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Discussions on the mathematics of the cube
5x5 sliding puzzle can be solved in 205 movesSubmitted by Ben Whitmore on Fri, 01/26/2018 - 17:46.Consider a 5x5 sliding puzzle with the solved state
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24We can solve the puzzle in three steps. First solve 1,2,3,4,5,6,7, then solve 8,9,10,11,12,16,17,21,22, and finally solve the 8 puzzle in the bottom right corner. Step 1 requires 91 moves: depth new total 0 18 18 1 6 24 2 13 37 3 27 64 4 54 118 5 117 235 6 231 466 7 443 909 » 7 comments | read more
God's algorithm for the <2R, U> subset of the 4x4 cubeSubmitted by Ben Whitmore on Wed, 01/24/2018 - 22:00.Here I'm using sign notation, so 2R is the inner slice only. There are 10 edges, 10 centres in sets of 2, 2, 2 and 4, and 4 permutations of the corner pieces for a total of 4*10!*10!/(2!2!2!4!) = 274,337,280,000 positions. From July 4th 2017 to July 6th 2017, I ran a complete breadth first search of this puzzle in around 60 hours. God's number is 28.
Depth New Total 0 1 1 1 6 7 2 18 25 3 54 79 4 162 241 5 486 727 6 1457 2184 7 4360 6544 Do we have a 3x3x3 optimal solver for stm metric?Submitted by cubex on Thu, 08/10/2017 - 06:46.I thought it might be interesting to run an optimal solver using the slice turn metric (including face turns) on some pretty patterns. I don't remember anyone releasing an optimal solver that uses stm but maybe there is one by now?
Also is it true we don't know if using slice turns plus face turns could reduce God's Number from 20 to less than 20? More details about my new programSubmitted by Jerry Bryan on Thu, 06/08/2017 - 14:55.Introduction On 02/23/2016, I posted a message about a new program I had developed that had succeeded in enumerating the complete search space for the edges only group. It was not a new result because Tom Rokicki had solved the same problem back in 2004, but it was important to me because the problem served as a testbed for some new ideas I was developing to attack the problem of the full cube. I am now in the process of adapting the new program to include both edges and corners. In this message, I will include some additional detail about my new program that was not included in the first message. » 4 comments | read more
Pattern databases for the 5x5 sliding puzzleSubmitted by stannic on Sun, 04/02/2017 - 11:37.In 2002, Korf and Felner [1] used pattern databases to solve optimally 50 random instances of the 5x5 sliding puzzle. They used a static » 3 comments | read more
A cubic graph with cubic diameterSubmitted by stannic on Mon, 03/06/2017 - 03:53.The Fifteen puzzle is sometimes generalized to a sliding puzzle on an arbitrary simple connected graph G with n vertices in the following way. » 4 comments | read more
Is there a way to evenly distribute face turns for 12 flip?Submitted by cubex on Mon, 11/14/2016 - 22:10.Back in Jan 1995 Mike Reid found this process for the 12 flip:
R3 U2 B1 L3 F1 U3 B1 D1 F1 U1 D3 L1 D2 F3 R1 B3 D1 F3 U3 B3 U1 D3 24q This process has 24 q turns, so I'm wondering could there be a 24 q turn process that evenly distributes the turns so that each side turns 4 q? The idea just seemed elegant to me, 6 faces each turning 4 q turns. Mark Super Group Cosets of the Centers SubgroupSubmitted by B MacKenzie on Mon, 06/20/2016 - 06:52.Continuing my work with the 3x3x3 super group, I have written a coset solver for cosets of the pure center cubie subgroup. This subgroup is made up of the 2048 even parity center cubie configurations composed with the identity edge and corner configurations. The super group may be partitioned into cosets of the pure centers subgroup, g * [CTR] , where g is an element of the super group and [CTR] is the centers subgroup. The centers subgroup is a normal subgroup of the super group, g * [CTR] = [CTR] * g, and the standard cube group is the quotient group of the super group and the centers subgroup. » 10 comments | read more
Finally hitting depth 13 consistently with my 5x5x5 solverSubmitted by NoLongerUnsolve... on Fri, 06/17/2016 - 18:52.![]() It finally occurred to me why my hash table was sometimes not finding the shortest solutions 100% of the time. When I upgraded my computer to one with 128 GB of RAM, I had enough to load more positions into RAM. The number of hash table entries exceeded 4.2 billion, which is more than 32 bits. I never adjusted all of my access code to use 64-bit indices which were now necessary. All I had to do was change the data type, and, lo and behold, it found a 13-move solution to this arrangement, which previously it was reporting required 14 moves! A very happy day for me. Super Cube States at DepthSubmitted by B MacKenzie on Wed, 06/15/2016 - 10:36.Super Cube States at Depth I've been working with the super cube group (the 3x3x3 cube with center cubie orientation). There are two earlier threads here dealing with this group, Lower bounds for the 3x3x3 Super Group and Supergroup knowledge. Neither of these contain any states at depth information. To test my model I have performed a breadth first states at depth enumeration of the group out to depth 10 in the qtm. Can anybody confirm these numbers for me? » 3 comments | read more
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