Solid Polyomino Constructions

Abstract

A plane polyomino of order n is defined by Solomon W. Golomb [4] as a set of n congruent squares that are simply connected, edge-to-edge. A three-dimensional, or solid, polyomino of order n is a set of n congruent cubes that are simply connected, face-to-face. These are easily made by gluing like cubes together in all permissible patterns. No formula has been discovered which gives the number of different polyominoes of order n, plane or solid. The solid ones of orders one to five are shown in FIGURE 1, each being identified by a number. The set contains 41 pieces and a total of 186 individual or unit cubes. Martin Gardner has discussed polyominoes in his Mathematical Games section of Scientific American and also in each of his three books [1, 2, 3]. Chapters thirteen in [1] and [3] are concerned with plane polyominoes. Chapter six in [2] deals with the well-known Soma cube of Piet Hein consisting of the pieces numbered 4, 6, 7 and 9-12 in FIGURE 1. Gardner points out that there are more than 230 essentially different ways of stacking these pieces in a 3 x 3 x 3 cube (or simply, a 3-cube) but to date no one knows precisely how many. A similar cube occurs in Steinhaus [5, p. 168] where pieces numbered 6, 11, 12, 30, 33 and 40 are used. It is easy to prove that there are just two solutions to the Steinhaus cube. The 30 and the 40 go together successfully in just one way. Then the 33 has only two possible positions after each of which the positions of 6, 11 and 12 are uniquely determined. Steinhaus remarks: Two pieces are congruent by symmetry. (Was it possible to avoid it?). He refers to numbers 11 and 12. We do not know his answer but it is certainly possible to avoid using either 11 or 12. Indeed, a 3-cube can be constructed using, in addition to the four polyominoes 6, 30, 33, 40, any of the pairs 8, 11 or 8, 12 or 9, 11 or 9, 12 or, for that matter, Y T1Z~~~~~ZZ~~~IflT 1 2 3 4 5