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Discussions on the mathematics of the cube
Watermelon Rubik's CubeSubmitted by Jerry Bryan on Mon, 08/10/2009 - 11:07.I trust that I may be forgiven for being slightly off topic. After all, a watermelon Rubik's cube is not very mathematical. But still, it's an interesting concept.
http://www.watermelon.org/FeaturedRecipe.asp I am in no way connected with the National Watermelon Promotion Board. FTM Antipodes of the Edge GroupSubmitted by Bruce Norskog on Tue, 07/21/2009 - 11:23.I have done my own independent breadth-first search of the edge group using the face-turn metric. I used symmetry/antisymmetry equivalence classes to reduce the number of elements in the search space. I confirm the "Unique mod M+inv" values for this group/metric that Rokicki reported in 2004. I reduced the "coordinate space" for the search to 5022205*2048=10285475840 elements by using symmetry/antisymmetry equivalence classes of the edge permutation group. (This gives a much more compact overall coordinate space than using an edge orientation sym-coordinate, at a cost of more time required to calculate representative elements. This allowed me to keep track of reached equivalence classes with a ~1.3 GB bitvector in RAM and 5022205 KB disk files to keep track of distances.) » 8 comments | read more
God's Algorithm out to 13f*Submitted by rokicki on Wed, 07/15/2009 - 14:51.Just finished running out to a distance of 13 in the face turn metric.
First, the positions at exactly that distance: d mod M + inv mod M positions -- ------------- -------------- --------------- 0 1 1 1 1 2 2 18 2 8 9 243 3 48 75 3240 4 509 934 43239 5 6198 12077 574908 6 80178 159131 7618438 7 1053077 2101575 100803036 » 8 comments | read more
God's Algorithm out to 14q*Submitted by rokicki on Wed, 06/24/2009 - 09:48.I've computed the count of positions out to 14 quarter turns.
First, positions at exactly the given distance: d mod M + inv mod M positions -- ------------ ------------- -------------- 0 1 1 1 1 1 1 12 2 5 5 114 3 17 25 1068 4 130 219 10011 5 1031 1978 93840 6 9393 18395 878880 7 86183 171529 8221632 8 802788 1601725 76843595 » 19 comments | read more
God's Algorithm out to 12f*Submitted by rokicki on Tue, 06/23/2009 - 10:22.I just completed exploring all positions of the cube
out to depth 12 in the face turn metric.
The first table is the count of positions with exactly the given depth. d mod M + inv mod M positions -- ------------ ------------ -------------- 0 1 1 1 1 2 2 18 2 8 9 243 3 48 75 3240 4 509 934 43239 5 6198 12077 574908 6 80178 159131 7618438 7 1053077 2101575 100803036 » 5 comments | read more
Twenty-Nine QTM Moves SufficeSubmitted by rokicki on Mon, 06/15/2009 - 20:35.With 25,000 QTM cosets proved to have a distance of 25 or less,
we have shown that there are no positions that require 30 or more quarter turns to solve. All these sets were run on my personal machines, mostly on a new single i7 920 box. These sets cover more than 4e16 of the total 4e19 cube positions, when inverses and symmetries are taken into account, and no new distance-26 position was found. This indicates that distance-26 positions are extremely rare; I conjecture the known one is the only distance-26 position. In order to take the step to a proof of 28, I would need a couple » 16 comments | read more
Inappropriate linksSubmitted by cubex on Sun, 05/10/2009 - 21:00.Any inappropriate link (i.e. not math and/or puzzle related) will be deleted. I'd like to keep the forum completely free of ads with the sole exception of ads for books about puzzles, or at least limited to materials appropriate for the site.
For newbies or younger readers: If you find some of the posts are too difficult to understand please go ahead and ask questions! The people here are willing to help explain things. Mark Interesting Problem/Puzzle/GameSubmitted by dukerox7593 on Sun, 05/03/2009 - 22:32.ok here is the game:
your opponent has a secret number that is 4 digits long. the digits are 0-9 and no digit can be repeated in the number (in other words all 4 numbers are different) examples: 1234, 1948, 4950 you have to guess the number when you guess a number: your opponent gives you back 2 numbers (x,y) number x is the amount of numbers in the right place number y is the amount of numbers right but in the wrong place anyone have an algorithm to solve this problem using the 2 numbers given back? subgroup enumerationSubmitted by B MacKenzie on Sun, 04/12/2009 - 15:07.I've been playing around with the Rubik's cube subgroup generated by the turns: (U U' D D' L2 R2 F2 B2) which I refer to as the D4h cube subgroup after the symmetry invariance of the generator set. This, I believe, is the subgroup employed by Kociemba in his two phase algorithm. Anyway, I have performed a partial enumeration of the subgroup and its coset space. I was wondering if anyone might be able to confirm these numbers as a check on my methodology. Six Face/D4h Coset Enumeration (q turns) Depth Cosets Total 0 1 1 1 4 5 » 9 comments | read more
Twenty-Five Random Cosets in the Quarter Turn MetricSubmitted by rokicki on Sun, 03/01/2009 - 13:42.As the next step in my exploration of the quarter turn metric, after
finishing a proof of an upper bound of 30, I decided to run 25 cosets
all the way (until I have optimal solutions for all positions).
Unlike my runs in the half turn metric in November of 2008, I went
ahead and made the search phase run in parallel. In addition, I
acquired a newer, faster machine (a Dell Studio XPS with an Intel i7
920 processor).
I decided to run the same 25 cosets I ran for the half turn metric. The results are summarized in the table below. Sequence 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 » 15 comments | read more
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