SOMA-CUBE
Elwyn R. Berlekamp
John H. Conway
Richard K. Guy
Winning Ways II (Games in Particular)
Chapter 24: Pursuing Puzzles Purposefully
Soma: p 735
Notation (The seven Pieces of Soma)
1=W 2=Y 3=G 4=O 5=L 6=R 7=B
1 1 1 1 1 2 2 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1
White Yellow Green Orange bLue Red Black
The Hidden Secrets of Soma: p 737-739
The 3*3*3 cube has 8 vertex cells, 12 edge cells, 6 face cells and
1 central cell.
Now the respective pieces can occupy at most
W Y G O L R B
1 2 2 1 1 1 1
of the vertex cells, so just one piece, the DEFICIENT one, must occupy
just one les vertex-cell than it might. The green piece can't be
deficient without being double so, and therefore:
the Green piece has its spine along an edge of the cube.
Now let's color the 27 cells of the cube in two alternating colors,
Flame for the 14 FaVored cells, F and V,
Emerald for the 13 ExCeeded ones, E and C.
Then in one solution that we know, the respective pieces occupy
W Y G O L R B
2+2+3+2+2+2+1 = 14 F, V cells.
1+2+1+2+2+2+3 = 13 E, C cells,
but the Yellow, Orange, bLue, and Red pieces, and we now know also the
Green piece, MUST occupy these numbers in every solution, and therefore
so must the White and the Black, since an interchange of the colors in
either or both of these would alter the totals.
The White piece occupies 2 FV cells, 1 EC cell.
The Black piece occupies 1 FV cell and 3 EC ones.
For the placing of a single piece within the box, these considerations
leave only the positions of Fig. 6 (which all arise). You'll see that
up to rotations of the cube, the placement of any single piece is
determined by whether or not it is deficient and whether or not it
occupies the central cell.
The hidden secrets of Soma make it quite likely that one of the first few
pieces you put in may already be wrong, when of course you'll spend a lot
of time assembling more pieces before such a mistake shows its effect.
S The Yellow piece has two possible positions:
S normal (not central, not deficient) -> 219 solutions
S deficient (not central, deficient) -> 21 solutions
The complete list of 240 Soma solutions was made by hand by J. H. Conway
and M. J. T. Guy one particular rainy afternoon in 1961. The SOMAP in
the Extras enables you to get 239 of them, when you've found one - and
located it on the map!
Fig. 6 (short form)
W Y G O L R B
normal * * * * * * *
central * *
deficient * * * *
central & deficient * * * * *
Extras: The Somap p 801-804
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- Michael J. Whiniham, Charles W. Trigg;
Parity and Centerness Applied to the SOMA Cube,
Journal of Recreational Mathematics 6:1 (Winter 1973) 61-66