Qubeks logic game - is it solvable with random initial state

[Guest]

Hi,
I've been making a new logic game for the last few months - a 3D puzzle game called QUBEKS. You can find more details about it on the web page www.qubeks.com and you can play the 2D game in the user area, but I will also try to explain the game concept here.

Imagine a cube with six faces where each of the faces can be assigned a single color state from an ordered set of colors. Cube can be rotated around two horizontal axes x & y and one vertical axis z, perpendicular to the cube faces. When rotating the cube around one of the horizontal axes, the two perpendicular faces will change their color to the next one in the color set, if the face was rotated clockwise with regard to the axis of rotation, or will go back to the previous color if rotated counterclockwise. In this case there are always two opposite faces involved, which change color states in opposite directions.

Only exception is rotation around the vertical axis: in this case only the top face will change its color state, while the bottom face which cannot be seen because the cube is resting on it, will not change state.

When the color state reaches the beginning or the end of the ordered color set, it is going to loop correspondingly to the end or the beginning of the set.

The math problem I have is that I want to start the Qubeks game with randomized color states on all faces. However, at this moment I am not sure if I can always solve a randomly shuffled Qubeks game within finite number of moves, so in my algorithm (both in the physical prototype and in the online game on the website) I always start from a situation where all faces of the Qubeks cube have the same color state and I randomly shuffle it from here. In this way I am sure that there is a solution if I go backwards in the shuffling sequence and select the opposite moves (shuffling sequence can be seen in the online 2D Qubeks game).

What kind of math theory (game theory, set theory, combinatorial theory etc. ) should I use if I want to prove that with this set of rules I could solve all randomly selected combinations of color states on the Qubeks cube? Or should I go the "computational way" and try different combinations until I find one that I cannot solve with a computer, so that I prove at least the opposite - that not all combinations are solvable. But then how can I be sure that there is no solution, it is maybe only a weakness of the solver algorithm? Or I should just try all possible moves, but this might take long time and computational resources? I am a bit in doubt in which direction to go, because I would like to understand even better the math theory behind the Qubeks cube puzzle.

I hope somebody out there could help with the mathematics theory of the Qubeks game.

Best regards,
Petar

[Guest]

As with any new problem it's hard to say what approach will lead to the answer. From my own experience I solved the Rubik's cube in the following way:

First I played around with it and got to grips with the fundamentals of what was going on, then I started coming up with little procedures to do various useful things (like swap the positions of two side cubes, or change the orientation of some cubes without changing any positions). Eventually I had enough procedures that I could solve the cube. However, I wanted to be sure I could always solve it, there were configurations that I hadn't seen before that I had no way of dealing with (for example all the pieces in the right position and orientation except for one piece which is incorrectly orientated). I played with it for a while and never came across any problematic configurations and I eventually became convinced they couldn't appear. The next step was proving it. My approach was to come up with a set of invariants, I was inspired by the proof that the 14-15 puzzle is unsolvable see
https://en.wikipedia.org/wiki/15_puzzle
I managed to show that the problematic configurations would never be seen because they had a different set of invariants to the solvable configurations.

Presumably you could try doing something similar for your puzzle (perhaps do some computational searching to speed up finding the "useful little procedures"). It may be easier to just try all the moves computationally. If the number of colours isn't too large you could just list all the possible states and connect two states if there is a valid move that transforms one into the other, then check if all states are reachable or whether you have a disconnected set of states.

In terms of mathematics, I guess a bit of combinatorics will be helpful in calculating the size of search spaces (so you can tell if a computational search is a viable option). Probably group theory would be the most relevant area to look at, but if you are a novice I don't think it will be all that useful.

Hope this helped,

R. Baber.