SOMA Crystal
SOMA News 18 May 2003
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The complete "SOMAP" is found


The original SOMAP
Back on a rainy afternoon 1961 J.H.Conway and M.J.T.Guy produced the complete list of all the 240 Soma solutions.
Later this list was presented as a chart, called the "SOMAP" in the book Winning Ways.

I obtained a permission to reproduce the SOMA relating chapter in 1999, and you may read the article in the newsletter
N990201.HTM Winning Ways for your mathematical plays.
Although without the SOMAP map, because it was of too bad quality.
I selected to present a list of all solutions, instead of the "SOMAP"

Then in 1972 the SOMAP reappeared in SOMA Addict Newsletter \NADDICT\NADDICT.HTM (Select page Vol 2 No 2 Page 2)
However, the "SOMAP" was only shown partly, and in a very small scale, where it was hardly readable.

Now - however (30 Apr 2003), the complete "SOMAP" has surfaced, and in this News letter the finder William (Bill) Kustes (of Louisville, Kentucky, USA.) will explain it's history and working.

Bill wrote in his initial letter to me

Have you ever looked at the SOMAP created by Conway & Guy, and featured (partially)in the last issue of the SOMA Addict Newsletter?
It shows connections between all 240 solutions of the cube that can convert one solution to another by rearranging two or three SOMA pieces within the cube (always using the 7-piece standard cube).
I got a full copy of the map from Martin Gardner many years ago (8x11 paper, small print).
It is a challenge to follow because the creators used color designations for the pieces, instead of the 1-7 numbers.
I made a large version of the SOMAP years ago, using piece numbers and following the connections all over the 240 solutions, but the paper got wet and the inks all ran together or washed off.
At present I am working on it again, making a 2x4 foot copy as I verify the piece-swap connections.
The SOMA Addict Newsletter did a good job of explaining the map coding in vol. 2 no. 2 page 2 with a full-page discussion of the map, including the piece color/number assignments, piece dexterity rules, and map coding explanations.
I'm just using those guidelines, substituting piece numbers for piece colors.
Unfortunately, the newsletter only showed part of the map, and humorously suggested we finish it on our own; hence I made the effort to get the full map from Gardner.
(Precious piece of paper, huh?)

It really does add a lot to the mathematical richness and history of the SOMA cube: puts the spotlight back on the cube itself as a puzzle (with 240 solutions), not just as a source of pieces to make other figures with.

Thorleif: Thank you for the SOMAP, maybe I could use that as a basis for making a cleaned up version. Why would I do that? Well, I like my graphics to be readable, but at the same time I need to keep the size a little low, because I have so many figures and only limited server space. (All Soma takes about 11.5 MB right now, and the SOMAP size dropped from 520KB to 66KB)

Looking at the SOMA Addict writeup, I see that there is a lot of extra verbiage in the piece. I will write up a condensed explanation of the SOMAP, Presented here - below.
The only copy I have of the complete SOMAP is the 8 x 11 stained page (Our basement flooded in 1997,; I dried and saved the SOMAP).




The SOMAP


The SOMAP Edited for clarity on the web

Commentary by Bill Kustes, with excerpts from SOMA Addict Newsletter

The SOMAP was created by John Horton Conway and M.J.T.Guy to connect all 240 solutions of the SOMA cube. The map was partially presented in Parker Brothersí SOMA Addict Newsletter in 1972. The map shows a coded description of each cube solution and connects each to other solutions by rearranging two or three pieces within the cube. It really adds a lot to the mathematical richness and history of the SOMA cube: puts the spotlight back on the cube itself as a puzzle (with 240 solutions), not just as a source of pieces to make other figures with.

Conway and Guy used a multi-colored SOMA cube to generate the map, with a single letter designation for each piece. Thus:

Brown=B= piece1; Yellow=Y=2; Green=G=3; Orange=O=4; Blue=U=5; Red=R=6; Black=A=7.

But the map itself is only typed as black on white.

Cube players know that the 7 SOMA pieces can provide 9 corners to the cube, but any cube only has 8 corners. Thus one piece in the cube will not supply one of its corners, and is called the DEFICIENT piece. Note that piece 3 can only supply 0 or 2 corners. If piece 3 is placed to supply no corners, then the other 6 pieces can only supply 7 corners, and the cube cannot be made. Thus piece 3 will always be placed along a cube edge to supply 2 corners.

