  For a Hamilton cycle in a rectangular $m \times n$ grid, what is the greatest
number of turns that can occur? We give the exact answer in several cases and
an answer up to an additive error of $2$ in all other cases. In particular, we
give a new proof of the result of Beluhov for the case of a square $n \times n$
grid. Our main method is a surprising link between the problem of 'greatest
number of turns' and the problem of 'least number of turns'.
