10 Jan 2001
Mathematical Solitaires & Games
edited by Benjamin L. Schwartz
Comments to the article:
EXCURSIONS IN RECREATIONAL MATHEMATICS
Parity and Centerness Applied to the SOMA Cube
Michael .J. Whinihan
Charles W. Trigg
Twelve different shapes can be formed by combining not more than four cubes, all the same size and joined at their faces. The seven of these shapes, one 3-cube and six 4-cubes, left after eliminating all straight shapes (rectangular parallelepipeds) constitute the SOMA puzzle invented by Piet Hein. The challenge of the puzzle is to use the seven pieces to construct any one of a profusion of specified forms.
Rather than use Hein's numerical designation of a piece, we have substituted the capital letter which has the same position in the alphabet as the number has in the set of ordered positive integers. Thus, piece 1 becomes piece A, piece 2 becomes piece B, and so on. This heresy is committed in order to reduce confusion among the piece number, the number of cubes in the piece, and the two indices which are introduced below.
An orderly approach to the construction of a form can be facilitated by applying the measures of parity and centerness. These indices may be used to identify restricted locations of a piece in a construction and the impossibility of certain constructions and components.
The following discussion deals with the application of parity and centerness to the assembling of the seven SOMA pieces into a 3 x 3 x 3 shape, the SOMA cube.
Parity, an independent property of a single piece or a combination of pieces, derives from the "excess color" technique of Jotun Hein .
The cubes composing each SOMA piece or combination of pieces can be colored alternately black and white so that no two faces in contact have the same color. The parity of the piece or combination of pieces is defined as the number of black cubes minus the number of white cubes.
The parities of the seven pieces, A, B, C, D, E, F, and G, are shown in Figure 1. Piece A could have either 2 black and 1 white or 1 black and 2 white cubes with corresponding parities of +1 or -1. Similarly, the parities of C and G each are +/-2. The other four pieces have parities of zero. Where necessary to distinguish between two pieces with the same shape but different parities, the parity is indicated in parentheses, as in A(+1) and A(-1).
1) E and F each can fit in the SOMA cube in two ways with
centerness 2. Thus, there are 28 distinct positions that
may be assumed by one of the 7 pieces in a 3 x 3 x 3 shape.
2) Referred to the SOMA cube with parity 1.
It is convenient always to take the parity of any completed construction as non-negative, which is to say, it has at least as many black cubes as white cubes. Accordingly, the assembled SOMA cube has a parity of 1, with a white cube at its center. In any orientation, the SOMA cube consists of the three layers in Figure 2.
Centerness. The centerness of a piece depends upon its location in the structure of which it is a part, and upon the number of corner cubes and edge cubes in the structure. The centerness measure of a piece is the number of cubes it contains that are not corner cubes nor edge cubes of the structure. Consequently, each piece may take on a variety of centerness values as it moves about in the structure.
The SOMA cube with a parity of 1 contains 12 white mid-edge cubes, 8 black corner cubes, 6 black face-center cubes, and 1 white cube-center cube. The 7 center cubes are indicated by dots in Figure 2.
Piece A(-1) may be located in the SOMA cube so as to contain 0, 1, or 2 center cubes. A(+1) may contain 1, 2, or 3 center cubes. The several possible centerness values of each piece are listed in Figure 1. Bracketed are those eliminated by the following analysis. The centerness of a piece in a particular position or discussion is indicated by a subscript, as in A(-1)2 or B0.
1. The SOMA cube's parity of 1 must be compounded from the parities of the non-zero pieces A, C, and G. Consequently, the parities of C and G are opposite and A has parity 1. It follows that A(-1) cannot appear in the SOMA cube. A(+1) must appear in one of three essentially different positions; at a corner of the middle layer, with its corner at the center of the middle layer, or with an end at the center of a top or bottom layer.
2. Every cube adjacent to the cube-center cube is a face center. Hence, if a piece contains the central cube it must have a centerness of at least 2.
C(-2) and G(+2) together with the other five pieces have a minimum centerness of 7, which is the maximum for the SOMA cube. Since the minimum for G(+2) is 2 it is the one that contains the cube-center. But, if it does, it also contains 3 other centers, raising the total centerness to the impossible 9. This eliminates C(-2) and G(+2) from SOMA construction.
3. C(+2) and G(-2) together with the other five pieces have a minimum centerness total of 5. Hence, no piece can exceed its minimum centerness by more than 2. This eliminates B3 and C(+2)4.
4. If B2 replaces B0 the minimum centerness total is raised to 7. But B2 must lie along an edge of a middle layer, so some other piece contains the cube-center cube and has a centerness of at least 2. This raises the centeness total to an impossible value, so B2 is eliminated.
5. It is now established that C and B lie in outside layers.
6. If G(-2) contains the cube-center cube, then A(+1)1, C and D1 lie in outside layers.
Impossible Components of the SOMA Cube
There are portions of a 3 x 3 x 3 cube that could have been made by parallel assembly of a number of 3 x 1 x 1 parallelepipeds. Any such shape that also can be made from some of the seven SOMA pieces will have to contain either 3(4) or 3(4) + 3 cubes. Indeed, a 12-cube shape and a 15-cube shape are complementary . If one can be a component of the SOMA cube, the other can be also. If either is shown to be impossible as a component of the SOMA cube, then neither shape can be a component.
