Solution Numbers for Eight Blocks to Madness Puzzle

TL;DR

This paper investigates the solution counts and configurations for a puzzle involving MacMahon cubes, analyzing how collections of these cubes can be assembled into larger target cubes and identifying universal sets capable of constructing all 30 cube types.

Contribution

It provides a detailed enumeration of solutions for the 2x2x2 MacMahon cube puzzle, characterizes collections that can build multiple target cubes, and introduces nine new Minimum Universal sets.

Findings

Exactly two solutions for the specific 8-cube arrangement.

Maximum of five target cubes can be built from a collection of eight.

Identified nine new Minimum Universal sets of twelve cubes.

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.