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Solution Numbers for Eight Blocks to Madness Puzzle

Inga Johnson, Erika Roldan

TL;DR

The work analyzes the $2\times2\times2$ target puzzle built from the ${30}$ MacMahon colored cubes, focusing on the non interior-face matching version. It introduces a graph model ${M}$ that encodes corner-numbers as vertices and cubes as edges, enabling exact counting of a collection’s solution number for a given target; exhaustive computation yields that at most ${16}$ solutions occur and characterizes when this maximum is achieved. Across all ${\binom{30}{8}}$ collections, the results show many collections solve no target, while a subset solves multiple targets (up to ${5}$), with ${360}$ collections achieving this maximum; the study also identifies ${9}$ new minimum universal sets of ${12}$ cubes and conjectures a complete structural description for all such sets, revealing a rich symmetry under ${S_6}$. The combination of graph-theoretic modeling and algorithmic enumeration provides a comprehensive map of which eight-cube collections can realize which targets and how these properties scale as the pool of cubes grows. These insights advance understanding of MacMahon-based puzzles and offer computational tools and structural conjectures relevant to universal-set questions in colored-cube puzzles.

Abstract

The 30 MacMahon colored cubes have each face painted with one of six colors and every color appears on at least one face. One puzzle involving these cubes is to create a $2\times2\times2$ model with eight distinct MacMahon cubes to recreate a larger version with the external coloring of a specified target cube, also a MacMahon cube, and touching interior faces are the same color. J.H. Conway is credited with arranging the cubes in a $6\times6$ tableau that gives a solution to this puzzle. In fact, the particular set of eight cubes that solves this puzzle can be arranged in exactly \textit{two} distinct ways to solve the puzzle. We study a less restrictive puzzle without requiring interior face matching. We describe solutions to the $2\times2\times2$ puzzle and the number of distinct solutions attainable for a collection of eight cubes. Additionally, given a collection of eight MacMahon cubes, we study the number of target cubes that can be built in a $2\times2\times2$ model. We calculate the distribution of the number of cubes that can be built over all collections of eight cubes (the maximum number is five) and provide a complete characterization of the collections that can build five distinct cubes. Furthermore, we identify nine new sets of twelve cubes, called Minimum Universal sets, from which all 30 cubes can be built.

Solution Numbers for Eight Blocks to Madness Puzzle

TL;DR

The work analyzes the target puzzle built from the MacMahon colored cubes, focusing on the non interior-face matching version. It introduces a graph model that encodes corner-numbers as vertices and cubes as edges, enabling exact counting of a collection’s solution number for a given target; exhaustive computation yields that at most solutions occur and characterizes when this maximum is achieved. Across all collections, the results show many collections solve no target, while a subset solves multiple targets (up to ), with collections achieving this maximum; the study also identifies new minimum universal sets of cubes and conjectures a complete structural description for all such sets, revealing a rich symmetry under . The combination of graph-theoretic modeling and algorithmic enumeration provides a comprehensive map of which eight-cube collections can realize which targets and how these properties scale as the pool of cubes grows. These insights advance understanding of MacMahon-based puzzles and offer computational tools and structural conjectures relevant to universal-set questions in colored-cube puzzles.

Abstract

The 30 MacMahon colored cubes have each face painted with one of six colors and every color appears on at least one face. One puzzle involving these cubes is to create a model with eight distinct MacMahon cubes to recreate a larger version with the external coloring of a specified target cube, also a MacMahon cube, and touching interior faces are the same color. J.H. Conway is credited with arranging the cubes in a tableau that gives a solution to this puzzle. In fact, the particular set of eight cubes that solves this puzzle can be arranged in exactly \textit{two} distinct ways to solve the puzzle. We study a less restrictive puzzle without requiring interior face matching. We describe solutions to the puzzle and the number of distinct solutions attainable for a collection of eight cubes. Additionally, given a collection of eight MacMahon cubes, we study the number of target cubes that can be built in a model. We calculate the distribution of the number of cubes that can be built over all collections of eight cubes (the maximum number is five) and provide a complete characterization of the collections that can build five distinct cubes. Furthermore, we identify nine new sets of twelve cubes, called Minimum Universal sets, from which all 30 cubes can be built.
Paper Structure (8 sections, 14 theorems, 12 figures, 3 tables)

This paper contains 8 sections, 14 theorems, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Main Results
  3. Background
  4. A Model for the Target Puzzle
  5. Solution Numbers and a Proof of Theorem 1
  6. All Eight Blocks to Madness puzzles and a Proof of Theorem 2
  7. Minimum Universal sets and Proof of Theorem 3
  8. Algorithm for Computations

Key Result

Theorem 1

Given a target cube, there are 133680 collections of eight cubes that can be used to build a $2 \times 2 \times 2$ model with the same external coloring as the target. For each of these collections we determine its solution number. In particular, 81 collections achieve the maximum solution number of

Figures (12)

  • Figure 1: The MacMahon colored cubes with names and tableau arrangement due to J.H.Conway.
  • Figure 2: The eight corner numbers of the cube Fb are $\{ 124, 146, 165, 152, 234, 253, 356, 364\}$.
  • Figure 3: The usable cubes for target Ba are highlighted with a shaded background in Conway's tableau. The color/font pairs $\textit{Ac, Cb}$; $\textbf{Ad, Db}$; $\texttt{ Ae, Eb}$; and Af, Fb indicate cubes pairs that share the same two corner numbers with the target Ba.
  • Figure 4: A geometric representation of the graph used to calculate solution numbers.
  • Figure 5: The graph used to determine the solution number for the target Ba and any collection of eight MacMahon cubes.
  • ...and 7 more figures

Theorems & Definitions (24)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Conjecture 4
  • Lemma 4.1
  • proof
  • Theorem 5
  • Theorem 6
  • Lemma 4.2
  • proof
  • ...and 14 more