SOMA Crystal
SOMA News 20 Dec 2020
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Bob Nungester about "SOMA Trade Game".

Back i july 2020. Bob Nungester sendt me some thoughts about the mathematical consequences of the SOMA-game mentioned already in the earliest booklets.
The "SOMA Trade Game"
So without hesitation, But alas, considerable delay. I will now let Bob Nungester present the information to you:

Bob:
Even the very first SOMA set from 1934/35 presents this game. From that original set (Newsletter 2003.03.05) we can read a text on the outside of the box.
The 7 SOMA-Pieces
can be combined to the cube
in over 100 Ways.
 If you switch a Piece
in one SOMA-puzzle with one
in another, ie. Nr.4 with Nr.7,
just not Nr1, you may still
always build the Cube.
                    SOMA can be played by
two or more. You start
simultaneously building the same
Figure with separate SOMA-
Games, or exchange a Piece
and build the Cube.
 The one, finishing
first, has won

I checked how many times each placement of each piece orientation occurs in the cube.
The TradeGame.xls file analyzes the SOMA game of two players trading a single piece to form two non-standard sets.


Using the Double Solver program and the Unique Solutions program
I analyzed each of the 30 possible nonstandard sets to see how many cube solutions are possible with each set.
The number of solutions varies greatly from only 13 (no piece 2, extra piece 4) E5:
up to 849 (no piece 5 or 6, extra piece 2) C8 or C9:.

In any game the two sets are complementary, so the second table in the spreadsheet shows the ratio of
solutions for these two sets.
For example, the set with no piece 7 and an extra piece 2 C10: has 635/18 = 35.28 times as many solutions
as the set with no piece 2 and an extra piece 7 H5:.

These tables show the best piece for the first player to give up is piece 7 since more solutions are
available than the other player's set regardless of which piece they trade back.
Piece 2 should never be given up since the other player will have from 9-35 times more solutions available.
Swapping piece 5 for piece 6 is interesting since the two sets are mirror images of each other.
That means they both have an identical number of solutions.


The cell formulas in the lower part are the divisions. D17=D5/C6   C18=C6/D5   and   E17=E5/C7   C19=C7/E5 etc.

The file, WinningWays.xls shows the number of solutions where each piece is in each of its possible positions.
These numbers can be derived from the solver program, or directly from an analysis of the SOMATYPE of each solution on the SOMAP.
A lot more about examining the SOMAP is found Here.


The cell formulas of row 10 are "Total   =SUM(B5:B8)   =SUM(C5:C8) etc.
In column F [5,6*] the pieces 5 and 6 are reflections of each other instead of reflections of themselves,
so occurrences of each can't be separated from the total of both.




Made by Bob Nungester <bnungester@comcast.net>
Adjusted by Thorleif Bundgaard <thorleif@fam-bundgaard.dk>

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