]>
Domain of the Cube Forum - Cube lovers returns
http://forum.cubeman.org
To promote mathematical discussions about the Rubik's Cube and related puzzles.enNew 5 spot pattern discovered for megaminx
http://forum.cubeman.org/?q=node/view/574
<img src='http://www.maxhost.org/other/5-spot-pattern-megaminx' width=600>
<p>
Thanks to Tom's web program at <a href="https://twizzle.cubing.net" target="_blank">twizzle</a> I've discovered a new megaminx pattern. Quite a long time ago I realized that slice patterns could be adapted to the megaminx. The early results can be seen here: <a href="http://www.cubeman.org/mega2.html" target="_blank">megaminx patterns</a>
<p>
The notation to generate the 10 spot with Tom's program is (2L 3u')36. With more experimentation we should be able to find many more.
<p>
MarkWed, 19 Jun 2019 15:16:19 -0400An unsolved problem: how many solutions are there to the 8-Color Cube?
http://forum.cubeman.org/?q=node/view/573
The <a href="http://www.randelshofer.ch/rubik/virtual_cubes/rubik/eight-colored_faces/eightcolor_cube_1.html"target="_blank">8-Color Cube</a> is an extremely elegant problem, both in appearance and concept;<br />
<br />
The cube is very easy to make at home: numbered stickers are available everywhere and the whole<br />
construction process takes only about 15 minutes.<br />
<br />
As you can see, Walter Randelshofer and myself have managed to find a number of extra solutions<br />
separate from the pre-existing design solution with its ”Superflip Centre” variant, however the real<br />
problem remains: how many solutions are there, in theory, to the 8-Color Cube?--this is the toughMon, 17 Jun 2019 16:27:42 -0400Skewb Star Special Challenge/Competition, with Special Prize
http://forum.cubeman.org/?q=node/view/572
SKEWB STAR <br />
Special challenge/competition 24 October 2018<br />
by Peter Tchamitch<br />
www.petertchamitch.se<br />
<br />
Question 1:<br />
How many solutions are there to this puzzle?--in other words, in how many different ways<br />
is it possible to physically orientate a solved octahedron and a solved cube/skewb in relation<br />
to each other?<br />
<br />
Question 2:<br />
How many color-matchings (please see definition below) are there in total, in other words<br />
what is the sum of all the various color-matching values for all the various solutions?<br />
A “color-matching” is an instance of one of the sides of an octahedron-pyramid having theFri, 14 Jun 2019 04:04:04 -0400cubezzz.dyndns.org/drupal is once again active
http://forum.cubeman.org/?q=node/view/571
Hi folks,<br />
<br />
These old URLs should all work now:<br />
<br />
http://cubezzz.dyndns.org/drupal/?q=node/view/563#comment<br />
http://cubezzz.duckdns.org/drupal/?q=node/view/563#comment<br />
http://cubezzz.homelinux.org/drupal/?q=node/view/563#comment<br />
http://forum.cubeman.org/?q=node/view/563#comment<br />
<br />
MarkTue, 11 Jun 2019 00:00:55 -0400Optimal solutions to the Eliac puzzle
http://forum.cubeman.org/?q=node/view/570
The Eliac is a complex deep-cut 2-gen circle puzzle:<br />
<br />
<img src='https://i.imgur.com/b5lE2kw.png' width=300><br />
<br />
The left circle rotates in increments of 90 degrees and the right circle rotates only by 180 degrees. There is a simulator of the puzzle <a href='https://www.stringham.me/eliac/'>here</a>.<br />
<br />
Using ksolve++ I made an optimal solver modified it slightly to turn it into a coset solver. The subgroup I used for the coset solver is the subgroup of positions where the 18 small triangles, 10 diamonds, and 2 squares are solved. There are 1600300800 arrangements of those 30 pieces and each coset has 3024000 solvable positions. Unfortunately since the puzzle is 2-gen, there isn't a good way to select a subgroup generated by a subset of the generators of the whole puzzle, which (as far as I can tell) is what is required in order to make the "pre-pass" trick work for sub-optimally solving cosets very quickly. So each coset needs to be solved optimally using a pure DFS, which takes quite a long time (about 1.5 hours on my laptop). Notice that the puzzle has a horizontal reflection symmetry so we only need to solve one coset in each symmetry class.Sat, 18 May 2019 17:13:03 -0400Remember to use http://forum.cubeman.org
