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The Knight's Tour problem consists of finding a Hamiltonian path for theknight on a given set of points so that the knight can visit exactly once everyvertex of the mentioned set. In the present paper we provide a $5$-dimensionalcounterexample to the well-known statement that it is not ever possible for aknight to visit once every vertex of $C(3,k):=\{\underbrace{\{0,1,2\} \times\{0,1,2\}\times \cdots \times \{0,1,2\}}_\textrm{k-times}\}$ by performing asequence of $3^k-1$ jumps of standard length, since the most accurate answer tothe original question actually depends on which mathematical assumptions we aremaking at the beginning of the game, when we decide to extend a planar chesspiece to the third dimension and above. Our counterintuitive outcome followsfrom the observation that we can alternatively define a $2$D knight as a piecethat moves from one square to another on the chessboard by covering a fixedEuclidean distance of $\sqrt{5}$ so that also the statement of Theorem 3 in[Erde, J., Gol\'enia, B., & Gol\'enia, S. (2012), The closed knight tourproblem in higher dimensions, The Electronic Journal of Combinatorics, 19(4),\#P9] does not hold anymore for such a Euclidean knight, as long as a $2 \times2 \times \cdots \times 2$ chessboard with at least $2^7$ cells 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 of $2$ from $(1,1,1,1)$,and then we show a closed Euclidean knight's tour on$\{\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\}\subseteq\mathbb{Z}^7$.

Proving the existence of Euclidean knight's tours on $n \times n \times \cdots \times n$ chessboards for $n < 4$