Created by Pavel Klavík
Mathematics behind puzzle with dice where dice are flipped along their sides on a grid. The goal is to get dice from one configuration into another one. It is discussed when it is precisely possible and when it is impossible.
For the count, it is possible to map a vertex to all others (8 possibilities) and then further rotate the cube along this vertex in three ways, so 24 in total.
Every rotation maps each pair of opposite vertices to a pair of opposite vertices. There are four pairs of opposite vertices which can be permuted arbitrarily. Therefore, the structure of rotations is isomorphic to S4.
Rotating the cube to the front.
Depicted green and purple in the picture.
These generators transform one parity into the other.
Rotating the cube to the right.
We have a die placed on a grid and we want to rotate it to neighboring grid fields and to return it back while changing its rotation. When it is possible? When the starting rotation and the target rotation belong to the same parity.
https://math.stackexchange.com/questions/502449/the-rotation-of-dice-on-a-grid
Since the number of rotations when the starting and target positions are the same is always even.
Corresponding to moving against red, along blue, along red and against blue.
For that, we consider two simple sequences of moves which clearly preserve the position. These sequences correspond to rotating a die in place along a diagonal going through two opposite vertices. There are four such rotations but we just need two to connect all 12 states.
We just look how they rotate the cube and they can be combined to get from every possible rotation to every other.
Corresponding to moving along red, along blue, against red, against blue.
Slightly more symmetric drawing.
Zero Escape: Virtue's Last Reward is a 2012 adventure game developed by Chunsoft.
https://store.steampowered.com/app/477740/Zero_Escape_The_Nonary_Games/
