Some Forbidden Rectangular Chessboards with Generalized Knight’s Moves

Abstract

The m × n chessboard is an array with squares arranged in m rows and n columns. An (a,b)-knight’s move or generalized knight’s move is a move from one square to another by moving a knight passing a squares vertically or asquares horizontally and then passing b squares at 90 degrees angle. A closed (a, b)-knight’s tour is an (a, b)-knight’s move such that the knight lands on every square on the m × n chessboard once and returns to its starting square. In this paper, we show that (i) the (a + b) × n chessboard admits no closed (a, b)-knight’s tours if n ∈ [2b + 1, 4b − 1] where 1 ≤ a < b or if n ∈ [4b + 1, 5b] where 1 ≤ a < b < 2a, and (ii) the (2a + 1) × n chessboard admits no closed (a, a + 1)-knight’s tours if n = 4a+4 where a ≥ 1, or if n = 6a+6 where a > 3, or if n = 6a+8 where a > 3.