The Invisible Solutions of the Rubik’s Cube

Abstract

SummaryThe center cubies on the Rubik’s Cube can change orientation when the puzzle is brought from one solved state to another. The set of all possible reorientations of the center cubies creates what we call the invisible solutions group. We investigate the size and structure of the invisible solutions group for the Rubik’s Cube, Rubik’s Revenge, and the Professor’s Cube.MSC: 20-01 AcknowledgmentsThe authors would like to thank the Adrian Tinsley Program for Undergraduate Research and Creative Scholarship for funding and the anonymous referees for the helpful comments.Notes1 Online version of the article contains color diagrams.Additional informationNotes on contributorsAllen CharestALLEN CHAREST received a Bachelors in Mathematics and Secondary Education from Bridgewater State University in 2019. He currently works as STEAM Math Teacher at Greater Lawrence Technical School and is fascinated by group theory and nature.Ben CotéBEN COTÉ (MR Author ID: 951394, ORCID 0000-0003-2844-1935) received a Ph.D. in Mathematics from the University of California, Santa Barbara in 2016. He currently teaches at Western Oregon University. When not investigating recreational mathematics, he enjoys camping and gardening with his wife Brittany and sons Levi and Oliver.Ward HeilmanWARD HEILMAN received a Ph.D. in Mathematics from Northeastern University. He has been at Bridgewater State University (Mass.) since 1996. He is active in social justice, and fascinated by axioms, cryptology, basketball, Huxley, Kerouac and most recently Thomas Paine.