, length of diagonals on each face, and the length of the main diagonal are all integers?

**update 2012-07-12** Since the question has returned to the front page, I'm taking the liberty to add some links that I found after Scott Carnahan's comments. (Scott deserves the credit, really, but I thought the links belonged in the answer rather than in the comments.)

- [On perfect cuboids](https://www.math.leidenuniv.nl/~rvl/ps/cuboids.pdf), by Ronald van Luijk, master thesis, 2000.

- [The surface parametrizing cuboids](http://arxiv.org/abs/1009.0388), by Michael Stoll and Damiano Testa, arXiv.org:1009.0388.