SOMA Crystal
SOMA News 6 october 2018
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Examining the SOMAP in greater depth.


The SOMAP has intrigued us ever since it appeared in the SOMA Addict vol 2 no 2. on page 2 [ SOMA Addict ]
In the years to follow, the SOMAP featured in our Newsletters:

    Newsletter 2003.05.18 The complete "SOMAP" is found by William Kustes.
    Newsletter 2013.07.14 New way of drawing SOMA (re. SOMAP), by Furman Smith.
    Newsletter 2016.05.09 A way to solve the SOMAP, by Merv Eberhardt.
    Newsletter 2017.07.11 A lot more about SOMAP, by Merv Eberhardt.
    Newsletter 2018.10.06 An interesting note about the history of finding SOMAP, by Bob Nungester.
    Newsletter 2020.06.10 A new way to visualize the SOMA cube, by Bob Nungester.

And NOW Bob Nungester
who is also the programmer of our great new solver programs Newsletter 2017.09.01 The American SOMA Solver program.
has put a tremendous effort into understanding the properties of the SOMAP.
The new work started 2018-08-10 When Bob wrote a letter to Merv Eberhardt and Thorleif Bundgaard detailing some thoughts,
noting that the SOMAP contained reflections and an error in the associated file.
This work then continued for the next two months, digging further into the secrets of the SOMAP.

On the way, Bob also note that the original SOMAP work was actually done by hand, using juat paper and a pencil. - Wow. !

The complexity of this work is high, so I will let Bob explain for himself.



Secrets of the SOMAP.


Iíve been interested in the SOMA cube since getting one as a child in the 1960ís. My main interest has been creating and solving single- and double-set figures, but recently the focus returned to the basic 3x3x3 cube and the SOMAP. After reading Merv Eberhardtís newsletters and viewing the complete SOMAP provided by William Kustes I started to analyze it in more detail. If youíre not familiar with SOMAP nomenclature and the basics of whatís represented on the SOMAP, you need to read Mervís newsletters before reading this one.

The first question is what the map of the other 240 reflected solutions looks like. It didnít take long to realize that just holding the SOMAP up to a mirror shows those 240 solutions in a reflected SOMAP. As noted by John Conway in the last issue of the SOMA Addict, itís necessary to interchange occurrences of R and U, and take the complement of the dexterity, but all the information and links are there.

The only connections shown on the SOMAP to reflected space are from B2f and B4f (the reflected pair in the lower-right quadrant). The obvious question is how many other connections exist and can they be shown to improve the appearance of the SOMAP?

Before drawing the SOMAP, a table of connections must have been generated when John Conway and Michael Guy first found all 240 solutions. Now that Merv has followed the moves and published all the solutions, itís possible to regenerate a table of all possible 2- and 3-piece moves using a computer. Comparing each solution to all others, including reflections, generated the attached spreadsheet.
    Table of all possible 2- and 3-piece moves. (An Excel file that will download.)

The main finding from the spreadsheet is that the ONLY 2-piece moves between real and reflected space are the ones from B2f and B4f, so these are very special solutions. While holding the SOMAP up to a mirror and looking at both the SOMAP and reflected SOMAP, the best way to represent the B2f/B4f solutions is to remove them from the paper and let the lines connected to them become strings. Putting them on top of each other, they should be placed directly on the mirror surface. In this way there are no 2-piece connections that go through the mirror. This special pair of solutions exists exactly at the border between real and reflected space (cue The Twilight Zone music).

The other interesting finding relates to the Diamond solution R7d. Two references to the SOMAP have clues to two different 4-piece moves between R7c and R7d. None of the 480 solutions has more than a single 2- or 3-piece move to any other solution, but when 4-piece moves are considered there is at least this one example that has two.
Generating another spreadsheet of all 4-piece moves connected to R7d (An Excel file that will download.) shows there is even a THIRD different 4-piece move between R7c and R7d.

Thereís a lot more information associated with this, but the newsletter was getting so long I decided to summarize it all in a separate paper. If youíre interested, you can read the (.pdf) paper here.

Best SOMA wishes.
  Bob Nungester.





A few speculations on the way.

Along the way of this story, many emails traveled between us. Some of these mails were complex pre-thoughts that went into the investigations.
Whereas others were short stops on the way.
Below - I show the content of a few of these mails:
Between: Bob Nungester, Merv Eberhardt, Thorleif Bundgaard, Ed Vogel, Jan de Groot
Note that not all mails are included, that would be too much. And some are carefully edited to just show the SOMAP relevant information.



7 Aug 2018 From: Bob Nungester to Merv Eberhardt

After getting done working on Spotted Soma I finally got interested in SOMAP and read all newsletters on Thorleif's site in detail.
I see in N170711 that you've identified every solution in the SOMAP! Congratulations!

