SOMA Crystal
SOMA News 9 Jan 2001
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The Torsten Sillke figure collection.

From Torsten Sillke, Bielefeld, Germany, Europe.
Mail addres: TORSTEN.SILLKE@LHSYSTEMS.COM
I have received this collection of SOMA figures.

Torsten has been interested in Nonominoes for more than 20 years, here he writes about many of his figures, and as we shall see, there are an interesting number of combinations in these shapes.

Torsten writes:


Figures for the SOMA puzzle, made with a base count of 13 cubes, and with double symmetry.

These figures have a height of two cubes This totals 26 cubes, and the 27'th cube is now deposited in all the possible positions on the top. These positions are denominated using small letters, although ONLY when they are possible to create using SOMA pieces.

Diagonalsymmetry (double diagonal)

    x          a:  7    x x          a: 22      x x
  x x x c      b: 77    x x x        b: 89    x   x
    x a b      c: 55      x a d      c: 90    x x x x x
    x x d x    d: 72        x b e    d:119        x   x
        e      e: 48          x c    e:205        x x

Rotational symmetry (90 Grad)

    x                       x x
    x x x x    b:  3    x   x
    x x b      c: 44    x x x x x
  x x x c      d: 23        x   x
        d                 x x

Axial symmetry (double orthogonal)

  x x x x x    b:  1          x x x        b:  5
    x x b      d: 22      x x x x b c d    c: 28
  x x x d e    e:  8          x e f        d: 27
                                           e:  2
                                           f: 10

  x x   x x    a: 76        x x x x x      a:  2
  x x a b c    b:159        x   a   b      b: 24
  x x   d e    c:143        x x c d e      c:  8
               d:218                       d: 21
               e:204                       e: 34

            x x x x x x x x x x x x x
            
  
            x                 x               x
  x x x x x x x x x x x       x x x x x x x x x
            x                 x               x

    x           x                 x       x
  x x x x x x x x x           x x x x x x x x x
    x           x                 x       x

        x   x                   x     x     x
  x x x x x x x x x             x x x x x x x
        x   x                   x     x     x

      x   x   x                 x x       x x
    x x x x x x x                 x x x x x
      x   x   x                 x x       x x

          x                           x
          x                       x   x   x
  x x x x x x x x x               x x x x x
          x                       x   x   x
          x                           x

      x       x                     x   x
      x       x                     x   x
      x x x x x                   x x x x x
      x       x                     x   x
      x       x                     x   x



Quadratic symmetry
                                    x
      x                             x
    x x x                           x
  x x x b x    b: 84          x x x x x x x
    x x x                           x
      x                             x
                                    x

Now follows figures for the SOMA puzzle, consisting of a symmetrical base of 7 cubes. These figures have a height of 4 cubes. This totals 28 cubes. The 28'th cube is then deleted from all its possible locations in the top layer. These positions are denominated using small letters, although ONLY when they are possible to create using SOMA pieces.

 x x      a: 15          a      a:1060
 x   d    b: 17        x b d    b:  59
 a b c    c: 29        x c e    c: 288
          d: 58                 d:1108
                                e:1403

 x x      a: 208       x x x    b: 3
 x a x    b:1653       x        d: 2
   b c    c: 855       b x d

 x            a: 16                      a: 17
 x x a b c    b: 11         x       d    b: 12
         d    c: 10         x x a b c    c: 10
              d: 33                      d: 33

       x x    a: 6          x x          a: 20
   a b x      b: 4            x          b: 19
       c d    c: 2            a b c      c: 22
              d: 2                d      d: 54

       c d    a: 54           x          a: 50
   x a b      b: 42           x x        b: 36
 x x          c: 54             a b      c: 59
              d:136               c d    d:114

              a: 56         x x x        b: 3
 x x   c d    b: 46             x        d: 2
   x a b      c: 44             b x d
              d:127

         d    a: 1
         c    b: 9
       a b    c: 3
   x x x      d:18

                 x x x       x x x x x           x
   x   x           x             x           x x x x x
 x x x x x       x x x           x               x

                                 x
       x           x             x
 x x x x x       x x x x x       x
       x               x         x x x x

                                 x
       x             x           x x x x x
   x x x x         x x x         x
       x         x x
       x           x                 x x x x x x x

Here are some other SOMA figures, ALL are noted in height notation "Pyramid notation"(red.) (In this notation, is noted how many cubes there is in each position. There are NO hidden holes.)

