It’s well-known {that a} knight positioned on one sq. of a chessboard can get to some other sq., however a bishop can solely attain half the squares from a set start line. Another query on this website handled a brand new kind of chess piece with a special manner of shifting. I’m attempting to generalise all of those concepts.

A knight’s transfer takes you between reverse corners of a $2times3$ rectangle of squares. A bishop’s transfer takes you between reverse corners of a $2times2$ rectangle. So let’s outline a General (a brand new, generalised breed of chess piece) to be a bit that may transfer from its present place to the other nook of an $mtimes n$ rectangle. For what values of $m$ and $n$ is it attainable for a General to begin from one sq. and attain each different sq. on the chessboard?

(Possible additional generalisations embody altering the chessboard to $Mtimes N$ slightly than $8times8$; on the lookout for a ‘General’s tour’, the General visiting every sq. precisely as soon as; or on the lookout for a ‘General’s tour’ with the General visiting every sq. precisely as soon as and ending on the identical sq. it started on.)