No. Create a 3D coordinate system with one of the cube's vertices as origin and the 3 corresponding edges as directions.That coordinate system stays fixed even if the cube moves. The cube's 8 vertices have the standard coordinates for a 3D-cube ([0,0,0], [0,0,1], ...,[1,1,1]). If you roll the cube over one of its edges on that plane, that edge will remain in place and the other edges will come to rest still parallel to the coordinate sytem. So the 4 vertices that are exactly one edge away from the edge that is rolled over will have coordinates that are (+-1,0,0) or (0,+-1,0) or (0,0,+-1) from the corrresponding vertix that stays where it is. So these vertices (although they moved) still have integer coordinates. It is easy to see that the 2 remaining vertices also have interger coordinates after the move. So no matter how often you roll the cube, it's vertices will always have interger coordinates. The points with integer coordinates can be divided into 2 classes: Those whose sum of coordinates is even and those whose sum is is odd. Vertices joined by an edge of a cube with interger coordinates and edges parallel to the coordinate system have a sum of cordinates that differ by exactly one, hence those vertices belong to different classes. One can easily see that from knowing the class of one particular vertix the classes of all the other vertices of the cube follow: The three neighbours belong to the other class, their 3 neighbours (the opposite vertices of the 3 squares the original vertex belongs to) are in the original class and the last vertex (opposite to the original vertex) is again in the opposite class. In one move of rolling the cube over one of it's edges, the vertices of that edge stay put, so do not change from one class to another in the move. Since their class does not change, the class of the other vertices does not change as well in the move, since they are all determined by knowing the class of just one vertex. In other words, the class of a vertex does not change after any number of rolls. That's why the task asked about in the problem description is impossible: Same face up and rotation by 90° means that each of those top face vertices would have to change it's class during the series of moves, which is impossible. Note: The invariance disussed here holds true if there is not only one plane on which to roll, but many planes (or parts of planes) that are parallel to the xy,yz or xy coordinate planes and which are bounded by integer coordinates. In other words, if the "landscape" can be build from putting unit cubes together face by face.