You also note in N160509 that you'd like to map these against the 240 previously published solutions, which you normalized (and we used for Spotted Soma work).
You found that many of them must map to reflections due to R and U swaps and dexterity issues.

Since each solution has a mirror image, there are 2^240 ways to construct a set of 240 solutions and 240 mirror images.
However, since there's a one-to-one linking of each solution and its reflection, it's easy to have a computer program generate a list of SOMAP solution 001-240 along with the corresponding 1999 solutions and their reflections (from 001-240 and 001R-240R).



10 Aug 2018 From: Bob Nungester

After finding Merv's complete list of solutions to the SOMAP I got interested in his suggestion to index the list of cube solutions on Thorleif's site.
I noticed in the SOMAP notation that many lists of solutions with the same letters and deficiency have subscripts of a all the way up to p.
In those sequences some of the solutions have one designation and others have the reflected designation, such as A5f,g,h and A2a,b,c,d,e.
I went through the list in detail to follow those sequences and noticed one error in the file.

The file lists A1c,f,i and the reflected designation A6a,b,c,d,g,h. They both contain a "c" subscript, but neither has an "e".
I looked at the actual SOMAP and found A1e located near the right top, at the end of three parallel lines pointing up and right.
All that's needed is to change the label A1c to A1e on the first line of solutions.

I'm going to go ahead and normalize the solutions and generate an index.
I've already got a program that will index any two lists of 240 solutions (also calculating reflections), so all that's needed is to normalize the solutions.



12 Aug 2018 From: Bob Nungester
After a lot more work than I thought it would take, I've got the SOMAP solutions normalized and indexed.
The reason it's not attached to this message is there are three errors in the SOMAP solutions that need to be fixed.
I tried fixing two of them using connections shown on the SOMAP, but the indexing program showed all I did was make a duplicate of other solutions.
I guess there are several ways to do an oy transition.

Anyway, it's obvious once you know where to look that solutions U0h, BR2c, and BR4b have several errors.
Some pieces have extra cubelets and others don't have enough.
Let me know if you have corrections for these, or I'll work on using the SOMAP to connect from nearby linked solutions.



12 Aug 2018 From: Thorleif Bundgaard
Great - Bob

I think your trip into this huge amount of data is beyond my comprehention.
But who knows, maybe you will find some inner beauty, presenting the solutions that way.
Though the shere amount of data is ... unimaginable.

Another thing I had been wondering, was how the SOMAP really looks.
Imagining that it was plotted in 3 dimensions. And having some graphic program that allowed us to rotate and zoom. And finally to pick a solution.

When I look at the SOMAP, it seems as if many of the connections might be easier if 3D was used, instead of the flat 2D of the piece of paper they originally had.



14 Aug 2018 From: Bob Nungester

Yes, I agree that the SOMAP would look better in 3-D, but that would be hard to create or visualize.
I just started looking at SOMAP recently and noticed several interesting things:

1) With seven pieces there are 21 possible combinations of 2 pieces (7+6+5+4+3+2+1).
Of these, 18 have two or three ways to fill their combined space and are used to move between solutions on the SOMAP.
The interesting thing is that an only 17 are used on the map, The combination gr exists as well, but I don't see it anywhere on the map.
Why is that? Eliminating all solutions containing a gr configuration could be used to define which solutions are listed and which are reflections,
but that doesn't eliminate all possibilities. All other combinations with r or u are used, so some other method was used to determine which solutions are referred to as reflections.
I haven't checked yet if any listed solutions have a gr combination contained in them. If so, that's another link.

2) In the Soma Addict Conway states that the SOMAP is not complete, and that is easily seen once you understand the moves.
It looks like 14 of the move types involve two possible configurations, so one is represented at each end of the link.
However, three of them have 3 possibilities each; ab, br, and bu.
This is the reason for many triangular links between three solutions throughout the map with these moves.
However, ANY ab, br, or bu move has to form such a triangle of solutions, and many of these links are not shown.
All 240 solutions are shown so there are many more links to complete the map. I haven't followed any of these yet.

It was very difficult to figure out the specific configuration of several of the piece combinations used in the map.
I had to look at Merv's solutions to see them. I entered them all in your standard .htm file format (see attachment).
You need to open them in the Double Soma solver to see them since the old single-set solver doesn't display solutions from a file.
Just open the file in Double Soma and click on View Solutions From File. I think it includes each possible 2-piece move configuration from the SOMAP.
I'll probably go on to the 3-piece configurations, but not right now.



