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Algorithms 2012, 5(2), 273-288; doi:10.3390/a5020273

Imaginary Cubes and Their Puzzles

Graduate School of Human and Environmental Studies, Kyoto University, Yoshida-Nihonmatsu, Sakyo-ku, 606-8501, Kyoto, Japan
Received: 16 November 2011 / Revised: 15 April 2012 / Accepted: 8 May 2012 / Published: 9 May 2012
(This article belongs to the Special Issue Puzzle/Game Algorithms)
Download PDF [4034 KB, uploaded 9 May 2012]


Imaginary cubes are three dimensional objects which have square silhouette projections in three orthogonal ways just as a cube has. In this paper, we study imaginary cubes and present assembly puzzles based on them. We show that there are 16 equivalence classes of minimal convex imaginary cubes, among whose representatives are a hexagonal bipyramid imaginary cube and a triangular antiprism imaginary cube. Our main puzzle is to put three of the former and six of the latter pieces into a cube-box with an edge length of twice the size of the original cube. Solutions of this puzzle are based on remarkable properties of these two imaginary cubes, in particular, the possibility of tiling 3D Euclidean space.
Keywords: imaginary cubes; hexagonal bipyramid; triangular antiprismoid; assembly puzzles; 3D tessellation imaginary cubes; hexagonal bipyramid; triangular antiprismoid; assembly puzzles; 3D tessellation
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Tsuiki, H. Imaginary Cubes and Their Puzzles. Algorithms 2012, 5, 273-288.

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