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Domain of the Cube Forum - Cube lovers returns
http://cubezzz.dyndns.org/drupal
To promote mathematical discussions about the Rubik's Cube and related puzzles.enPattern databases for the 5x5 sliding puzzle
http://cubezzz.dyndns.org/drupal/?q=node/view/555
<p>In 2002, Korf and Felner [1] used pattern databases to solve optimally 50 random instances of the 5x5 sliding puzzle. They used a static <nobr>6+6+6+6</nobr> partition of tiles (described below), along with its reflection in the main diagonal. In a 2004 paper by Felner, Korf and Hanan [2], the authors describe in a footnote the way they handled the empty tile as 'not trivial'; the empty tile was taken into account when precomputing the database, but then the tables were compressed by discarding the information about the empty location. The authors do not provide the distribution of values from the pattern databases, but do provide maximum values and the number of nodes generated when solving random instances of the 5x5 sliding puzzle.</p>Sun, 02 Apr 2017 12:37:51 -0400A cubic graph with cubic diameter
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<p><a href="https://en.wikipedia.org/wiki/15_puzzle" target="_blank">The Fifteen puzzle</a> is sometimes generalized to a sliding puzzle on an arbitrary simple connected graph <i>G</i> with <i>n</i> vertices in the following way. <nobr><i>n</i> − 1</nobr> movable pieces numbered 1, 2, ... <nobr>(<i>n</i> − 1)</nobr> are placed on vertices of the graph <i>G</i>. At most one piece is placed on each vertex. One vertex of <i>G</i> is left unoccupied. A move consists in choosing a vertex <i>v</i> adjacent to the currently unoccupied vertex <i>v</i><sub>0</sub> and 'sliding' the piece at <i>v</i> along the edge (<i>v</i>; <i>v</i><sub>0</sub>). The aim is to restore the order so that piece numbered <i>i</i> occupies vertex numbered <i>i</i>, for <nobr><i>i</i> = 1 .. <i>n</i> − 1</nobr>. In the case of the Fifteen Puzzle, the underlying graph <i>G</i> is the <nobr>4 × 4 grid graph</nobr>, and a degree-2 vertex is left unoccupied in the goal configuration.</p>Mon, 06 Mar 2017 03:53:47 -0500Is there a way to evenly distribute face turns for 12 flip?
http://cubezzz.dyndns.org/drupal/?q=node/view/553
Back in Jan 1995 Mike Reid found this process for the 12 flip:<br />
<br />
R3 U2 B1 L3 F1 U3 B1 D1 F1 U1 D3 L1 D2 F3 R1 B3 D1 F3 U3 B3 U1 D3 24q<br />
<br />
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. <br />
<br />
MarkMon, 14 Nov 2016 22:17:19 -0500Super Group Cosets of the Centers Subgroup
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<p align="justify">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.</p>Mon, 20 Jun 2016 07:52:11 -0400Finally hitting depth 13 consistently with my 5x5x5 solver
http://cubezzz.dyndns.org/drupal/?q=node/view/551
<img src ="http://i.imgur.com/cq9J6SC.jpg"><br />
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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!<br />
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A very happy day for me.Fri, 17 Jun 2016 19:52:37 -0400Super Cube States at Depth
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<p align="justify">Super Cube States at Depth</p>
<p align="justify">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,
<a href="http://cubezzz.dyndns.org/drupal/?q=node/view/536" target="_parent">Lower bounds for the 3x3x3 Super Group</a>
and <a href="http://cubezzz.dyndns.org/drupal/?q=node/view/120" target="_parent">Supergroup knowledge</a>.
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?