14 Aug 2018 From: Bob Nungester

It is done! I was able to navigate a few links on the SOMAP and correct the errors in the U0h, BR4b and BR2c solutions in the newsletter.
Attached is the file of normalized SOMAP solutions in both the standard 1-7 piece number format and the B,Y,G,O,U,R,A format used in SOMAP.
I also indexed the SOMAP solutions against the list of solutions Merv had already normalized.
Each of the three index files compares one file against the other, or against itself in the case of the SOMAP to SOMAP index.
Each index compares the first set of 240 against the other set of 240 plus it's 240 reflections. Comparing SOMAP to itself shows that no solution is a reflection of another.
For indexing I left B2f out of the SOMAP set (which has 241 total) since it is known and shown on the SOMAP as a reflection. With this taken out all solutions are shown to be unique.

I also created a file in Thorleif's format that shows all two- and three-piece transitions on the SOMAP.
This was a great help to me when navigating between solutions without having to figure out how to do a difficult piece manipulation each time.
I may have missed one or two, and a couple of them have more than one form. You'll see them in the file.

Anyway, I'm done working on SOMAP for at least a week or so. As you know, the SOMAP shows a minimum amount of links to allow navigating the map.
I'm interested in exploring where all the other links go, such as every bo, br, ab, and roy link that all have three forms.
In every case these must link four separate solutions together. And the guy and ayx transitions have four possibilities so they link five solutions directly to each other.
Most of these extra links aren't shown on the SOMAP, but it'll be interesting to see where they go.

Onward with SOMA,



15 Aug 2018 From: Bob Nungester

Oops, I noticed a couple of mistakes.
I was in a rush to get on the road yesterday, so as soon as the indexing program put out meaningful data I went ahead and sent the files.
Today I did some basic checking and noticed the two files indexing one set of solutions against the other don't always agree.
Any link in one file should show up as the same link in the other file. Most agree, but many are off by one and will point to the next solution, not the correct one.
I think this is related to the removal of B2f from the SOMAP set. Somehow the titles got out of sync with the solutions. I'll correct that and send the updated files.
Next time I'll do more detailed checking before claiming it's done.
Also, I notice in talking about solutions that have three or four forms; these link three or four SOMAP solutions, not four or five.

Anyway, the SOMAP is a fascinating study in finite group theory.
I've just started reading about that and found this appears to be an example of that branch of mathematics.
The SOMA Addict mentions that Richard Guy (Michael Guy's father) had written a paper about the SOMA cube in NABLA when he was in Malaya.
It would be great to find that paper. NABLA is a publication of the Singapore Mathematical Society (Mathematical Society of Malaya and Singapore at the time),
so maybe there's a way to find copies of the NABLA journal from some time in the 1960's when the journal was named as NABLA.
See http://www-history.mcs.st-and.ac.uk/Societies/Malaysian.html

That's all I've found so far trying to track down the paper.
Have either of you read the book Winning Ways for your Mathematical Plays?
This book by Conway and Guy has a final section with an analysis of the SOMA cube, according to Wikipedia.
I'm going to get a copy of that and see what insights it provides. I need to do some work today and get under my wife's car to change the oil.
No time for important things like SOMA.

An interesting note about the history of finding SOMAP

On 2019-02-24 Bob Nungester discovered a very interesting thing about the creation of SOMAP.
(Sorry Bob, I'm all too late adding this note, its now 2020-06-16)

Bob wrote: As you may recall, in the article regarding the SOMAP in the last issue of the SOMA Addict
There's mention of an article by Richard Guy published in the NABLA Journal (from the University of Malaya in Singapore).
All 4 issues of the Original SOMA Addict.     See page 14, middle column, just below the SOMAP.

Prof. Donald Knuth has sent me a copy of the "relevant pages" from the December 1960 NABLA article.
    The pages from NABLA Journal.

The portion relating to SOMA is quite short, but it does state that:
"the number of essentially different solutions so far found is 234, and the total number is probably not much more than this".
Richard Guy also notes: "I have spent some time in linking solutions together, so that each is obtainable from another by removing just two pieces and replacing them with a different orientation. I have not linked all the 234 solutions so far found, but I can do so if in some cases the rearrangement of three pieces is allowed."

That means he [Richard Guy] had at a minimum started the list of moves depicted on the SOMAP, and may have started the SOMAP itself. This shows work on the SOMAP started at least 12 years before the 1972 SOMA Addict article. It's also interesting to note that the SOMA Addict article mentions "... originally, Yellow was Silver and Brown was White ..." referring to the piece colors, which is what's shown in the NABLA article (S instead of Y, and W instead of B).
Anyway, after all this time we've finally got what's probably the first published information relating to work on the SOMAP.




15 Aug 2018 From: Thorleif Bundgaard

No worries Bob.

