My Math Forum Qubeks logic game - is it solvable with random initial state?
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 September 18th, 2015, 05:21 AM #1 Newbie   Joined: Sep 2015 From: Denmark Posts: 3 Thanks: 0 Qubeks logic game - is it solvable with random initial state? 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 Qubeks 3D puzzle game 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 there. 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
 September 18th, 2015, 05:59 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,620 Thanks: 2609 Math Focus: Mainly analysis and algebra I should think that there would be parity issues so that at least half of the possible arrangements turn out to be impossible. You need to investigate the cyclic groups of different squares under different operations (rotations) to determine that sort of thing. Last edited by v8archie; September 18th, 2015 at 06:02 AM.
 September 18th, 2015, 06:21 AM #3 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms I don't understand the game well enough to give an answer, but the field you need is called group theory. Possibly enumerative combinatorics, especially the Pólya enumeration theorem, will also help.
 September 21st, 2015, 01:52 PM #4 Newbie   Joined: Sep 2015 From: Denmark Posts: 3 Thanks: 0 Qubeks - now with videos Thank you for your answers. They are really giving me some of the information I need. You had an interesting statement about the existence of solutions for the Qubeks puzzle, or better to say non-existence of solutions in certain cases. This somehow approves my approach of shuffling the cube by starting from a known solution and mixing randomly from there by following the game rules, in order to be sure that puzzle is solvable. I'm gonna stick to that! The proposed approach to proving solvability by group theory seems very challenging within the limited time I have, so I think I would prioritize finishing the product development and starting crowd-funding campaign on Kickstarter before I do a deep dive in the math theory. For those who didn't understand the Qubeks puzzle or didn't want to read the very long description on www.qubeks.com , I have put several videos of the prototype on the webpage > Qubeks 3D puzzle game <. I hope you will like it - feel free to comment! You can still play the 2D online Qubeks game free of charge in the user area, after registering for a free account. If the number of color states is 1, the game is solved by default so I disabled this choice. 2 and 3 states are not very challenging, so I would encourage you to try the puzzle with increased number of states and shuffling depth.
 October 2nd, 2015, 08:56 AM #5 Newbie   Joined: Sep 2015 From: Denmark Posts: 3 Thanks: 0 Hi, I am very excited to share some interesting news with the math community helping me out with the new puzzle: the Qubeks cube puzzle which I have introduced recently on www.qubeks.com has been listed as a project on the most popular crowd-funding website Kickstarter. In this way I intend to raise the awareness of the puzzle solvers around the world about this new logic game, but also get the necessary fundings to finish the product development and introduce the Qubeks to the market. You can visit and support the Qubeks project on Kickstarter using the following link: https://www.kickstarter.com/projects...be-puzzle-game I will of course encourage you to contribute to the Qubeks project and obtain some of the exciting Qubeks cube rewards on Kickstarter, which will definitely support towards reaching the funding goal of the campaign. But you can also help turning Qubeks cube puzzle into reality by spreading the word about Qubeks and the Kickstarter campaign throughout your social network. Together we can make it happen! Regards, Petar

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