  2            1 2            2 2 2 2          1 2 1 1
2 2 2 2        2 2 2 2        2 2 3 2        1 1 1 1 1
  2 3 2          2 1 2          2 2 2        2 1 1 1 2
  2 2 2 2        2 2 2 2          2 2        1 1 1 1 1
      2              2 1                     1 1 2 1

TS1: 7         TS2: 15        TS3: 429       TS4: 1


    1 1 1      1 1 1 2 1
  2 2 2 1      1 1 2 3 2
2 2 1 2 1      1 1 1 2 1        4 3            1 1 2 2
1 1 2 2            1 1 1      4 5 4          3 3 3 3 3
  1 2              1 1 1      3 4            2 2 1 1

TS5: 27        TS6: 7         TS7: 87        TS8: 88


1 1 2 1 1      2 2 2 2 2        2 2 2        2 1 2 1 2
3 3 3 3 3      1 2 1 2 1      3 3 3 3 3      2 2 3 2 2
1 1 2 1 1      2 2 2 2 2        2 2 2        2 1 2 1 2
                                                                                    
TS9: 3         TS10: 66       TS11: 15       TS12: 62


1 2 2 2 1        4 4          2 1 1 1 2      2 1 1 1 2
2 3 1 3 2      4 3 4          2 3 3 3 2      3 2 3 2 3
1 2 2 2 1      4 4            2 1 1 1 2      2 1 1 1 2

TS13: 77       TS14: 208      TS15: 32       TS16: 5


                                1 2
                              1 3 3 2          1 2 3
2 2   2 2        2 2 2 2        1 3 3 2      3 3 3 3 3
2 2 3 2 2      2 2 3 2 2          1 3 1        3 2 1
2 2   2 2      2 2 2 2              1

TS17: 76       TS18: 477      TS19: 70       TS20: 281


                   2 1 1        2 2 2          1 1 1
  1 2 3            2 1 1      1 1 1 1 2      2 1 1 1 2
3 3 3 3 3      2 2 3 2 2      1 1 1 1 2      2 1 1 1 2
  1 2 3        1 1 2          1 1 1 1 2      2 1 1 1 2
               1 1 2            1 1 1          1 1 1

TS21: 151      TS22: 2        TS23: 2        TS24: 3


  1 1 1
1 2 1 2 1      2 3 2          2 2 2          1 2 3
1 2 1 2 1      3 3 3 2        2 3 3 3        3 3 3 3 3
1 2 1 2 1      2 3 2          2 3 2              3 2 1
  1 1 1          2              3

TS25: 10       TS26: 394      TS27: 84       TS28: 50


               2 2 2 1
3 2 1          2 2 2 1        2 1   1 2      1 2   2 1
3 3 3 3 3      2 2 2 2        3 3 3 3 3      3 3 3 3 3
    1 2 3      1 1 2 1        2 1   1 2      1 2   2 1

TS29: 254      TS30: 652      TS31: 14       TS32: 2


1 2   2 1      2 1   1 2      1 2   2 1        2 3 2
2 3 5 3 2      2 4 3 4 2      2 4 3 4 2      3 3 3 3 3
1 2   2 1      2 1   1 2      1 2   2 1        2 3 2

TS33: 23       TS34: 3        TS35: 6        TS36: 243


2 2 3          3 2 2                2            1 2 2
2 3 2          2 3 2            1 2 2 2          1 2 2
3 2 2 2        2 2 3 2        1 1 1 2 2 2    1 1 3 1 1
    2 2            2 2          1 1 1 2      2 2 1
                                  1 1 1      2 2 1
                                    1
TS37: 81       TS38: 45       TS39: 26       TS40: 3


2 2 2 2        2 2 2 2        2 2 3 2        2 2 2 2
2     3        2     2        2     2        3     2
2     2        2     3        3     2        2 3   2
2 3 2 3        2 2 3 3        2 2 2 3          2 3 2

TS41: 1        TS42: 14       TS43: 1        TS44: 1


    2 1            2 1            2 2            1 2 2
  2 2 1 1        1 2 2 1        1 1 2 2          2   2
2 2 1 2 2      2 2 1 2 2      2 1 1 1 2      1 2 3 2 1
1 1 2 2        1 2 2 1        2 2 1 1        2   2
  1 2            1 2            2 2          2 2 1