It follows immediately that 3 x 1 x 1, 3 x 2 x 1, and 3 x 3 x 1 parallelepipeds are impossible components.
A group of three 4-cube pieces can be chosen in C(6, 3) or 20 ways.
The 3 x 2 x 2 parallelepiped R in Figure 3 has 0 parity. Eight of the groups of three pieces have this parity, but only three of them can be assembled into R. That is, R = B + E + F = C + E + G = C + F + G. If R is to be a component of the SOMA cube, one of its long edges must be an edge of the cube. Hence, R has a centerness of 5. The 3-cube A(+1) can be added to R in two ways.
Placed in contact with the faces a, a of R, the piece A has a centerness of 2. Hence, the other three pieces in the cube must have a total centerness of 0. The minimum total centerness of any three pieces is 1. Consequently , this assemblage cannot be part of the SOMA cube.
In its other possible attitude in contact with R at one of the a faces, A has a centerness of 1. Now, if B, E, and F constitute R, the only position G can assume is in a corner of the SOMA cube. Thus, the remaining two pieces must have a total centerness of 1. But D and C together have a minimum centerness of 2. If either C, F, G or C, E, G constitute R, then E or F, whichever is not in R, can occupy only positions with a centerness of 1. The remaining pieces must have a combined centerness of 0. But B and D together have a minimum centerness of 1. Consequently, neither arrangement of R and A can be a component of the SOMA cube.
If the bottom layer of S in Figure 3 were a middle layer of the SOMA cube, two straight 3-cube spaces would be isolated. Therefore, the bottom layers of S and of the SOMA cube must coincide. Whereupon, S has a parity of 2 and a centerness of 4. Thus S contains C(+2) and neither G(-2) nor A(+l). But C and D together cannot fit into S, so S = B + C + E = B + C + F. Then the complement of S must be composed of A, D, G, and either E or F. The only place that G(-2) can fit is in a corner of the cube. To avoid isolating a 1-cube space, D must be placed in the top layer of the cube so as to leave a 3-cube space for A. The remaining 4-cube space can be occupied only by a duplicate of B, and not by E or F. Hence, S cannot be a component of the SOMA cube.
A SOMA Cube
T in Figure 3 has a parity of 0 and a centerness of 2. With T = C + F + G, the other four SOMA pieces can be assembled into the other two shapes of Figure 4. Together the three components constitute one of the many possible constructions of the SOMA cube.
Other Impossible Components
Other 12-cube shapes can be formed by cross-piling four 3 x 1 x 1 shapes, two to a layer. Some of these are shown in Figure 5.
The symmetry of U(0) requires that one of its corners be black. Its complement consists of S(2) with a 3 x 1 x 1 piece cross-piled on cube a. The only SOMA piece which can occupy the cross-pile is C(-2), which has been eliminated for use in the cube. Hence, U(O) is impossible as a SOMA cube component.
V may assume two positions in the SOMA cube. Positioned at the top of the cube, V(+2) has a slot on the top that can only be filled by B. But B cannot lie in a middle layer. Positioned at the bottom of the cube, V(-2) must contain G(-2), which can only be located in a lower comer. But, this isolates a 2-cube space in V(-2) which cannot be filled with a SOMA piece. Consequently, V cannot be a component of the SOMA cube. As a non-component, V = B + C + D = B + C + E = B + C + F .
In any orientation, the most widely separated cubes of W and of X will be oppositely colored. Thus they always may assume the position shown in Figure 5.
W has a parity of O and a centerness of 4, so must
contain both C(+2) and G(-2) or neither of them.
G(-2) must occupy a comer position. In any place
that C(+2) can be placed in the remaining part of W,
it will have a centerness of 2 which is impossible
for it in the SOMA cube. Consequently, this pair
of pieces must appear in the complement of W.
This complement consists of T with a 3 x 1 x 1
piece cross-piled at an end of its top layer.
This cross-piled piece can be occupied only by B,
which leaves a residual of the complement consisting
of a 9-piece bottom layer to which a 2-cube is attached.
If G(-2) is placed in a comer of the residual it
will isolate a top 1-cube space. If G(-2) is placed
in the middle of the residual, it will require A and
a duplicate of B to complete the residual.
Thus, W cannot be a component of the SOMA cube.
The same impossibility proof used for W applies to structure X.
The concepts of parity and centerness, as defined, are useful in analyzing constructions made with SOMA pieces. With this aid impossible positions for the various pieces in the SOMA cube have been established: three for A, two for B, three for C, and two for G. Also, six selected l2-cube structures, R, S, U, V, W, and X, some of which can be formed from SOMA pieces, have been shown to be impossible as components of the SOMA cube.
1. Introducing SOMA, Parker Brothers, Inc., Salem, Mass., pp. 38-40, 1969.
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