Using a portion of the Continuum Hypothesis, we prove that there is a Menger-bounded (also called o-bounded) subgroup of the Baire group Z, whose square is not Menger-bounded. This settles a major open problem concerning boundedness notions for groups, and implies that Menger-bounded groups need not be Scheepers-bounded.

The Rubik’s Cube is a familiar toy that has embedded itself into popular culture since its invention in 1974 by Ernő Rubik. It is especially popular among mathematicians, for good reason. In this paper, we will investigate some of the interesting mathematical structure underlying the Rubik’s Cube, which was documented by David Joyner in his book, Adventures in Group Theory [Joy]. Let us conside...