Here a "tour" is a sequence of moves that visit each square of the board. A square can be visited more than once if necessary.

For example, in a $8\times 8$ chess board, a knight which makes $2\times1$ type moves can tour the whole board from an arbitrary starting point, but a bishop which makes $1\times 1$ (or $k\times k$) type moves can not. What condition must $M,N,m,n$ satisfy so that a full board tour from any starting point is possible?

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    $\begingroup$ I assume that the piece which moves "along" the diagonals of a rectangle jumps from corner to corner without "visiting" the squares in between ? The standard name for that piece is $(m,n)$-leaper. "Tour" is the standard name for a sequence of moves that visits each square exactly once, as in the classic problem of the knight's tour. What (I think) you are asking, in plain language: On which rectangular chessboards can an $(m,n)$-leaper get from any square to any other square? $\endgroup$ – bof Jul 24 at 8:39
  • $\begingroup$ Since you allow squares to be visited repeatedly, there is no need to say "from any starting point"; if you can do it from one starting point, you can do it from any starting point. $\endgroup$ – bof Jul 24 at 8:48
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    $\begingroup$ You may as well assune that $m,n$ are relatively prime, and not both odd; otherwise your "tour" wil be impossible no matter how big $M$ and $N$ are. $\endgroup$ – bof Jul 24 at 8:51
  • $\begingroup$ @bof Not in every case. A bishop, for example can reach every light square from every other light square, but not from a dark square. $\endgroup$ – Peter Jul 24 at 15:47
  • $\begingroup$ @Peter What does that have to do with my comment that, if you can "tour the whole board" from one starting point, you can do it frim any starting point? A bishop cannot tour the whole board, no matter where it starts. $\endgroup$ – bof Jul 24 at 20:50

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