  We present properties and invariants of Hamiltonian circuits in rectangular
grids. It is proved that all circuits on a $2n \times 2n$ chessboard have at
least $4n$ turns and at least $2n$ straights if $n$ is even and $2n+2$
straights if $n$ is odd. The minimum number of turns and straights are
presented and proved for circuits on an $n \times (n+1)$ chessboard. For the
general case of an $n \times m$ chessboard similar results are stated but not
all proofs are given.