The piece supplying the block in the center of the cube is called the CENTRAL piece (naturally), and it is possible for the same piece (piece 5 or 6) to provide the central block and a corner block at the same time. It is also possible for a piece (piece 1, 4, 5, 6, 7) to provide the central but no corner block in the cube. Note that piece 2 cannot be the CENTRAL piece and still provide a corner.

Letís look at a few of the SOMAP cube codes. RU1d=651d tells that piece 6 is DEFICIENT, piece 5 is CENTRAL. YU2c=252c tells that piece 2 is DEFICIENT, piece 5 is CENTRAL. A4n=74n tells that piece 7 is DEFICIENT and CENTRAL.

Now on to the DEXTERITY VALUES. Of the 7 pieces, only 1, 2, and 4 can be placed right or left-handed on the face of the cube. (Piece 1 or 4 inside the cube is neither.) If 1, 2, or 4 are placed on the cube face so that they appear as an alphabet letter (L, L or Z), then the piece number is added to the total DEXTERITY VALUE. Piece 1 is checked with one edge at the bottom of the cube face. If the piece(s) appear as a backward alphabet letter on the cube face, then the piece number is NOT added to the total DEXTERITY VALUE.

Letís look again at some cube codes. RU1d tells that 6 is DEFICIENT, 5 is CENTRAL, dexterity is 1 (only piece 1 looks like an alphabet letter; the others are backwards), and it is solution d. YU2c tells that piece 2 is DEFICIENT, piece 5 is CENTRAL, dexterity is 2 (only piece 2 looks like an alphabet letter; the others are backwards), and it is solution c. A4n tells that piece 7 is DEFICIENT and CENTRAL, dexterity is 4 (only piece 4 looks like an alphabet letter; the others are backwards), and it is solution n. U3b tells that piece 5 is DEFICIENT and CENTRAL, piece 1 and 2 look like alphabet letters, and piece 4 is backwards, for a total DEXTERITY VALUE of 3.

The solid lines show every 2-piece rearrangement linking two cube codes. The dotted lines show enough of the 3-piece rearrangements to join the map together and allow the SOMA addict to move from one part of the map to another. One solution, the diamond R7d, requires a 4-piece rearrangement of pieces 1, 2, 5, and 6 to get to R7c.

If you count the cube codes, you will total 241 solutions, because B4f and its reflection B2f are both shown in the map. Thus the SOMAP is a complete catalog of all 240 SOMA cube solutions.

Now letís work through two complete examples. At the center bottom of the map we see RU5b---by---RU6d. RU5b=655b, which means piece 6 is DEFICIENT, 5 is CENTRAL, DEXTERITY VALUE is 5 (pieces 1 and 4 look like alphabet letters on the face). ---by--- tells to swap pieces 1 and 2. So RU5b---by---RU6d means 655d---12---656d, and the final DEXTERITY VALUE is 6.

A second example. At the top right part of the SOMAP we see 01l---you---U3g. This link is a 3-piece rearrangement of pieces 2, 4, and 5. By the numbers then we get 41l---245---53g, which means 4 is DEFICIENT and CENTRAL, DEXTERITY VALUE is 1; rearrange pieces 2, 4, and 5; the result is piece 5 is DEFICIENT and CENTRAL, DEXTERITY VALUE is 3.

How do you get started following the SOMAP? Use the 7 pieces to assemble the cube. (All solutions are shown elsewhere on this website.) Determine which piece is not supplying a corner. This piece is DEFICIENT. Determine which piece supplies the center block. This piece is CENTRAL. (Remember piece 2 or 3 cannot be central.) Look for pieces 1, 2, and 4 on the faces of the cube. Any that look like alphabet letters are added to generate the total DEXTERITY VALUE. Now formulate the cube code and look for it among the 240 solutions on the SOMAP. If you find a match you can follow the 2- and 3-piece rearrangements to get to all other cube solutions. If you do not find a match, then you may have made a cube code reflection. Take your cube apart, make a new one, and try again.

As you work through the map, write down the solution after every five or six links. The written solutions will serve as a cube code to restart from when you get lost or continue from after an interruption.

Have fun!




Submitted by William (Bill) Kustes <wkustes@sud-chemieinc.com>
Edited by Thorleif Bundgaard <thorleif@fam-bundgaard.dk>

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