]> http://cubezzz.homelinux.org/drupal To promote mathematical discussions about the Rubik's Cube and related puzzles. en http://cubezzz.homelinux.org/drupal/?q=node/view/166 <p> I've been reading a little bit about graph theory, and I wish I knew more about it.&nbsp; So pardon what may perhaps be a bit of naiveté about graphs on my part. </p> <p> I suspect that everyone who reads this forum is familiar with Cayley graphs and how they relate to Rubik's cube space.&nbsp; One of the best references in this regard may be found at <a href="http://www.jaapsch.net/puzzles/cayley.htm">http://www.jaapsch.net/puzzles/cayley.htm</a>&nbsp; I also suspect who everyone reads this forum is familiar with the concept of doing a breadth first search while storing all the results as a way of investigating a space such as the Cayley graph for Rubik's cube. </p> Sat, 14 Nov 2009 14:27:36 -0500 http://cubezzz.homelinux.org/drupal/?q=node/view/165 As you know one of the breakthrough of cube computing is Silviu's successful depth calculation of all symmetrical positions. This breakthrough used a two phase algorithm that has the symmetrical positions as its target group. Regular solvers will simply stop once they hit the position they want to solve but Silviu's idea is to never stop until all positions in the target group are hit... This way of solving is very fast and it is what Rokiki has used to calculate the "group that fixes the edges".<br /> <br /> My question is can this be applied to arbitrary target groups whose elements share something in common? for example let's say my target group is some conjugacy class of the cube. Thu, 12 Nov 2009 09:06:11 -0500 http://cubezzz.homelinux.org/drupal/?q=node/view/164 I modified kociemba's optimal solver so that it calculates the depth of sequences in a Depth-first search manner. <br> <br> The moves order I used is URFDLBU'R'F'D'L'B'. <br> <br> Here are the initial results after an hour or so of running : <PRE> ./optiqtm initializing memory. initializing tables......................................................... loading pruning table (538 MB) from disk................ U -1 UU --2 UUR ---3 UURU ----4 UURUU -----5 UURUUR ------6 UURUURU -------7 UURUURUU --------8 UURUURUUR ---------9 UURUURUURU ----------10 UURUURUURUU -----------11 UURUURUURUUR ------------12 UURUURUURUURU -------------13 UURUURUURUURUU -------------14 UURUURUURUURUUR ---------------15 UURUURUURUURUURU ----------------16 UURUURUURUURUURUU ---------------15 UURUURUURUURUURUR ---------------15 UURUURUURUURUURUF ---------------15 UURUURUURUURUURUD ---------------15 UURUURUURUURUURUL -----------------17 UURUURUURUURUURULU ----------------16 UURUURUURUURUURULR ----------------16 UURUURUURUURUURULF ------------------18 UURUURUURUURUURULFU -----------------17 UURUURUURUURUURULFR -----------------17 UURUURUURUURUURULFF -----------------17 UURUURUURUURUURULFD -----------------17 UURUURUURUURUURULFL -----------------17 UURUURUURUURUURULFB -------------------19 UURUURUURUURUURULFBU ------------------18 UURUURUURUURUURULFBR ------------------18 UURUURUURUURUURULFBF ------------------18 UURUURUURUURUURULFBD --------------------20 UURUURUURUURUURULFBDU -------------------19 UURUURUURUURUURULFBDR -------------------19 UURUURUURUURUURULFBDF -------------------19 UURUURUURUURUURULFBDD -------------------19 UURUURUURUURUURULFBDL -------------------19 UURUURUURUURUURULFBDB -------------------19 UURUURUURUURUURULFBDU' -------------------19 UURUURUURUURUURULFBDR' -------------------19 UURUURUURUURUURULFBDF' -------------------19 UURUURUURUURUURULFBDL' -------------------19 UURUURUURUURUURULFBDB' -------------------19 UURUURUURUURUURULFBL ------------------18 UURUURUURUURUURULFBB ------------------18 UURUURUURUURUURULFBU' ------------------18 UURUURUURUURUURULFBR' --------------------20 UURUURUURUURUURULFBR'U -------------------19 UURUURUURUURUURULFBR'F -------------------19 UURUURUURUURUURULFBR'D -------------------19 UURUURUURUURUURULFBR'L -------------------19 UURUURUURUURUURULFBR'B ---------------------21 UURUURUURUURUURULFBR'BU --------------------20 UURUURUURUURUURULFBR'BR --------------------20 UURUURUURUURUURULFBR'BF --------------------20 UURUURUURUURUURULFBR'BD --------------------20 UURUURUURUURUURULFBR'BL --------------------20 UURUURUURUURUURULFBR'BB --------------------20 UURUURUURUURUURULFBR'BU' --------------------20 UURUURUURUURUURULFBR'BR' --------------------20 UURUURUURUURUURULFBR'BF' --------------------20 UURUURUURUURUURULFBR'BD' --------------------20 UURUURUURUURUURULFBR'BL' --------------------20 UURUURUURUURUURULFBR'U' -------------------19 UURUURUURUURUURULFBR'R' -------------------19 UURUURUURUURUURULFBR'F' -------------------19 UURUURUURUURUURULFBR'D' -------------------19 UURUURUURUURUURULFBR'L' -------------------19 UURUURUURUURUURULFBR'B' -------------------19 UURUURUURUURUURULFBF' ------------------18 UURUURUURUURUURULFBD' ------------------18 UURUURUURUURUURULFBL' --------------------20 UURUURUURUURUURULFBL'U -------------------19 UURUURUURUURUURULFBL'R -------------------19 UURUURUURUURUURULFBL'F ---------------------21 UURUURUURUURUURULFBL'FU --------------------20 UURUURUURUURUURULFBL'FR --------------------20 UURUURUURUURUURULFBL'FF --------------------20 UURUURUURUURUURULFBL'FD --------------------20 UURUURUURUURUURULFBL'FL --------------------20 UURUURUURUURUURULFBL'FB --------------------20 UURUURUURUURUURULFBL'FU' ----------------------22 UURUURUURUURUURULFBL'FU'U ---------------------21 UURUURUURUURUURULFBL'FU'R ---------------------21 UURUURUURUURUURULFBL'FU'F ---------------------21 UURUURUURUURUURULFBL'FU'D ---------------------21 UURUURUURUURUURULFBL'FU'L ---------------------21 UURUURUURUURUURULFBL'FU'B ---------------------21 UURUURUURUURUURULFBL'FU'U' ---------------------21 UURUURUURUURUURULFBL'FU'R' ---------------------21 UURUURUURUURUURULFBL'FU'F' ---------------------21 UURUURUURUURUURULFBL'FU'D' ---------------------21 UURUURUURUURUURULFBL'FU'L' ---------------------21 UURUURUURUURUURULFBL'FU'B' ---------------------21 UURUURUURUURUURULFBL'FR' --------------------20 UURUURUURUURUURULFBL'FD' --------------------20 UURUURUURUURUURULFBL'FL' --------------------20 UURUURUURUURUURULFBL'FB' --------------------20 UURUURUURUURUURULFBL'D ---------------------21 UURUURUURUURUURULFBL'DU --------------------20 UURUURUURUURUURULFBL'DR --------------------20 UURUURUURUURUURULFBL'DF --------------------20 UURUURUURUURUURULFBL'DD --------------------20 UURUURUURUURUURULFBL'DL --------------------20 UURUURUURUURUURULFBL'DB --------------------20 UURUURUURUURUURULFBL'DU' --------------------20 UURUURUURUURUURULFBL'DR' --------------------20 UURUURUURUURUURULFBL'DF' --------------------20 UURUURUURUURUURULFBL'DL' --------------------20 UURUURUURUURUURULFBL'DB' --------------------20 UURUURUURUURUURULFBL'B -------------------19 UURUURUURUURUURULFBL'U' -------------------19 UURUURUURUURUURULFBL'R' -------------------19 UURUURUURUURUURULFBL'F' -------------------19 UURUURUURUURUURULFBL'D' -------------------19 UURUURUURUURUURULFBL'L' -------------------19 UURUURUURUURUURULFBL'B' -------------------19 UURUURUURUURUURULFU' -----------------17 UURUURUURUURUURULFR' -----------------17 UURUURUURUURUURULFD' -----------------17 UURUURUURUURUURULFL' -------------------19 UURUURUURUURUURULFL'U ------------------18 UURUURUURUURUURULFL'R ------------------18 UURUURUURUURUURULFL'F --------------------20 UURUURUURUURUURULFL'FU -------------------19 UURUURUURUURUURULFL'FR -------------------19 UURUURUURUURUURULFL'FF -------------------19 UURUURUURUURUURULFL'FD -------------------19 UURUURUURUURUURULFL'FL ---------------------21 UURUURUURUURUURULFL'FLU --------------------20 UURUURUURUURUURULFL'FLR --------------------20 UURUURUURUURUURULFL'FLF --------------------20 UURUURUURUURUURULFL'FLD --------------------20 UURUURUURUURUURULFL'FLL --------------------20 UURUURUURUURUURULFL'FLB --------------------20 UURUURUURUURUURULFL'FLU' --------------------20 UURUURUURUURUURULFL'FLR' --------------------20 UURUURUURUURUURULFL'FLF' --------------------20 UURUURUURUURUURULFL'FLD' --------------------20 UURUURUURUURUURULFL'FLB' --------------------20 UURUURUURUURUURULFL'FB -------------------19 UURUURUURUURUURULFL'FU' -------------------19 