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
boards? Matthew Cook has conjectured that the asymptotic solution to this problem
is
(Cook 1997).
The algebraic moat-crossing problem asks if it is possible to walk to infinity on the real line using only steps of bounded lengths and
steps on the prime numbers. The answer is negative (Gethner et al. 1998).
However, the Gaussian moat problem that asks whether it is possible to walk to infinity
in the Gaussian integers using the Gaussian
primes as stepping stones and taking steps of bounded length is unresolved. Gethner
et al. (1998) show that a moat of width exists.
boards? Matthew Cook has conjectured that the asymptotic solution to this problem
is 
(Cook 1997).
exists.