It is often that pressing the send button seem to reveal a ton of mistakes in a mail.
I still think it is great that you are doing this.
I dont understand the SOMAP myself, but it's still interesting that so much can be correlated.

No I have'nt got the mentioned document by "Richard Guy" - "a paper about the SOMA cube" in NABLA.

My guess is that it is this one "Richard K. Guy. A combinatorial problem, Nabla (Bull, Malayan Math. Soc), 7:68-72, 1960."
But it does'nt seem to be online available.

If you stumble across it, then I would like a scan/copy.



17 Aug 2018 From: Bob Nungester to Merv Eberhardt

Good to hear from you again.
Yes, I've seen Jan de Groot's website in the past and it is quite impressive.
That's yet another of the 2^240 ways a set can be organized with 240 solutions and their reflections.

I now have an index program that will index any set of 240 solutions against another set.
It generates reflections for the second set and then indexes the first 240 set against the second's 480.
The sets have to be in a specific format (SOMAP attached), and titles are optional.
It notes any duplicate solutions or missing solutions, so it can be used to check the validity of any set.
Let me know if anyone would like a copy.

I've gotten interested in SOMAP now, and I'm wondering if you can provide any insight to a few questions.
I did find the Winning Ways for your Mathematical Plays book and immediately saw it's the page on Thorleif's site I've read about 100 times in the past.
I guess I should read his clear references when looking at newsletters.

Here are the questions:
How did Conway and Guy keep track of solutions to make sure reflections would not pop up and be added to the set?
Was it by use of Deficient, Central, and Dexterity? If so, what were the rules to exclude reflections in their search?
Why do sequences of solutions (a, b, c, ... up to q) skip back and forth between a particular type and its reflection,
such as U0a,b,e,f,g,h and R7c,d? Is this part of the answer to question 1?
It looks like every possible symmetric pairing of two pieces is included in the moves except gr.
That configuration is the reflection of a gu. All the other reflected combinations, ub/rb, ua/ra, uy/ry are there, but no gr.
Why is that? I suppose it might be that any gr move makes a reflection of an existing solution,
but it seems strange only one of the reflected pairs does not exist as a move on the SOMAP.

I'm trying to understand as much as possible about the structure of the SOMAP and then see if there are other ways of mapping the data.
SOMAP shows the minimum amount of links to allow navigation of the map, but many more links exist.
For example, any bu or br configuration exists in three forms that link three solutions together.
Many of these are shown, but many are not. The first step is to see if the third linked solution is a reflection of an existing one.
Anyway, maybe there are other ways to present the map so it's even more symmetrical.
Well, that'll keep me busy for quite a while pondering the complexities of the amazing SOMAP.



18 Aug 2018 From: Merv Eberhardt

A few other comments:

The bu, br, and ab triangles are fascinating since the two-pieces can be rotated.

It was a challenge to figure out the direction of rotation on the SOMAP.

Another interesting rotation is found in the YU3a, YU3b, YU3c, YU3d, YU3e, and YU3f "spoke."

As presented it has three two-piece swaps, i.e., bg, gu , and bu.

It can also be presented as two triangles with a three-piece rotation; like the P003 [https://www.fam-bundgaard.dk/SOMA/FIGURES/P001025.HTM]



19 Aug 2018 From: Bob Nungester to Merv Eberhardt

I'm just starting to understand SOMAP.

I found that SOMAP is an interesting study in mathematical group theory.

Both Conway and Guy were heavily into that.

Group theory deals with sets and groups within the set.
You already know these facts, but stated in group theory the set of 27 small cubes has various groups of the 7 pieces that are symmetrical,
and can be varied to form 2 or more constructions (solutions) of the 3x3x3 cube. When he found each group, Conway placed it in each of it's possible configurations,
and each placement generated a different solution. The moves on the SOMAP represent the group actions and their effect on the properties of Deficient, Central and Dexterity.
He was enumerating the occurrence of all these groups within the set, which is a standard method used in group theory.
I know almost nothing about group theory, but just looking at the vocabulary and some examples I saw that's what he was doing.
The list of moves documented in the .htm file I sent is the list of all the groups he found.
It could be missing a few since there may be different versions of some of the groups in addition to the two noted for byg.

Anyway, this may lead to different ways to draw the SOMAP. Each move (or group action) represents a cycle between two or more configurations of a group.
The two-configuration groups aren't too interesting since they just make a connection between two solutions.
The three- and four-configuration groups, however, create triangles or squares of linked solutions.
Conway also found a pair of group moves that can be alternated with each other to form a pentagon of 5 solutions, and even the combination you note of three two-piece moves
(or one three-piece move and its two configurations) that forms a hexagon of 6 solutions.

