  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 paper, we provide a $5$-dimensional
alternative to the well-known statement that it is not ever possible for a
knight to visit once every vertex of $C(3,k) := \{0,1,2\}^k$ by performing a
sequence of $3^k-1$ jumps of standard 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 extend a planar chess
piece to the third dimension and above. Our counterintuitive outcome follows
from the observation that we can alternatively define a $2$D knight as a piece
that moves from one square to another on the chessboard by covering a fixed
Euclidean distance of $\sqrt{5}$ so that also the statement of Theorem~3 in
[Erde, J., Gol{\'e}nia, B., \& Gol{\'e}nia, 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 a $2 \times
2 \times \cdots \times 2$ chessboard with at least $2^6$ cells is given.
Moreover, we show a classical closed knight's tour on $C(3,4)-\{(1,1,1,1)\}$
whose arrival is at a distance of $2$ from $(1,1,1,1)$, and we finally
construct closed Euclidean knight's tours on $\{0,1\}^k$ for each integer $k
\geq 6$.
