moat problem

There are two versions of the moat-crossing problem, one geometric and one algebraic. The geometric moat problems asks for the widest moat Rapunzel can cross to escape if she has only two unit-length boards (and no means to nail or otherwise attach them together). More generally, what is the widest moat which can be crossed using n boards? Matthew Cook has conjectured that the asymptotic solution to this problem is O(n^(1/3)).