Which Chessboards have a Closed Knight's Tour within the Rectangular Prism?

Abstract

A closed knight's tour of a chessboard uses legal moves of the knight to visit every square exactly once and return to its starting position. In 1991 Schwenk completely classified the $m\times n$ rectangular chessboards that admit a closed knight's tour. In honor of the upcoming twentieth anniversary of the publication of Schwenk's paper, this article extends his result by classifying the $i\times j\times k$ rectangular prisms that admit a closed knight's tour.