UURUURUURUURUURULFL'FR' -------------------19 UURUURUURUURUURULFL'FD' -------------------19 UURUURUURUURUURULFL'FL' -------------------19 UURUURUURUURUURULFL'FB' -------------------19 UURUURUURUURUURULFL'D ------------------18 UURUURUURUURUURULFL'B ------------------18 UURUURUURUURUURULFL'U' ------------------18 UURUURUURUURUURULFL'R' ------------------18 UURUURUURUURUURULFL'F' ------------------18 UURUURUURUURUURULFL'D' ------------------18 UURUURUURUURUURULFL'L' ------------------18 UURUURUURUURUURULFL'B' ------------------18 UURUURUURUURUURULFB' -----------------17 UURUURUURUURUURULD ------------------18 UURUURUURUURUURULDU -----------------17 UURUURUURUURUURULDR -------------------19 UURUURUURUURUURULDRU --------------------20 UURUURUURUURUURULDRUU -------------------19 UURUURUURUURUURULDRUR -------------------19 UURUURUURUURUURULDRUF -------------------19 UURUURUURUURUURULDRUD -------------------19 UURUURUURUURUURULDRUL -------------------19 UURUURUURUURUURULDRUB -------------------19 UURUURUURUURUURULDRUR' -------------------19 UURUURUURUURUURULDRUF' -------------------19 UURUURUURUURUURULDRUD' -------------------19 UURUURUURUURUURULDRUL' -------------------19 UURUURUURUURUURULDRUB' -------------------19 UURUURUURUURUURULDRR ------------------18 UURUURUURUURUURULDRF ------------------18 UURUURUURUURUURULDRD ------------------18 UURUURUURUURUURULDRL ------------------18 UURUURUURUURUURULDRB ------------------18 UURUURUURUURUURULDRU' ------------------18 UURUURUURUURUURULDRF' ------------------18 UURUURUURUURUURULDRD' ------------------18 UURUURUURUURUURULDRL' ------------------18 UURUURUURUURUURULDRB' ------------------18 UURUURUURUURUURULDF -----------------17 UURUURUURUURUURULDD -----------------17 UURUURUURUURUURULDL -----------------17 UURUURUURUURUURULDB -----------------17 UURUURUURUURUURULDU' -----------------17 UURUURUURUURUURULDR' -----------------17 UURUURUURUURUURULDF' -------------------19 UURUURUURUURUURULDF'U ------------------18 UURUURUURUURUURULDF'R </PRE> Sun, 01 Nov 2009 18:22:57 -0500 http://cubezzz.homelinux.org/drupal/?q=node/view/163 I was wondering if the number of commutator elements of the cube is known?<br /> <br /> Commutator elements are of the form ABA'B' where A and B are some sequences.<br /> <br /> It seems that the subgroup generated by the commutator elements is half the size of the cube, but is it the case that every element of the commutator subgroup is a commutator element? if not how many are they and how far they are from solved? Sat, 31 Oct 2009 19:35:14 -0400 http://cubezzz.homelinux.org/drupal/?q=node/view/161 I was reading the following :<br /> <br /> http://en.wikipedia.org/wiki/Conjugacy_class<br /> <br /> and wondered if the number of conjugacy classes of the cube is known? Sun, 18 Oct 2009 06:03:06 -0400 http://cubezzz.homelinux.org/drupal/?q=node/view/160 Ok, I think you all can solve the cube without thinking. So here is a new challenge. First scramble the cube. Next, make the 2x2x2 cube on your 3x3x3 cube. After that, solve the rest of the cube by using three faces only. I can tell you that I know plenty of tricks to solve the cube, but only just enough remain to when you are restricted to three mutually adjacent faces. Tue, 06 Oct 2009 12:33:16 -0400 http://cubezzz.homelinux.org/drupal/?q=node/view/158 Let's assume that there is a linear formula that gives us P(n) using P(k) where 0 <= k <= n-1. P(n) is the number of positions at depth n. <BR> <BR> We have then the following formula: <BR> <BR> P(n) = sum(R(k)*P(n-k)) for 1 <= k <= n <BR> <BR> Calculating R(k) using rokiki's results we deduce : <BR> <BR> <pre> 1 12 = 12*1 2 114 = 12*12 -30*1 3 1068 = 12*114 -30*12 +60*1 4 10011 = 12*1068 -30*114 +60*12 -105*1 Mon, 28 Sep 2009 09:13:03 -0400 http://cubezzz.homelinux.org/drupal/?q=node/view/157 <p> I've been thinking about writing a program to calculate and count duplicate positions - roughly speaking, those positions that are half way through an identity.&nbsp; What I have in mind will probably be a more time consuming program to write than I would prefer.