Generalised knight's tours (Q405114)

Summary: The problem of existence of closed knight's tours in \([n]^d\), where \([n]=\{0, 1, 2, \dots, n-1\}\), was recently solved by \textit{J. Erde} et al. [Electron. J. Comb. 19, No. 4, Research Paper P9, 17 p. (2012; Zbl 1266.05078)]. They raised the same question for a generalised, \((a, b)\) knight, which is allowed to move along any two axes of \([n]^d\) by \(a\) and \(b\) unit lengths respectively.{ }Given an even number \(a\), we show that the \([n]^d\) grid admits an \((a, 1)\) knight's tour for sufficiently large even side length \(n\).