The SOMAP shown in the SOMA Addict has pencil-drawn letters identifying various areas of the map.
The article notes there are "118 solutions (A), 16 solutions (B), 11 (C) .. . down to 15 isolated solutions (Z)".
Do you know what these groupings represent? This is probably how Conway kept track of solutions so he didn't duplicate any of the 240, but it's only one way to make an interesting graph.
All the group actions throughout the cube can be generated and sorted with a computer program (not easy),
but sorting the solutions by group action instead of by deficient, central and dexterity may make a more interesting graph.
Each ab, bu, br, roy and ray move have three configurations forming a triangle of solutions, and guy and bay moves have four configurations forming squares.
Many of these links are not shown on the SOMAP, so this may lead to a different way of linking the 240 solutions.

I'm a newbie regarding SOMAP and group theory, but I'll be working on it for a while and see what develops.
I'll copy you and Thorleif on anything interesting that pops up, and I look forward to any suggestions, insights or comments you may have.



19 Aug 2018 From: Merv Eberhardt

I'm attaching a SOMAP_old.docx file that has the pencil-drawn letter groupings (A-Y) for the whole SOMAP.
There maybe some hidden significance that you can benefit from.

I'm copying Ed Vogel, a mathematician who has deep interest in the group theory aspect of the SOMAP.

I'd would like so very much to meet with JHC to get his autograph on my SOMAP solutions.
RKG was 100 years old September 2016.
Being British mathematicians they are not forthcoming with hints and explanations.



21 Aug 2018 From: Ed Vogel

Hi Guys, glad to see people continue to have interest in SOMAP.
I have been focused more on getting BLOKL into some boutique manufacturable mode and get samples out to educators (see attached).
That being said I have made two videos of my observations on finding a manual algorithm that constrains the solution and space and sorts solutions into a tree graph.

SOMA Cube Solution Methods 1 https://youtu.be/ZWs7A9yNJ3w

SOMA Cube Solution Methods 2 https://youtu.be/72kyEFsUOIU

As far as group theory goes I have had some conversation with David Nash author of "A Friendly Introduction to Group Theory" about SOMA Cube solutions.
He is tied up with teaching but does agree there are interesting insights into symmetry operations in the SOMA Cube.
He found the Klein Four Group example particularly entertaining.

The introductory chapter on group theory of this book "Symmetry and Spectroscopy An Introduction to Vibrational and Electronic Spectroscopy" may provide the best insights I have read yet into thinking about SOMA in terms of groups. (a steal at ~$5.00):
https://www.amazon.com/gp/offer-listing/B07BMJQFSM/ref=tmm_pap_used_olp_sr?ie=UTF8&condition=used&qid=&sr=

In closing I think finding a way to make the SOMAP more "viewable and doable" would be a great gift to math education and could end up a travelling museum/education center exhibit.

Look forward to seeing the latest discoveries.



22 Aug 2018 From: Merv Eberhardt

Looking at the Christoph PETER-ORTH article, he states -37 that his solutions were based on the placing of the 4/T-piece in normalized position then placing the 2/Tripod-piece.
This may result in some sort of grouping or tree structure. Just a thought.

I have noted that his 240 solutions as well as the 240 solutions from the SOMA Newsletter N990201
2019-02-01 Newsletter. have numerous reflections compared to my SOMAP solutions 2017-07-11 Newsletter.


1 Sep 2018 From: Merv Eberhardt

Here is an interesting Excel file for your enjoyment.
It relates the 241 SOMAP solutions to the Jan de Groot cube solutions.
From there it relates to the 240 cube solutions of the 1999 SOMA News Letter solutions and the 1984 article by Christoph PETER-ORTH.

The asterisk in the latter two references relate to reflections of the SOMAP and JDG cases.
111 reflections in the 1999 SOMA News Letter listing and 97 reflections in the CPO listing.
In case you have not noticed, the CPO solutions can be broken down into eleven (11) SOMAtypes:
1-37 for A, 38-70 for O, 71-89 for B, 90-103 for BR/BU, 104-124 for YR/YU, 125-175 for R/U, and 176-240 for UR/RU.



21 Sep 2018 From: Bob Nungester

I'm finally catching up on SOMAP correspondence. Regarding the CPO solutions, I analyzed his methodology as part of studying ways to generate a tree graph of solutions, as suggested by Ed.
CPO's method used a similar but different method from Central, Deficient and Dexterity.
He referred to Center, Edge, Face and Vertex as shown in Table 2 of his paper.

