Soma puzzle

[lpetrich]

The  Soma cube is a puzzle invented by mathematician Piet Hein in 1933.

One has seven pieces consisting of smaller cubes that one tries to assemble into a larger cube. I'll call those smaller cubes cells here.

The pieces with their names, how many cells, and cell arrangements:

V 3
##
#

L 4
###
#

T 4
###
 #

Z 4
 ##
##

P 4
X#
#

A 4
##
X

B 4
#X
#

# is a cell, X is a cell with another cell on top of it.

This is a 3D version of 2D polyominoes, something that I mentioned in HAKMEM - collection of mathematical curiosities and useful algorithms

There are some missing ones, however.

#

##

###

####

##
##

 

[lpetrich]

Here is how many are possible, starting with 1 cell.

A000162 - OEIS - Number of 3-dimensional polyominoes (or polycubes) with n cells.
1, 2, 8, 29, 166, 1023, 6922, 48311, 346543, 2522522, 18598427, 138462649, 1039496297, 7859514470, 59795121480

A038119 - OEIS - Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification).
1, 2, 7, 23, 112, 607, 3811, 25413, 178083, 1279537, 9371094, 69513546, 520878101, 3934285874, 29915913663


Pieces A and B are mirror images of each other.

How many solutions: 240, within rotations and reflections (24 rotations, 24 reflections)

There are some constraints that one can find for solutions.

One of them is that the long direction of the T must be at an edge, meaning that the two ends in that direction are at corners.

Here is a proof. The T can contain either 0 or 2 corners, the L 0, 1, or 2 corners, and the five remaining pieces 0 or 1 corner.

The best case is 9 corners, more than 8 corners, but if the T is not at the edge, then the best case is 7 corners. So the T must be at an edge.


Another constraint is parity. Imagine an infinite 3D grid of Soma cells, and give one of them the value +1. Its neighbors across from each face have a reversed of its sign, or -1. Extend this to every cell. All the cells with +1, or else -1, form a face-centered cubic lattice, much like the sodium and chloride ions in table salt, NaCl.

For our target, a 3-cube, let us set one of the corners to +1. Then:

• Corners (8): +1
• Edge centers (12): -1
• Face centers (6): +1
• Overall center (1): -1

Total: +1

So this cube's parity sum is 1.

The Soma pieces have parity sums

• 2: T P
• 1: V
• 0: L Z A B

which can be + or - depending on how they are placed relative to the grid.

Since the long direction of the T must lie along an edge, the T makes a contribution +2. That means that the others must make contribution -1, and that is only possible for P -2 and V +1. That means that the P's center must be at an edge center or the overall center, while the V's center must be at a corner or a face center. The others' locations are not constrained by this method.


This might be easier to picture in two dimensions. Consider dominoes. Try to make shapes with them flat on a surface and touching each other. What shapes are possible? A domino has 2 cells, so a domino shape must have an even number of cells. Using this parity argument, a domino has a parity sum of 0, and every constructible domino shape has a party sum of 0.

Thus, this is constructible:
####
####
####

but this isn't:
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####
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