Discussions on the mathematics of the cube

Bonus problem related to the Skewb Star Special Challenge/Competition, with Special Prize, of 14 June 2019

So far, no one has sent in the answers to the two questions posed in the Skewb Star Special Challenge/Competition which I posted here on 14 June 2019, so I thought that I would avail myself of this window of opportunity to add a further bonus problem.

As I´m sure everyone immediately realized, the whole point of the Special Prize of the competition, the Skewb Star Xtreme, SSX, together with the Wolf Tooth Xtreme, WTX, is that solving these cubes is a practical application of knowing all of the solutions to the Skewb Star as well as how to alternate between them, in other words of having found a way to answer the two questions of the competition.

New 5 spot pattern discovered for megaminx

Thanks to Tom's web program at twizzle 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: megaminx patterns

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.


An unsolved problem: how many solutions are there to the 8-Color Cube?

The 8-Color Cube is an extremely elegant problem, both in appearance and concept;

The cube is very easy to make at home: numbered stickers are available everywhere and the whole
construction process takes only about 15 minutes.

As you can see, Walter Randelshofer and myself have managed to find a number of extra solutions
separate from the pre-existing design solution with its ”Superflip Centre” variant, however the real
problem remains: how many solutions are there, in theory, to the 8-Color Cube?--this is the tough

Skewb Star Special Challenge/Competition, with Special Prize

Special challenge/competition 24 October 2018
by Peter Tchamitch

Question 1:
How many solutions are there to this puzzle?--in other words, in how many different ways
is it possible to physically orientate a solved octahedron and a solved cube/skewb in relation
to each other?

Question 2:
How many color-matchings (please see definition below) are there in total, in other words
what is the sum of all the various color-matching values for all the various solutions?
A “color-matching” is an instance of one of the sides of an octahedron-pyramid having the

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Optimal solutions to the Eliac puzzle

The Eliac is a complex deep-cut 2-gen circle puzzle:

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 here.

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.

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{4,3,3} 3 symmetry

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).

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.

The 2018 Computer Fewest Moves Challenge is Underway

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.

For those who are interested, visit:


Lower bound for the diameter of the 3-gen subgroup <U,F,R>

What are the known bounds for the diameter of this subgroup? This thread 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:
U R F2 R U2 F2 U' F' R U F U R' F R2 U F R U2 R F'
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:
Depth   Positions