Closed Knight's Tour on (m,m,r)-Ringboards

Wasupol Srichote, Ratinan Boonklurb, Sirirat Singhun

Abstract

It is well known that a legal knight's move is the resulting of moving two squares horizontally or vertically on the board and then turning and moving one square in the perpendicular direction. That is, if we start at (i,j), then the knight can move to one of eight squares: (i±2,j ±1) or (i±1,j ±2) (if exist). A closed knight’s tour is a legal knight's move that visit every squares on a given board exactly once and return to its starting position. A closed knight’s tour over a rectangular board or a three- dimensional cube have been studied widely. Some researchers turn their attention to investigate a closed knight’s tour over a ring board of width r, (m,n,r)-ringboard. For m,n > 2r, the (m,n,r)-ringboardis defined to be an m × n chessboard with the middle part missing and the rim contains r rows and rcolumns. In this paper, we give necessary and sufficient conditions for the (m, m, r)-ringboard to have a closed knight’s tour.