I didn't notice the correlation with SOMA types, but I do understand the CPO methodology pretty well.
It's difficult to describe with all the various ways to number and color pieces, so I'll describe this with his number notation followed by Thorleif's numbering and the SOMAP color in parenthesis. CPO first placed piece 4 (3 or G) in its only possible position, and then placed piece 2 (7 or A) in its first branch orientation.
Per table 2, the first configuration used was CFEE (note that's Central), which results in 37 solutions.
When A is Central it must also be Deficient, so these all have SOMA type A. Note there are commas in his list of solutions to note the division of solutions from each of the 11 branches in table 2.

Next, he moved on to branch 2 with piece 3 (4 or O) Central, resulting in 33 solutions of type O, since O must be Deficient when it is Central.
The next branch has 7 (1 or B) central, producing 19 solutions of type B (B is Deficient if Central also).
Branches 4-11 then alternate between piece 5 (6 or R) and piece 1 (5 or U) being Central.
This R ,U ,R ,U ,R ,U ,R ,U altering as Central results in the BR, BU, YR, YU, R, U, UR, RU progression you note.
The Central correlation holds up, but the Deficient piece varies because he's stepping through 11 branches of a matrix using orientations of Face, Edge, Center and Vertex.
Because of that the link to Deficiency starts to break down, but still leaving a nice correlation to SOMA types.

I'm continuing to study the SOMAP and will write up a summary of that in the next few days.



22 Sep 2018 From: Bob Nungester to Merv Eberhardt

It is interesting how the CPO solutions relate closely to the SOMAP types, but with many reflections.
You're probably aware of most of the comments that follow, but I'm going to include them in my article about SOMAP when It's finished, so they're written more for the general public.

The SOMA cube actually has 480 solutions, arranged in 240 pairs of reflected solutions.
In any pair, each piece mirrors to itself other than pieces 5 and 6 which are reflections of each other.
These simply switch places in a reflected solution.
The question when generating any set of 240 solutions is which solution of each pair is called normal and which is a reflection?
Each of the 240 pairs has two choices, so there are a total of 2^240 ways to create a unique set of 240 solutions.
It's pretty much impossible to even imagine a number that large. If each unique set of 240 is represented by a grain of sand, think about how many grains of sand there are in a handful next time you're at the beach.
Now imagine all the grains of sand on all the beaches on earth. That's about 7 x 10^22 grains of sand.
Imagine if there was an earth around each star in the galaxy.
There are about 250 billion stars in the galaxy, so it would contain 1.7 x 10^34 grains of sand.
There are about 250 billion galaxies in the known universe, so the universe would contain 4.4 x 10^45 grains of sand.
Regarding sets of SOMA solutions, 2^240 is equal to 1.8 x 10^72, so we'd still need 4 x 10^26 (that's 400 trillion, trillion) universes to hold that many grains of sand. That's a lot!

The few sets of solutions that have been used are generated by some method or rule that is used to generate them.
It appears the SOMAP solutions were generated by starting with one solution (unknown which one) and then moving to other solutions by finding configurations of two pieces that fill the same space when the pieces are swapped.
The example below shows the three configurations of a br move, which occurs in numerous places in the lower center of the SOMAP.


A very simple rule was used by Christoph Peter-Orth in his article "All solutions to the SOMA cube puzzle".
He noted that when all solutions are normalized with the T piece laying flat along the front bottom edge, piece 7 (his different numbering system calls it 2) must touch either the left or right face.
This can be used to define a normal solution from a reflected one.
His "piece 2 touches left" rule is a very simple way to narrow 2^240 possibilities down to one.

Anyway, back to SOMAP ......... (more to follow)
-----------------
Also, I recently wrote a program to index any set of 240 solutions to any other.
Attached is the index between SOMAP and the CPO solutions.
A quick sampling shows yours and mine appear to agree exactly, which is a good check.
Note that it excludes B2f since one of the reflected pair (B2f/B4f) had to be removed to make a 240-solution set.



23 Sep 2018 From: Merv Eberhardt

The SOMAP will remain a mystery. RGC and RKG are not about to give up their masterpiece.

To me the "GENESIS" solution is RU2a.
It is at the end of a string of "a" solutions and the swaps are basis two-piece swaps.
My thought process is documented in SOMA News Letter, N160509, "A Way to Solve the SOMAP."
[https://www.fam-bundgaard.dk/SOMA/NEWS/N160509.HTM]
And it leads to all the other SOMAP solutions.

Regarding "normalization."
I first came across the term in an article on www.digplanet.com that no longer exists.
It is not specifically used in the CPO article.
After my article on normalization was published, Ed Vogel brought to life the CPO article and its use of the "T" piece as the basis for those 240 solutions from the CFEV analysis.
Has anyone tried to follow the program that followed the solutions?
That might identify any "tree" structure.
Any "googles" on "SOMA normalized position" will cite my two articles in the SOMA News Letters.



