SOMA Crystal
SOMA News 2 Jan 2004

Mirroring and SOMA

From: Edwin Hathaway

Sent: Tuesday, December 30, 2003 10:26 PM

My interest in Soma dates back to the Scientific American math page on Piet Hein. Sept 2003, I started to work on making Soma into a software puzzle
( I assume that this has been done numerous times before).
I am also a user of IBM's APL2 for Workstations and at age 76 retired so I had the time.
While working on the program, I did work out 240 solutions to the SOMA cube and a scheme to store them in a particular order. I also developed a CEFO (Corners, Edges,Faces, and Origin=Cube Center) system to study what solutions were possible. i.e. the Tripod piece can assume the following four CEFO's 1 3 0 0, 0 2 1 1 , 1 1 2 0, and 0 0 3 1. Of course the last two CEFO's produce no solutions. I proved this by exhaustively listing all possible CEFO's for each piece and then tried every combination of piece CEFO's to test that they add up to 8 12 6 1, 8 corners, 12 edges, 6 faces and 1 center.

Almost everyone, when referring to puzzle solutions adds the obligaory phrase, after eliminating rotations and reflections. Certainly there are 24 rotations of any cube and these must be eliminated.
HOWEVER, SOMA is DIFFERENT. It already contains two pieces that are mirror images of each other. namely A and B or 5 and 6 in number notation. The result is that if you mirror any SOMA cube, you will end up with pieces that are incorrectly marked 6 and 5 or B and A. If at this point, you re-label these two pieces, to 5 and 6 or A and B, you will have a new solution to the SOMA cube which is not a reflection of the starting SOMA cube solution.

In recording my solutions, I picked a particular standard format for them. For solutions with the P or 7 (I call it the tripod) centered on a cube corner, I always rotate the cube until the P is in the bottom, right, rear corner. (Cube positions 18,24,26,27 )
The cube still has 3 possible orientations in this position. I chose to SPIN the cube on its major diagonal ( the diagonal that runs from position 1 to position 27 or top,left,front to bottom right rear. You can create the spin with two standard rotations of the cube e.g. rotate cw on horizontal axis, and then rotate ccw on the vertical axis) I used the SPIN to be sure that piece T or 3 was visible in a cube drawing as seen looking down on the '3'-corner of the cube. Thus, the center of piece 3 or T is always located on cube positions 2, 6, or 8. The significance of piece 3 or T is apparent when you checkerboard the cube.

If you start with a solution in this or any standard position and then mirror the cube across any plane and then re-label pieces 5 and 6, and restore the cube to its standard position, you will see that you have a new unique solution.
I refer to this transformation as the 5-6-reflection.

My standard format for solutions with the 7 or P centered on the center of a cube face was to rotate the cube until the center of piece 7 or P was located on cube position '17' and the three arms are located on 'cube positions 18 , 26 and 14' ; Position 14 is the center of the cube.

Now for the STARTLING NEWS. In accruing my 240 solutions to SOMA, I was amazed at how easy it was to find new solutions. I had expected it to get harder as I got near the total of 240. It was very easy especially using my CEFO scheme.

Therefore, I tried for a new solutions and almost immediately found a 241st and its 5-6-reflection for 242 solutions. I now have 48 solutions for the Soma cube with piece 7 centered on a cube face and 194 solutions with piece 7 centered on a cube corner

In comparing any new solution to those previously recorded, I first orient the new solution in standard format. Then, my task is simplified by IBM's APL2 for Workstations. It provides an easy way to compare arrays using a command similar to matrix multiplication. I can elaborate on this if anyone cares.

For the record, I'm using an old AMD K6-2 processor with something like 350m instruction speed and Windows 98. To compare and 242 x 27 array with itself looking for duplicate rows takes less than a minute.

