The corner poset with an application to an n-dimensional hypercube stacking puzzle

For any dimension n >= 3, we establish the corner poset, a natural triangular poset structure on the corners of 2-color hypercubes. We use this poset to study a problem motivated by a classical cube stacking puzzle posed by Percy MacMahon as well as Eric Cross's more recent "Eight Blocks to Madness." We say that a hypercube is 2-color when each of its facets has one of two colors. Given an arbitrary multiset of 2-color unit n-dimensional hypercubes, we investigate when it is possible to find a submultiset of 2(n) hypercubes that can be arranged into a larger hypercube of side length 2 with monochrome facets. Through a careful analysis of the poset and its properties, we construct interesting puzzles, find and enumerate solutions, and study the maximum size, S(n), for a puzzle that does not contain a solution. Further, we find bounds on S(n), showing that it grows as Theta(n2(n)).

Title	The corner poset with an application to an n-dimensional hypercube stacking puzzle
Creator	Berkove, Ethan
Creator	Shettler, J.
Publisher	Australasian Journal of Combinatorics
Academic Department	Mathematics
Division	Natural Sciences
Organization	Lafayette College
Date Issued	2022
Date Available	2022-08-12
Type	Article
Language	English
Bibliographic Citation	Berkove, E. and J. Shettler (2022) "The corner poset with an application to an n-dimensional hypercube stacking puzzle." Australasian Journal of Combinatorics 84 (1): 86-110.
Standard Identifier	Handle 10385/w95052009
Permalink	http://hdl.handle.net/10385/w95052009
Rights Statement	Creative Commons - Attribution
Rights Holders	Berkove, Ethan
Rights Holders	Shettler, J.