TS45: 6        TS46: 4        TS47: 8        TS48: 1


3 2 2 2 2      2 3 2 2 2      2 2 3 2 2      2 2 2 2 2
2       3      2       3      2       3      2   3   2
3 2 2 2 2      2 3 2 2 2      2 2 3 2 2      2 2 2 2 2

TS49: 1        TS50: 1        TS51: 1        TS52: 2


2 2 2 2 2        2 1 2 1 2 1 2
  3 2 2          1 1 1 1 1 1 1
2 2 2 2 2        2 1 1 1 1 1 2

TS53: 1          TS54: 3


2 1 2 1 1 1 2    1 1 2 1 2 1 1    1 1 2 1 2 1 2
1 1 1 1 1 1 1    1 1 1 1 1 1 1    1 1 1 1 1 1 1
2 1 2 1 1 1 2    2 1 2 1 2 1 1    2 1 2 1 2 1 1

TS55: 2          TS56: 1          TS57: 1


2 2     3 3      1 1     1 1      1     1     1
2 2 2 + 2 3      2 2 1 + 2 2 1    2 1 + 2 1 + 2 1
2 2 2            2 2 1   2 2 1    2 2   2 2   2 2

TS58: 52         TS59:6           TS60:1

Tetromino towers made from SOMA pieces:

These SOMA figures have a base coverage of 4 cubes, and a height of 1 to 6 cubes. You will see that there is NO tower reaching the maximum height of 6.! This lies in piece no. 7, that always cuts the tower in 2 partial towers, as long as it is NOT placed at the end (Top / Bottom) of the tower. The numbering of the SOMA pieces is according to 'Martin Gardner'. Piece no. 1 is NOT used in these tower constructions They are thus partial puzzles. as the number of cubes is then a multiple of 4.

  x x         1: -
  x x         2: -
              3: 2    (2,5,6):1  (3,5,7):1
              4: 5    (2,3,5,7):1  (2,4,5,6):1  (3,4,5,7):3
              5: 4    (2,3,4,5,7):4
              6: -

  x x         1: 1    (4):1
    x x       2: 1    (5,6):1
              3: 3    (2,5,6):1  (3,5,7):1  (4,5,6):1
              4: 6    (2,3,5,7):1  (2,4,5,6):2  (3,4,5,7):3
              5: 3    (2,3,4,5,7):3
              6: -

  x           1: 1    (2):1
  x x x       2: 1    (2,5):1
              3: 2    (2,5,6):1  (3,5,7):1
              4: 4    (2,3,4,7):2  (2,3,5,7):2
              5: 3    (2,3,4,5,7):1  (2,3,5,6,7):2
              6: -

    x         1: 1    (3):1
  x x x       2: 1    (2,5):1
              3: 2    (2,3,5):2
              4: -
              5: -
              6: -

  x x x x     1: -
              2: -
              3: -

These are combinations of a Tetromino and a Pentomino always having a height of 3 cubes.
The:

        x
      x x x
        x
cannot be constructed.

In paranthesis you see the solutions, In which 5,6 are together or not present. This gives a higher degree of symmetry in the combinations.

                                x x   x           x x       x
                                x x   x x x     x x       x x x
                                    =
  x x x x x       4(4)     |        0             0         0
                           |
  x x x           8(5)     |        0             1(1)      0
  x                        |
  x                        |
        x x      29(11)    |        6(1)          7(2)      4
    x x x                  |
                           |
  x x x         113(42)    |       21(5)         26(10)     8
  x x                      |
          x      17(8)     |        4(1)          4(1)      0
        x x                |
      x x                  |
  x               3        |        0             0         0
  x x x                    |
  x                        |
      x   x       8(5)     |        1             1         0
      x x x                |
  x              14(4)     |        2             4(2)      0
  x x x                    |
    x                      |
        x         7(2)     |        0             0         0
    x x x x                |
                           |
  x x x x        15(7)     |        2             2         4
  x                        |
        x x       8(5)     |        1             1         0
        x                  |
      x x                  |




Translated and Edited by Thorleif Bundgaard.

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