Recent content by Herbert Kociemba

For a even better approximation it is possible to show that the absolute difference between the linear function 3/Log[3/2]*(N-1) -3/2 and the asymptotic branching factor goes to 0 (with order O(1/N)). Here is a table of 3/Log[3/2]*(N-1) -3/2 for N= 3..10: {13.2978, 20.6967, 28.0956...

From ChatGPT I got a really nice hint how to find the asymptotic branching factor for a NxNxN cube given my generating function for the OBTM which is gf[n_, x_] := 3/(6 - 4 (3 x + 1)^(n - 1)) - 1/2 The asymptotic branching factor is the quotient of the number of algorithms of length k+1...

I think I did it in a similar way. For Gigaminx for example I used the 12 variables m1(1), m1(2), m2a(1,1), m2a(1,2), m2a(2,2), m2b(1,1), m2b(1,2), m2b(2,2), m3(1,1,1), m3(1,1,2), m3(1,2,2), m3(2,2,2).

I agree with these numbers of cases. Took me some time to write a program which generate the corresponding equation systems, wonder how you manged to do this so fast. My results for Teraminx gave for the coefficients for the linear recursion {108, -1728, -25344, -161280, -645120, -1720320...

You can compute it online for example , for GigaMinx here. The degree of the reccurence relation is the maximal number of commuting faces which is 3 for MegaMinx and 6 for GigaMinx. So such a relation exists and can be determined uniquely. But indeed I let the software do the job for me in...

Yes and no. The program has no symmetry reduction implemented, that's why already for Teraminx the program is too slow to give results within a reasonable time. Teraminx has 8708 different "categories", up to 9 moves may commutate with each other. Gigaminx has "only" 900 categories which of...

I cannot prove this but it is highly unlikely that a distance 20 position has a unique solution maneuver. In Tomas Rokicki's database are estimated about 20% of all 20 move positions and for almost all of them all neighbors (= one move distance, so each position has 18 neigbors) are 19 move...

Since we do not know the configurations which are at distance 20 from solved this approach does not work.

A proof would probably mean that you have to solve all 4.3 10^19 different configurations (reduced by 48 symmetries) within 20 moves. I do not think that this open question is important enough to justify an effort which would be presumably higher than that for the God's number proof since the...

I just wrote a general framework in Mathematica which generates the number of "canonical sequences" for any arbitrary undirected graph. For Gigaminx the graph has 24 vertices (one for each face or slice) and 120 edges and can be easily derived from the graph for the Megaminx which is the same as...

This is the right way. But which symmetries do you use? For corner orientation co you may use only those 16 symmetries (or 8 if you do not use reflections) which fix the UD-axis. Without reflections these are 8 rotations. The same holds for UD slice coordinate ud. But edge orientations is even...

This cube has no symmetry. Indeed Cube Explorer checks symmetries on the cubie level. Cube moves like R, D2 etc are handled in the same way as the 48 possible cube rotations and reflections. All are represented in the same format as a cube state on the cubie lev, though for cube reflections...

It would be much easier for me and less error-prone if you enter the cube you have in mind into Cube Explorer with the facelet editor and supply me with a generator.

If you like such patterns you can easily find and create them with Cube Explorer. The screen shot shows how to use it to create the pattern above. And as you can see there are two different patterns (1. and 2.) which have the design above.