Knight's tours on rectangular chessboards using external squares (Q2018959)

Summary: The classic puzzle of finding a closed knight's tour on a chessboard consists of moving a knight from square to square in such a way that it lands on every square once and returns to its starting point. The \(8 \times 8\) chessboard can easily be extended to rectangular boards, and \textit{A. Schwenk} [Math. Mag. 64, No. 5, 325--332 (1991; Zbl 0761.05041)] characterized all rectangular boards that have a closed knight's tour. More recently, \textit{J. Demaio} and \textit{T. Hippchen} [Math. Mag. 82, No. 3, 219--225 (2009; Zbl 1227.97064)] investigated the impossible boards and determined the fewest number of squares that must be removed from a rectangular board so that the remaining board has a closed knight's tour. In this paper we define an extended closed knight's tour for a rectangular chessboard as a closed knight's tour that includes all squares of the board and possibly additional squares beyond the boundaries of the board and answer the following question: how many squares must be added to a rectangular chessboard so that the new board has a closed knight's tour?