TO: Thorlief
Thanks for pointing me to Conway's ' Winning Ways' site. His reputations makes me withdraw my claim to 242 solutions until I can compare and see where I may have gone wrong. It'll take me a while to translate his color designations to your names of pieces and mine. Since your numbering system is the same as his, I will label the pieces 1VW, 2LY, 3TG, 4ZO,5AL,6BR,7PB, to help my feeble brain keep track of the pieces. I'll also have to translate his letter arrays which are layered top to bottom with my APL arrays which are layered front to back. Many, Many thanks for your Site and pointing me to the right information.

TO: Dr. John Conway,
A half night's reflection (pun intended) leaves me sure that you know all that I've included in the following e-mail to myself. In an age when half the world is shrieking 'Jihad', I tend to look aghast at purists in any form. You might have difficulty facing the puzzle solver who through some different brain wiring produced a complete set of your 'Reflected' solutions and claimed that your solutions were all invalid because they were mere reflections of his-er-hers.

Academics in general are often overly impressed with themselves (present communicants excluded). I couldn't even get a single math prof to look at the power of the APL2 programming language. It enabled me to uncover your private joke of insisting that there were only 240 solutions to the SOMA cube in the last 48 hours during which I produced my 241st and 242nd solutions, and exchanged 3 e-mails with my new friend, Thorlief Bundgaard. It was he who pointed me to your 'Winning Ways' chapter on his WebSite. I assume that the title, 'Winning Ways' is also a multi-level pun.

Astunded Punster
Edwin Hathaway

Puzzles should generaly not be reflected or rotated but the Soma Cube is a special case because two of its constituent pieces are mirror images of each other. If you simply reflect a solution in any mirroring plane, the resulting cube is not a solution because the two pieces red and blue have changed shape. However, if after mirroring, we relabel or recolor the two miscreant pieces, we will have a new solution. I have drawings of the first piece listed in your list of 240 solutions.
Picture 1 shows your first solutions:

SOMA Crystal
Picture 2 shows this solution mirrored thru a diagonal plane running from the front left edge to the rear right edge. You will note that the pieces bLue and Red are incorrectly shaped.

SOMA Crystal
However if we relabel the two miscreants as shown in picture 3, we will have a brand new solution.

SOMA Crystal
I have done a similar mirroring of all 240 of your pieces and found that the special R-L-Reflection produced 240 new solutions, none of which matched any of your original solutions. My checks included transforming all your 240 original solutions and their R-L-Reflections into my standard format.

For pieces with the Black tripod centered in a corner, I rotate the cube until the Black tripod is in the bottom right rear corner, and then spin the cube until the Green tee is on the front of the cube.

For pieces with the Black tripod centered on a cube face, I rotate the cube until the center of the tripod is on the center face of the bottom of the cube and two of its legs bracket the bottom right rear corner. No spin resolutions is possible in this case.

Having established all the original pieces and all 240 R-L-Reflections in my standard format, I am able to use IBM's APL2's '+.=' array command to verify that each of the 480 pieces are distinct. To do this, I translated all your solutions to my numbering system. I also merged your 240 solutions into my 242 solutions and ended up with 360 unique solutions. The R-L-Reflection should produce the full 480 solutions.

  Hi Thorlief,                                         Jan 3, 2004
  In order to work with Conway's solutions, I found it
  easier to translate it to your notation with its meaningful
  letters, L Z V T  etc.  I thought I would pass it along to
  you.  You may find it helpful.  If not, discard it.  I'm
  attaching it to this e-mail.

  Do you know if the question of reflection of the SOMA 
  solutions has come up before?  The peculiarity of SOMA 
  is that two of its pieces (A and B) are mirror reflections 
  of each other and so after the relection, these two pieces 
  must be relabeled.  The result is that there is no true 
  reflection of the entire solution.  

  Someone in the U.S. has created a wooden Soma puzzle
  which is skewed in 2 directions creating a rhombohedron.
  It only has one solution:
  L VV
  L VZ
  L LT

  P BB
  A ZZ
  A TT

  P PB
  P ZB
  A AT
  which is not a member of Conway's 240.
  Conway lists its  A-B-Reflection
  L L L
  V V L
  V Z T

  P B B
  A Z T
  A Z T

  P P B
  P Z B
  A A T
  You will have to look long and hard to find this 
  solution in Conway, because of rotations and
  different notation.
          Best Wishes and Thanks for work on the WebSite


Now in Jan 2004 Edwin continues his research, having found that some pieces seem to form partnerships.

