Discussions on the mathematics of the cube

## Proposed solutions to the Wolf Tooth Xtreme and Skewb Star Xtreme

This post is a follow-up to my posts of 13 August 2019, ”Bonus Problem”, and 26 September 2019 , ”The latest on the Bonus Problem”

Just to quickly recap: the problem in question was to find the best way to solve the custom hybrid cubes, the Wolf Tooth Xtreme and the Skewb Star Xtreme, which together constituted the Special Prize of the Skewb Star Special Challenge/Competition posted here on 14 June 2019

The top four pages of the presentation here show the state of affairs at the time of the 13 August 2019 post, a state of affairs which was subsequently summarized in the first paragraph of the 26 September 2019 post; in the rest of that post I then outlined, without going into detail, a practical improvement to the previous ”standard method” and added at the end that ”it remains to be shown formally exactly why the improved system described above works”

## Deficiency minus one presentation for the Tits Group

Note that the simple group T (named for Jacques Tits) of order 17971200

has a transitive permutation representation on 2304 points.

A general formula for the number of edges in a n-cube is n.2^(n-1) to

be found in this extract from "Beyond the Third Dimension" by Thomas Banchoff

https://www.math.brown.edu/tbanchof/Beyond3d/chapter4/section05.html

It is worth mentioning another source here for a general formula

for the number of pieces of different types on a d-dimensional Rubiks' cube

## A Generalization on the Shamir Method

I have not posted in a very long time, but I have continued to work on ideas for a better program to calculate God's Algorithm for the full cube. The time has long passed where a single desktop class machine could make further progress on the problem as a whole. The search space is just too large for that. Instead, I have been working on ideas for a program that could at least visit all the positions in a single coset no matter how far the positions in the coset are from Start.

I have such a program which works, but its performance is not acceptable. Therefore, I'm not going to report anything about it. Instead, I'm going to report on a plan for a new and similar program which I believe will have acceptable performance. I have developed most of the pieces that will be required for this new program, but it will take me a few months to put all the pieces together. The main idea in the new program is very old and is not original with me. The idea borrows heavily from a message to the original Cube-Lovers mailing list by Alan Bawden on 27 May 1987. Alan's message was based on a talk given by Adi Shamir.

## Megaminx "cube in a cube" solved in 42 moves

I've found a pattern for the megaminx which is analogous to the "cube in a cube" pattern for the Rubik's cube. It was originally found using a cyclically decomposable process, then many optimizations were applied. The software used to do this includes Twizzle (Tom Rokicki and Lucas Garron) and twsearch. To capture the video the extension nimbus was used on the google chrome web browser.

The sequence of moves in SiGN notation is:
BR2 U2' BR2' R2 U' F2 U2' L2 U' BL2 U2 BL2' U' L2' U' F2' U' R2' BR2 U2 BR
BL2' L2' R2' U2 R2' U' F2' R2 F2 U' F2' U R2' F2 U R2 U' R2 L2 BL2 BR2

The vid

## Jakub Stepo´s solutions to the two Skewb Star Competition problems

For those who don´t know about this competition, please refer to my post of 14 June 2019 entitled "Skewb Star Special Challenge/Competition, with Special Prize", and to my Winner Announcement post of 29 August 2019, in which, as you can see, it was stated that the Special Prize had been awarded but that the challenge itself was to remain open until the New Year; well, the New Year has now arrived, so please find below Jakub Stepo´s solutions to the two competition problems:

Question 1
========
Let’s say that the cube is solved and ﬁxed in position. We have to ﬁnd out which positions are permissible while having solved cube.

## The latest on the Bonus Problem posted on 13 August 2019

The so-called Standard Method referred to in the Bonus Problem post of 13 August 2019 involves
finding the easiest or most convenient Skewb Star or Wolf Tooth solution, depending on how the
SSX or WTX respectively were scrambled, and then, unless the characteristic valleys between the
corners of the Skewb Xtreme just happen to be spontaneously solved, proceeding from there to the
other Skewb Star or Wolf Tooth solutions in an orderly manner until the valleys are observed to be
correct. No matter how orderly, this method still involves trial and error, and the Bonus Problem
was basically asking for a way to eliminate that somehow. The SSX and WTX have only one

## The Special Prize of the Skewb Star Special Challenge/Competition of 14 June has been won but the challenge is still open

The winner of the Skewb Star competition posted here on 14 June is Jakub Stepo, who is
a member of the Cube Forum

The Special Prize has of course been sent to Mr. Stepo, but I would like to stress that the
challenge is still open and that everyone who sends in the correct answers by let´s say 31 December
2019 will have their names published here, as soon as the answers are received

After the tentative closing date of 31 December 2019, I was thinking of publishing here all the
actual worked solutions that may have been sent to me, and in this connection I can reveal ahead of time
that Jakub Stepo´s elegant and powerful solution will certainly be of great interest to Cube Forum

## 4x4 only two blocks from solved

Hello, I’m trying to solve the 4x4 for the first time but keep getiing the same mistake.
White is solved
Red is solved
Yellow is solved
Green is solved except left highest row: yellow/blue orange block
Bleu is solved except right highest row: orange/green/yellow block
Yellow is solved except the above two blocks.
These two blocks should be switch but I have no idea how.
If I start again, the same problem will appear.
Any ideas?

## 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.

Mark