
SOMA News 
2 Jan 2004
EMail. 
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
relabel 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 relabel 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 56reflection.
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
56reflection 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 K62 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 email 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 hiserhers.
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 emails 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 multilevel 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:
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.
However if we relabel the two miscreants as shown in picture 3, we will have
a brand new solution.
I have done a similar mirroring of all 240 of your pieces and found that the
special RLReflection produced 240 new solutions, none of which matched
any of your original solutions. My checks included transforming all your
240 original solutions and their RLReflections 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 RLReflections 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 RLReflection 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 email. 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 ABReflection 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 Ed##############################################################################
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
endforend 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 relabel 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.