Analysis of the checkerboard Soma Cube

Checkered

This article is the product of Anthony P. Rizzo "arizzo@sbdp.com". It discuss the question - Will Checkerboarding the soma cube, reflect in its solvability.?

This question arises again and again concerning the Soma Cube. How many of the 240 unique ways to form a cube result in a checkerboard pattern?
We can simply put the Soma pieces together into a cube, color the blocks in a checkerboard pattern and then take the cube apart. Looking at it this way, the answer to our question is obviously 240? When we take each cube apart, we have a set of the seven pieces each colored in a particular way. The real question now becomes, "How many different sets of the seven pieces are there that create the 240 unique and checkerboarded Soma cubes?

In "Winning Ways for your mathematical plays volume 2: Games in Particular", chapter 24, Berlekamp, Conway, and Guy present a figure titled "All Possible Positions for the Seven Soma Pieces". Newsletter 990201

Let's assume that the 8 vertex cells (V) and the 6 face cells (F) are colored dark and the 12 edge cells (E) and 1 central cell (C) are colored light.
By looking at the possible positions that each of the seven pieces can take in the cube we can easily determine that all the pieces except the number 2 or Yellow piece shaped like an L have only one way to color the blocks of the piece.
Piece 1 must be colored with the two ends dark and the corner light in order to fit in the normal, deficient and deficient & central positions.
If piece 1 could be placed in a central position, it would need to be colored with two light ends and a dark corner, but this position never occurs for piece 1 when forming a cube.
The number 2 piece can be colored in two different ways.
In the normal position, the long leg of piece 2 must be dark, light, dark and in the deficient position the long leg of piece 2 must be light, dark, light.
Piece 3 is always in the normal position and must be colored with the two ends and the projecting block dark and the middle block light.
Pieces 4, 5 and 6 can be simply rotated so that it doesn't matter if they are colored dark, light, dark, light or light, dark, light, dark.
Piece 7 is always in the normal or deficient & central positions and in both these positions it must be colored with the corner dark and the other three blocks light.
Therefore the answer is that there are two different sets of colored Soma pieces that produce all 240 unique and checkerboarded Soma cubes.

The figure in "Winning Ways" does not indicate the number of ways that each of the pieces actually occurs in a cube. Here is a breakdown of those counts:

            1     2     3     4     5     6     7
            W     Y     G     O     L     R     B

Normal     207   219   240   207   134   130   203

Central                             47    53

Deficient   14    21                36    29

Deficient   19                33    23    28    37
 & Central

If we start with a set where piece 2 has the long leg of the L colored dark, light, dark and it is placed in a normal position, then there are 219 ways to form a cube and all 219 of these ways are checkerboarded.
If we start with a set where piece 2 has the long leg of the L colored light, dark, light and it is placed in a deficient position, then there are 21 ways to form a cube and all 21 ways result in a checkerboarded cube. If the resulting cube is not checkerboarded, then pieces 4, 5, and/or 6 need to be rotated.

What this all means is that if we we place piece 2 in the normal or deficient position depending on its coloring, then the resulting cube will always be checkerboarded!
(Again, pieces 4, 5, and/or 6 will need to be rotated if the cube is not checkerboarded.)
If we have a set with piece 2 with the long leg of the L colored dark, light, dark and we create a cube and do not pay attention to whether it is in a normal or deficient position, there is a 219/240 chance or 91%+ that the resulting cube will be checkerboarded.

There are quite a number of web sites selling checkerboarded Soma cubes that say that there are 4 or "about" 4 ways out of the 240 ways to form a cube that is checkerboarded.
We can see from this analysis that the number of ways is either 21 or 219 depending on the coloring of piece 2 in their sets.

Anthony P. Rizzo lives in Fort Thomas, Kentucky, USA Now 49 years, is married to Joan, have 2 children, Lauren just out of college and Stephen who is a junior in college.
Anthony hold a degree in computer science from the University of Kentucky, and a Master in Business Administration from Xavier University of Cincinnati. Daily work is programming and network administrator. and one hobby is "solving math problems by developing alogrithms"

Submitted by Anthony P. Rizzo <arizzo@sbdp.com>
Edited by Thorleif Bundgaard <thorleif@fam-bundgaard.dk>

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