24 Sep 2018 From: Ed Vogel

"The question when generating any set of 240 solutions is which solution of each pair is called normal and which is a reflection?" this really is the crux of the puzzle of represesnting the solutions.

I think locking the Tee to 1,2,3,5 and then permuting the Crystal through the "left side" of remaining "cubelet" openings 4,7,810,11,13,14,16,17,19,20,22,23,25 and 26 provides a fairly simple description of a starting point.
All these solutions would be considered "normal" with in this definition.
I have found and recorded 160 plus unique solutions by hand in this fashion so I am guessing it probably does allow for 240 unique solutions that would then have mirror solutions in the permutation of the Crystal in "cubelet" openings 5, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27.

Bob - Does your computer program allow setting two pieces before it starts solving?



24 Sep 2018 From: Bob Nungester

Regarding the solver program the answer is yes, and no.
The Double SOMA Solver has the ability to use one or two sets of pieces and to exclude any pieces as well.
In other words it will use 0, 1 or 2 copies of each of the 7 pieces. Deselecting both copies of the T and Crystal and then entering a 3x3 cube with the spaces for the T and Crystal removed will result in a desired solution.
The "no" part of the answer is that the program is a figure solver, and it stops after finding one solution.
It doesn't cycle through many solutions to the same figure.

In addition to the solver, I recently wrote the program noted in an earlier email that summarizes the number of solutions at each node of a tree graph, based on the order of piece placement.
That program uses any set of 240 solutions as input and outputs a tree graph with a "mask" of current piece placements at each node.
So far I've just used it as a tool to generate summary data on the number of branches, nodes, and total solutions below each node.
The program is calculating the mask for each node, just not outputting it yet.
The data I sent was generated using the CPO set as input, so it gives all data for each of the 720 (6 factorial) ways of placing the 6 SOMA pieces.
The T piece always starts in the normalized position.
I'd be happy to give you a copy if you'd like, but it needs more development to output the node masks in some convenient way (that's a lot of node masks).
Attached is another copy of the output for placing pieces in order 3, 7, 2, 4, 5, 6, 1 (using Thorleif's piece numbering system).
That is the T followed by the 11 placements of the Crystal, followed by other pieces that give the best balance between number of branches and number of nodes on each level.
Again, this used the CPO solutions for input so they all have Crystal touching the left side.

Regarding the placement of the T and then the Crystal in all left possibilities, that is exactly what CPO did, and the listing of the 240 solutions is included in the paper.
It shouldn't be necessary to generate them by hand.
The 240 mirror solutions can be made by swapping the left and right values in each solution, and changing the designation of the two mirror pieces.



24 Sep 2018 From: Ed Vogel

Hi Bob, thanks for the insights.
I was't sure that was what CPO was doing for some reason. Great to know that is the case.
I will pass on trying to do any coding for the foreseeable future. I have too many projects going.



25 Sep 2018 From: Ed Vogel

I thought that some of the concepts presented in this article regarding "limit periodic" patterns might be of interest:
A Chemist Shines Light on a Surprising Prime Number Pattern
https://www.quantamagazine.org/a-chemist-shines-light-on-a-surprising-prime-number-pattern-20180514/



4 Okt 2018 From: Bob Nungester

I finally wrote up the analysis of the SOMAP, and here it is. It took longer than "one wet afternoon" to do it :-)
Originally it was just written as a newsletter or email, but it got so big I decided to do it more like a paper.
Sorry for the length, but it has a lot of info.
Let me know what you think.
I had trouble embedding the links, so I just quit for now. I can do that later.
I see the link to the spreadsheet opens a copy, but the titles on the left and top don't stay locked.
If you'd like a copy of the original spreadsheet I'll be happy to send it to you.
Thorleif, if you'd like to publish this as a newsletter I can send the original file or do any changes you'd like.



4 Okt 2018 From: Thorleif Bundgaard

I have been following your conversations on the side, it is really interesting (Though I don't really understand all of it)
And I see that you have produced a really fine set of documents.

It will be an honor for me to put these papers into a Newsletter.
And I will of course write the top lines saying "Hello .... This is about SOMAP....."

But again, I am not really understanding the argumentation.
So I would ask of you, if you would please write a short descriptive text, of approximately 10 to 200 lines.
Telling a little about the story that led you all to do the discussions, and a something about the ideas behind the plan.
What was the goal, what did you expect to find, was there an expected HARD area that took a longer time than the rest.?

That text will then be the body of the Newsletter. And I will include the links to the documents and spreadsheets.

