Analysis of the checkerboard Soma Cube
This article is the product of Anthony P. Rizzo "firstname.lastname@example.org".
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".
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 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
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
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
Therefore the answer is that there are two different sets of
colored Soma pieces that produce all 240 unique and checkerboarded Soma
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
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
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
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
(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
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 <email@example.com>
Edited by Thorleif Bundgaard <firstname.lastname@example.org>
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