&nbsp; So I wonder if I could ask Herbert Kociemba and/or mdlazreg to post a little something about the programs they have already written to find identities.&nbsp; It may well be that there is a much simpler approach to calculating duplicate positions than what I have in mind. </p> <p> What I have in mind is an iterative deepening depth first search beginning at the Start position.&nbsp; If that's all I did, the search would simply count 12ⁿ maneuvers for each distance from n, and it would not extract any useful information about how many duplicate positions there are for each n.&nbsp; To solve these problems, I propose to store all the duplicate positions and not to store those positions that are not duplicate.&nbsp; This would be for the quarter turn metric.&nbsp; The program I have in mind would not be able to handle the face turn metric. </p> Sat, 26 Sep 2009 17:45:38 -0400 http://cubezzz.homelinux.org/drupal/?q=node/view/156 In FTM the complete knowledge of the distribution of the symmetric subgroups (first table of http://kociemba.org/symmetric2.htm ) lets us not only give the odd and even entries in the distance table but also the distances mod 48, because all unsymmetric position contribute with a multiple of 48. So we get <pre> distance positions mod 48 0 1 1 18 2 3 3 24 4 39 5 12 6 22 7 12 8 40 9 3 10 4 11 20 Sat, 26 Sep 2009 08:32:28 -0400 http://cubezzz.homelinux.org/drupal/?q=node/view/154 Here's an easy puzzle for Rubik's Cube: What's the chromatic number of the Cayley graph for the quarter turn metric? <p> Here's a slightly harder puzzle: What's the chromatic number of the Cayley graph for the half turn metric? If you can't figure it out, can you figure out an upper bound? A lower bound? <p> This was discussed on speedsolving.com before, but I think it's a good enough puzzle to present here as well. Sat, 19 Sep 2009 15:22:12 -0400 http://cubezzz.homelinux.org/drupal/?q=node/view/153 I've finally managed to compute God's Algorithm out to 15q*. This took longer than I expected; I had difficulties using multiple cores because occasionally the memory consumption of the concurrently-calculated cosets would exceed my physical RAM; even though this was rare, it happened frequently enough to completely stall the computation. Also, the way memory was allocated and freed led to pretty intense memory fragmentation. <p> In any case, it is finally done; here are the results. First we have positions at exactly that depth: <pre> d mod M + inv mod M positions Sat, 19 Sep 2009 14:56:32 -0400 http://cubezzz.homelinux.org/drupal/?q=node/view/152 I wrote a program that counts cube positions by taking into account only the identities of length 4 and the identities of length 12. The results of this program are in the second column below: <BR> <BR> <PRE> d positions I4 positions I4&I12 positions ALL -- ------------ ---------------- -------------- 0 1 1 1 1 12 12 12 2 114 114 114 Tue, 15 Sep 2009 08:55:23 -0400 http://cubezzz.homelinux.org/drupal/?q=node/view/151 Sorry folks, I've been very busy lately and I just noticed the mysql database was badly corrupted on Sept. 6th, 2009. The database is updated daily, but all the backups on Sept. 6th and after are unusable. I think the only post lost was Tom Rokicki's. <br /> <br /> I try my best to make sure everything is working but this one slipped through the cracks. Somehow the mysql database ballooned in size to over 2 gigabytes. After that happened the subsequent databases were not backed up correctly.<br /> <br /> It would be a good idea for any posts to be buffered in some way before uploading to the forum, especially long ones. Wed, 09 Sep 2009 18:21:45 -0400 http://cubezzz.homelinux.org/drupal/?q=node/view/150 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.<br /> <br /> http://www.watermelon.org/FeaturedRecipe.asp<br /> <br /> I am in no way connected with the National Watermelon Promotion Board. Mon, 10 Aug 2009 12:07:02 -0400 http://cubezzz.homelinux.org/drupal/?q=node/view/147 <p>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.</p> <p>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.)</p> Tue, 21 Jul 2009 12:23:39 -0400