Proving the existence of Euclidean knight's tours on n x n x<middle dot><middle dot><middle dot> x n chessboards for n < 4

The Knight's Tour problem consists of finding a Hamiltonian path for the knight on a given set of points so that the knight can visit exactly once every vertex of the mentioned set. In the present, we provide a5-dimensional alternative to the well-known statement that it is not ever possible for a knight to visit once every vertex of C(3, k) :={{0,1,2}x{0,1,2}x<middle dot><middle dot><middle dot>x{0,1,2}|{z}} by performing a sequence of3(k)-1jumps of standard k-times length, since the most accurate answer to the original question actually depends on which mathematical assumptions we are making at the beginning of the game when we decide to extenda planar chess piece to the third dimension and above. Our counterintuitive outcome follows from the observation that we can alternatively define a2D knight as a piece that moves from one square to another on the chessboard by covering a fixed Euclidean distance of root 5 so that also the statement of Theorem 3 in [Erde, J., Golenia, B., & Golenia, S. (2012), The closed knight tour problem in higher dimensions, The Electronic Journal of Combinatorics, 19(4), #P9] does not hold anymore for such a Euclidean knight, as long as a2x2x<middle dot><middle dot><middle dot>x2chessboard with at least27cells is given. Moreover, we construct a classical closed knight's tour on C(3,4)-{(1,1,1,1)}whose arrival is at a distance of2from(1,1,1,1), and then we show a closed Euclidean knight's tour on{{0,1}x{0,1}x{0,1}x{0,1}x{0,1}x{0,1}x{0,1}}subset of Z(7).