</p>Wed, 15 Jun 2016 11:36:42 -0400Revisiting Korf's 3x3x3 Counts And Identifying Duplicate Positions
http://cubezzz.dyndns.org/drupal/?q=node/view/549
A while ago I generated a corners database for the 3x3x3 subset of positions that are applicable to the 5x5x5 cube. I used a move generator that generated 27 moves at depth 1 instead of the commonly reported 18. My move generator also spawned the middle slice turns along with the familiar turns of the outer faces. The total positions reported for the corners database agreed in the end.<br />
<br />
Recently I decided to implement the Korf move generator. In my mind, it is really more akin to a 2x2x2 move generator, since every move sequence that is generated can also be played out on a 2x2x2 cube. (Contrast that to some 5x5x5 moves which clearly have no counterpart of smaller cubes.) My move generator started with the solved cube, counted nodes as a function of depth, and placed each unique cube in a hash table, flagging all of the duplicate positions that came next.Fri, 10 Jun 2016 22:18:56 -04005x5x5 Solving Programs: The list is growing
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As of June 5, 2016, there are 3 different 5x5x5 solving programs that I know of: one is a robot that solves a physical 5x5x5 cube, one is a java program, and mine is written in C running under Windows 10. I know this forum caters specifically to those interested in "numbers" associated with the various cubes, and I do have some nodes-per-depth data that might be of interest to the group. As for those who might wish to submit test scrambles or other queries to the 5x5x5 programs, my guess is that this is outside of the domain of interest here. So as to not "step on any toes," I have a separate discussion board set up so as to not distract anyone from here. If you'd like more info about that, you can email me at edwardtrice at mail dot com.Sun, 05 Jun 2016 23:06:26 -0400New 3x3x3 Corners Data Including Centers
http://cubezzz.dyndns.org/drupal/?q=node/view/547
I think almost everyone here is familiar with the Corner Distance data that is used for pruning brute force searches for the 3x3x3 cube. This database features using one cubie as a "reference point" about which to orient the cube, and the relative disposition of the other corners is determined after hunting for that fixed frame of reference. <br />
<br />
I decided to create a database that:<br />
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A. requires no single point of reference<br />
B. contains all 24 possible rotated states of the cube's corner arrangements<br />
C. measures the distance the corners are from the solved state with respect to the fixed centersSat, 04 Jun 2016 09:55:14 -0400IP address for forum changed to 69.165.220.244
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Hi folks,<br />
<br />
My upstream has changed my ip address so now the numeric ip is 69.165.220.244. All the links should work as before once all the dns changes have propagated.Sat, 07 May 2016 01:00:31 -0400Cube archives changed
http://cubezzz.dyndns.org/drupal/?q=node/view/545
To improve security I've removed ftp access from the forum. We'll use http instead of ftp. I've also moved the Thistlethwaite files over so they can be accessed by http as well.Sat, 19 Mar 2016 08:43:00 -0400Confirmation of Results for Edges Only Cube, Face Turn Metric, Reduced by Symmetry
http://cubezzz.dyndns.org/drupal/?q=node/view/544
<p>
Even though it's only confirmation of some rather ancient results,
I would like to report that I have succeeded in replicating Tom
Rokicki's 2004 results concerning the edges only cube in
the face turn metric reduced by symmetry. My goal was not
really to solve that particular problem. Rather, it was
to use that problem as a way to prototype some ideas I have had
for improving my previous programs that enumerate cube space.
</p>
<p>
The base speed of my program is that
I am now able to enumerate about 1,000,000 positions per second
per processor core when not reducing the problem by symmetry,
and I am now able to enumerate about 150,000 symmetry representatives
per second per processor core when reducing the problem
by symmetry.
</p>Tue, 23 Feb 2016 12:42:25 -0500Refining Bruce Norskog's 4x4x4 five stage analysis
http://cubezzz.dyndns.org/drupal/?q=node/view/543
Bruce Norskog designed a <a href="http://cubezzz.dyndns.org/drupal/?q=node/view/62">five stage procedure</a> to solve 4x4x4 positions, where the last stage is solving inside the squares subgroup. When I rewrote the code in Java to be used for the official WCA scrambler program, I found a few improvements on coordinate representation and symmetry reduction that I think is worth sharing.<br><br>Except for the last stage where centers need to be solved, every other stage needs to place 8 centers from a single axis (RL, FB or UD centers) in their correct faces (RL, FB or UD faces), in one of the 12 positions that can then be solved using only half turns. However, as pointed by Shuang Chen in <a href="http://cubezzz.dyndns.org/drupal/?q=node/view/525">his analysis</a>, we don't need to store the exact colours of centers but only if two centers have the same or different colours. This reduces the number of center coordinates by 2.<br><br>Also, as opposed to the 3x3x3, 4x4x4 does not have fixed centers, which allows us to do more symmetry reduction. I'm representing a sym-coordinate as:<br>s1 * g * < H > * s1' * s2'<br>where g * < H > is a coset, s1 and s2 are symmetries from subgroups S1 and S2 of the group M of symmetries of the cube. s1 is the usual conjugated symmetry, and s2 correspond to a rotation of the cube, which is possible on the 4x4x4. Using carefully chosen subgroups S1 and S2 for each stage, more symmetry reduction is achieved. I will be using the Schoenflies symbols for subgroups of M in the following.Fri, 11 Dec 2015 08:03:06 -0500Router Problem
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There was a router problem last night, possibly due to memory corruption. I disconnected and reset everything on the router and DSL modem and now it's working again.<br />
<br />
Sorry about the outage. Just letting everyone know that the forum is still alive :)<br />
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MarkMon, 31 Aug 2015 12:18:17 -04004x4x4 upper bounds: 57 OBTM confirmed; 56 SST and 53 BT calculated.
http://cubezzz.dyndns.org/drupal/?q=node/view/541
I have replicated Shaung Chen's upper bound result of 57 moves in the OBTM (outer block turn metric), reproducing his numbers and calculating depth counts without symmetry reduction. His results are here: http://cubezzz.dyndns.org/drupal/?q=node/view/525
<p>
I have also calculated, with the same code, results in two other metrics: SST (single slice turn) and BT (block turn). The final results lower the upper bounds in these metrics to 56 and 53 (respectively).
<p>
The sequence of subsets chosen is identical to that used by Chen. The maximum distance for each stage, the sum, and the current lower bound is shown here:Tue, 03 Mar 2015 02:14:52 -0500