http://forum.cubeman.org/?q=node/view/569
Hi folks,<br />
<br />
This is a notice that http://cubezzz.dyndns.org/drupal will probably stop working tomorrow. Please use http://forum.cubeman.org instead.<br />
<br />
The similar http://cubezzz.duckdns.org/drupal should also work.<br />
<br />
More news to come soon.Sun, 12 May 2019 20:58:33 -0400{4,3,3} 3 symmetry
http://forum.cubeman.org/?q=node/view/568
Hello, I am new to this forum and this is my first post. I know that maybe its content will seem trivial to some of you, but I am afraid that I am not so well versed in mathematics and programming, so I try to do at least something within my limited capabilities (I am 15.34 at the moment). <br><br />
<br />
Based on Dan Hoey’s calculations, I was able to calculate the number of essentially different positions up to symmetry of the four-dimensional analogue of the Rubik’s cube. However, it is quite probable that I have made some mistakes, as I have done it only by hand. Nevertheless, from the patterns observed I was able to make some interesting generalizations, presented later in this post.Sun, 24 Feb 2019 06:09:21 -0500The 2018 Computer Fewest Moves Challenge is Underway
http://forum.cubeman.org/?q=node/view/567
For the first time, the FMC will have solvers in the "NxNxN" category for up to 9x9x9 competing to solve 10 random scrambles in as few moves as possible. Also of note: There are now four solvers in the 5x5x5 section.<br />
<br />
For those who are interested, visit:<br />
<br />
http://cubesolvingprograms.freeforums.net/Thu, 18 Oct 2018 07:53:28 -0400Lower bound for the diameter of the 3-gen subgroup <U,F,R>
http://forum.cubeman.org/?q=node/view/566
What are the known bounds for the diameter of this subgroup? <a href='http://cubezzz.duckdns.org/drupal/?q=node/view/190'>This thread</a> gives a lower bound of 26 QTM but doesn't discuss HTM. I seem to recall a lower bound of 20 HTM from somewhere, but no upper bound. However, I recently found a position that requires 21 HTM:
<pre>U R F2 R U2 F2 U' F' R U F U R' F R2 U F R U2 R F'</pre>
Is this the first one known? Out of all 186624 positions with all pieces correctly permuted, this position and the inverse are the only ones requiring 21. Here's the full distribution:
<pre>
Depth PositionsSun, 02 Sep 2018 23:00:35 -0400Mad Octahedron
http://forum.cubeman.org/?q=node/view/565
<p align="justify">I have written a computer simulation of the octahedral twisty puzzle. It is available as freeware on the Apple App Store:</p>
<a href="https://itunes.apple.com/hr/app/mad-octahedron/id1400458838?mt=12" target="_parent">Mad Octahedron</a>Thu, 21 Jun 2018 10:47:47 -0400Future URL recommendation for the forum
http://forum.cubeman.org/?q=node/view/564
Hello everybody :)<br />
<br />
In the future I think it would be a good idea if we all used URLs of the form:<br><br />
http://forum.cubeman.org/?q=node/view/563#comment<br><br />
rather than<br><br />
http://cubezzz.dyndns.org/?q=node/view/563#comment<br><br />
dyndns.org has raised their prices every year and I'm considering <br />
dropping their service. Unfortunately if we do that a lot of old URLs will stop <br />
working, so I'm open to any clever ideas on what is the best way to deal with <br />
this problem. I'm committed to keeping the maxhost.org and <br />
cubeman.org domain names working for the long term, but<br />
