Solid Polyomino Constructions
Solid Polyomino Constructions
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实心多骨牌结构
DOI：
10.1080/0025570x.1976.11976561
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发表时间：
1976
影响因子：
--
通讯作者：
Scott L. Forseth
中科院分区：
文献类型：
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作者：
Scott L. Forseth
所罗门W. Golomb [4]作为一组n个全等的正方形，它们是单连通的，边到边。一个三维的n阶多面体是一组n个全等的立方体，这些立方体是简单连接的，面对面。这些都很容易通过胶合像立方体在所有允许的模式。没有公式已被发现，这给了一些不同的polyominoes秩序n，平面或固体。图1中示出了阶数一到五中的实心阶数，每个阶数由数字标识。该套包括41件，共186个单独或单位立方体。马丁·加德纳在他的《科学美国人》的数学游戏部分以及他的三本书中讨论了多面手。[1]和[3]中的第十三章是关于平面多项式的。[2]中的第六章涉及著名的皮特·海因的索马立方体，它由图1中编号为4、6、7和9-12的块组成。加德纳指出，有超过230种不同的方式将这些碎片堆叠在一个3 × 3 × 3立方体（或简单地说，一个3立方体）中，但迄今为止没有人确切地知道有多少。类似的立方体出现在施泰因豪斯[5，p. 168]中，其中使用了编号为6，11，12，30，33和40的块。证明施泰因豪斯立方体只有两个解是很容易的。30和40只以一种方式成功地结合在一起。那么33只有两个可能的位置，在每个位置之后，位置6、11和12是唯一确定的。施泰因豪斯评论道：“两件作品因对称而全等。(Was有可能避免吗？）".他指的是11和12号。我们不知道他的答案，但肯定可以避免使用11或12。实际上，除了四个多角形6、30、33、40之外，还可以使用对8、11或8、12或9、11或9、12中的任何一个来构造3-立方体，或者就此而言，Y T1 Z ~
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