I assume that "Secrets of the SOMAP" is the central document.
So if you will mail to me the right versions of the other files that should be accesible frrom the Newsletter

By the way - which mail address is your "SOMA-official" mailbox. bob.nungester9@gmail.com or bnungester@comcast.net ??



12 Okt 2018 From: Bob Nungester

Here's a newsletter, the paper, and the two spreadsheets containing connections between all the solutions.
Feel free to add or edit anything in the newsletter, particularly adding links to the various documents and other newsletters mentioned.
Let me know if there's anything you'd like me to add or change.

You'll probably want to use a PDF of the paper, but I attached the original so you can first update the links so all the files are on your site.

If you have any questions about the SOMAP I can probably answer them now.
I learned a lot more than is in the paper, but between the newsletter and paper there's almost too much information already.



12 Okt 2018 From: Merv Eberhardt

It took me some time to recognize the layouts that you identified with the Newsletter 2017.07.11 SOMAP and then to reconcile them with your presentations. Check this out!
The layouts from the SOMA Newsletter N170711 should be corrected at entries BR2c BR4b U0h

Thanks Merv: This has been corrected in Newsletter 2017.07.11



13 Okt 2018 From: Bob Nungester

Thanks for checking that out! You are obviously correct.
4 B's in a solution is never correct, and I did indeed transcribe the BR6a instead of BR4b (next to it in file).
I swear I checked that all twice before sending it, but somehow I made those errors.
After importing your newsletter solutions to a text editor I eliminated all spaces for easier computer input, so I had to count over to find the desired solution after I'd done the transitions and corrected them.
I think I put the cursor between BR4b and BR6a and then read the solution to the right instead of to the left of the cursor (and wrote down a B instead of G).
Anyway, I verified my file is OK and I had just read and transcribed it wrong.
That was after a long day of staring at solutions and writing up portions of the paper. Thanks for noting the proper corrections.

I just wrote up a newsletter today and sent it to Thorleif with the original spreadsheet files so he can put them on his site.
Attached is a copy that Thorleif and I will edit to produce the final newsletter.



17 Okt 2018 From: Bob Nungester to Thorleif Bundgaard

Hold the presses! I've been in contact with professor Donald Knuth (author of "The Art of Computer Programming" books) and he has reviewed my paper!
I'll make a few changes and send you a new copy.

The newsletter and two spreadsheets don't change, but the paper needs minor revisions.

I see he also sent you a name request.
That's because he gave me a link to the pre-release version of the latest volume of his books, and in it he asks for any known credits for previous work on a list of exercises in the book.
The list includes exercise #408, regarding SOMA and the 1,285 3x9 nonominoes.
I sent him Courtney's and your names and a link to your 1999 newsletter, so you'll get credited in the book.
Here's a link to his pre-release book

http://cs.stanford.edu/~knuth/fasc5c.ps.gz

I'm also getting a coveted "Knuth reward check" (info on Wikipedia) due to an email I sent him regarding a separate analysis I found he had done on the SOMAP.
I didn't really expect a reply, but he responded that he is updating his answer to exercise #402 in the new book, based on my comments.
The check for $2.56 (a hexadecimal dollar) will be framed and put on the wall above my computer.
I must confess I wasn't aware of his books and didn't know who he was when I sent the first email, but now I see he's one of the greatest mathematicians in Computer Science and right up there with J. Conway, R. Guy and M. Guy.



7 Nov 2018 From: Ed Vogel

Hi to All,
just saw this article in Quanta:

Mystery Math Whiz and Novelist Advance Permutation Problem

https://www.quantamagazine.org/sci-fi-writer-greg-egan-and-anonymous-math-whiz-advance-permutation-problem-20181105/

I think SOMAP is a representation of the super permutation of SOMA cube puzzle solutions.
The article mentions some algorithms for estimating the best route in the Travelling Salesman problem.
Maybe these algorithms can be employed to sort SOMA cube solutions and arrive at an even more beautiful SOMAP.





28 Nov 2018 Bob Nungester released a video.
This video show a SOMA Cube mirror transition.
It is a demonstration of the only pair of the 240 solutions to the SOMA Cube that will actually transform to each other's reflection, in two 2-piece moves.
It is best viewed as a continuous loop.

To see this movie, press the ▶ in the left side of this picture.

The video is also available on YouTube: - https://www.youtube.com/watch?v=pgZ2JQKvEEE   "SOMA Cube mirror transitions".



Written by Bob Nungester <bnungester@comcast.net>
Review by Merv Eberhardt <merveberhardt@knology.net>
Review by Ed Vogel <ed_vogel@yahoo.com>
Review by Jan de Groot <groot64@gmail.com>
Placed on SOMAweb by Thorleif Bundgaard <thorleif@fam-bundgaard.dk>

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