I'm very unhappy with dyndns.Tue, 08 May 2018 17:47:42 -0400Three Million Positions in Four Metrics
http://forum.cubeman.org/?q=node/view/563
I optimally solved three million positions in four distinct metrics.<br />
These positions are distinct from the three million positions I ran<br />
some years back. Random numbers were generated with the Mersenne<br />
Twister algorithm. The four metrics I ran were quarter-turn metric,<br />
half-turn metric, slice-turn metric, and axial-turn metric (equivalent<br />
to the robot-turn or simultaneous-turn metric on the 3x3 cube).<br />
<br />
The generators for each metric are strict super- or sub-sets of the<br />
generators for the other metrics. The quarter-turn metric has 12<br />
generators, the half-turn metric has 18 generators, the slice-turnMon, 23 Apr 2018 11:52:52 -0400Presentation for the Mathieu Group M24 from dedge superflip
http://forum.cubeman.org/?q=node/view/562
Following on from a highly symmetric presentation I supplied for the miniature Rubiks cube group
<a href="http://cubezzz.homelinux.org/drupal/?q=node/view/177">Presentation for the 2x2x2 Rubiks cube group</a>
I investigated whether the Mathieu Group M24 could be similarily presented taking full advantage of the
plentiful symmetry inherent in the Rubiks Revenge cube (4x4x4). The answer was indeed yes - here is the presentation I found - again on three involutions:
<i>
<p style="font-size:17px;color:grey"
<pre><b>< a,b,c | a<sup>2</sup> = b<sup>2</sup> = c<sup>2</sup> = 1,
<p style="font-size:17px;color:magenta"
<b> (ab)<sup>6</sup> = [(bc)<sup>6</sup>] = [(ca)<sup>6</sup>] = 1,
<p style="font-size:17px;color:red"
<b> bacabacacabacababacabac = 1,
<p style="font-size:17px;color:green"
<b> (ababacbc)<sup>3</sup> = 1,
<p style="font-size:17px;color:blue"
<b> bababcbcbcbabab = cacabacacabacac</b> >
</i>
</pre>Sat, 21 Apr 2018 11:25:04 -0400Gear cube extreme can be solved in 25 moves
http://forum.cubeman.org/?q=node/view/561
Write the puzzle as the group <R,F,U,D> where R and F are 180 degree moves. We use a two-phase algorithm to first reduce the state of the puzzle to the subgroup <R3,F3,U,D>, and then finish the solve in the second phase. The subgroup <R3,F3,U,D> is the group of all positions where all of the gears are oriented, because R3 is the same as R' except the gear orientation remains unchanged.<br><br>
The first phase is easy to compute. There are only 3^8 = 6561 positions because each gear has only 3 different orientations, despite having 6 teeth.<br><br>
Phase 1 distribution:
<pre>Depth New Total
0 1 1
1 4 5
2 8 13
3 78 91
4 102 193
5 1064 1257
6 920 2177
7 3576 5753
8 592 6345
9 216 6561
10 0 6561</pre>
The second phase is harder. The number of positions is 24*8!^2/2 = 19,508,428,800, since it turns out that the permutation of the 3 unfixed edges on the E slice is completely determined by the permutation of the centres. This phase was solved with a BFS and took around 7 and a half hours to complete.Thu, 15 Feb 2018 23:07:51 -0500Kilominx can be solved in 34 moves
http://forum.cubeman.org/?q=node/view/560
Last night, I found <a href='https://www.speedsolving.com/forum/threads/kilominx-gods-number-bounds.67278/'>this thread</a> on the speedsolving forums which proves an upper bound of 46 moves. First, the puzzle is separated into two halves, which takes at most 6 moves. Each half is then solved in at most 20 moves (= 7 moves for orientation + 13 moves for permutation, after orientation is solved), for a total of 6+2*(7+13) = 46. xyzzy writes
<pre>The ⟨U,R,F⟩ subgroup, while much smaller than G_0, is still pretty large, having 36 billion states. It's small enough that a
full breadth-first search can be done if symmetry+antisymmetry reduction is used, but I will leave this for another time.</pre>Sun, 11 Feb 2018 12:58:51 -0500