So here's Edwin's description

January 25, 2004
Dear Thorlief, While studying the 480 possible solutions of SOMA, I was looking for notable characteristics and I came across some solutions that differed from each other by only 2 cublets of the 27 cublets in SOMA. It certainly was not something that my intuition had led me to expect. Perhaps you have noticed it before. There seem to be 3 patterns that can produce this condition.
One is fairly obvious. ‘V’ and ‘T’ positioned so each can rotate swapping one location of the V and T.
The other two patterns that can produce the 25 cublet match are with the ‘L’ and either the ‘A’ or the ‘B’. This requires that the L be swapped end-for-end and rotated a quarter turn. The A or B piece can then be moved so it vacates the location now used by the L and fills the location which the L vacated. This is all too wordy and may be ‘old hat’ to you.

I’ll include a drawing of the 3 patterns pairs with your letter designations and Conway’s colors except for Orange which I haven’t found a way to create on my screen. These conditions make good starting points for building a SOMA cube. Very Best Wishes and Thanks, Ed Hathaway, echotel

In March 2004 Edwin reflects to Mr Conway, about the number of cube solutions. Debating wether two visually identical solutions really are the same, if a piece has been rotated to its mirror position.

From: "Edwin Hathaway"
To: "John Conway"
Subject: Reflections on Soma Reflections
Date: Fri, 5 Mar 2004 08:23:50 -0500

Dr. Conway:
I have tried to call your attention to the omission in Chapter 24 of Vol. 2 of your book, "Winning Ways" as shown on Thorlief Bundgaard's website before this, but received no response from you.

According to Thorlief's website, you state:

"The complete list of SOMA solutions was made by hand by J.H. Conway and M.J.T. Guy one particularly rainy afternoon in 1961."

Your list only contains 240 solutions. The other 240 configurations of the SOMA Cube not listed are SOMA reflections of your 240 solutions. As I am sure you have noted, the SOMA Cube contains two pieces that are mirror images of each other, namely blue and red or 5 and 6 in number notation.

The result is that if you mirror any SOMA cube, you will end up with pieces that are incorrectly marked 6 and 5 or red and blue. If at this point, you re-label these two pieces, to 5 and 6 or blue and red, you will have a new solution to the SOMA cube which, although a reflection of the starting SOMA cube solution, is not a simple reflection if the pieces are labeled in any fashion. For instance, you may have been working with pieces that were painted the colors you use for identification.

Any puzzler starting out to work on building the SOMA Cube, has a fifty/fifty chance of getting one of your listed solutions or a SOMA reflection of one your solutions. Neither possibility is more correct than the other. As you listed your 240 solutions, for each solution that you picked, you rejected its reflection.

This was your personal choice, and does not establish your 240 solutions as the one and only correct list of solutions. The following diagram illustrates the situation.  The solution on the left is #81 in your list. I will be asking Thorlief Bundgaard if he will publish this diagram on his web site, if you choose not to respond.

Yours truly, Edwin B. Hathaway

  Consider the two solutions to the SOMA Cube shown above.
  One of them is a Conway solution. The other is not.
  Each is the mirror image of the other reflected across the
  major diagonal plane from the front right edge to the rear
  left edge of the cube. There is a problem, though. The
  reflection has changed piece 6, red, into piece 5, blue and
  vice versa. All 240 of Conway's solutions have a reflection
  which is also a solution to the SOMA Cube. Though a puzzler is
  obligated to eliminate reflections and rotations, he is not
  compelled to select Conway's choice over it's reflection.
  There are 480 solutions to choose from.

Submitted by Edwin B. Hathaway <>
Edited by Thorleif Bundgaard <>

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