Cube flips - Colegiul Național din Iași (Iași - Roumanie) Collège Sainte Véronique (Liège)

Titre du sujet
Cube flips
Établissement
Colegiul Național din Iași (Iași - Roumanie)
Année
2019-2020
Etablissement(s) jumelé(s)
Collège Sainte Véronique (Liège)
Résumé
On a chessboard (a lattice of 8x8 squares) place a cube (whose side is
equal to the side of the squares of the chessboard) in the lower left
corner. The cube faces are marked (say Up, Low, Left, Right, Front, Back) and the Up face is up and so on.
One can move the cube from a square to an adjacent square by tripping it over one of the bottom sides (this side is fixed and the cube is rotated 90 degrees around this side such that it lands in an adjacent square).
Can you find a sequence of trips that take the cube from the lower left
corner to the lower right corner and at the arrival it is in the same
position as in the start (the Up face is up, Low face is low etc.)
Generalizations: What about chessboards that have other dimensions than 8x8? What about other possible destinations on the chessboard or different initial positions?