The mathematics underlying Rubik's games I am interested in knowing (a little more) about the mathematics underlying some of Erno Rubik's games. I guess we all know his famous cube. I heard at some point that a solution was possible because it has the structure of a group. Is it true?
There is a similar game by Rubik, not so famous and easier to solve, consisting of a cylinder. I would like to know about the mathematical structure behind it. Is it also a group? Which is the reason for its being much easier than the cube, given that the parameters (colours, number of  slots or positions) are, I believe, the same?
Thanks in advance
 A: The main reason why group theory is useful for solving Rubik's cube is that the set of available moves is the same in each state of the cube.
There are other twisty puzzles where some moves are possible only in some state -- these are often referred to as "jumbling" puzzles by puzzle aficionados. As a simple example, imagine taking an ordinary Rubik's cube and gluing two of the little cubies in the solved state together, such that moves that take them apart are now disallowed. This resolves in a puzzle with fewer states than the original cube, which can nevertheless be harder to get a grip on because you can't mix and match combinations as freely as you usually can. And group theory is not particularly helpful for expressing this restriction.
(Actually, gluing one pair of cubies together is not so bad; you can keep that pair fixed in space and twist the cube around them, still yielding a group. And practically, many solution methods can fairly easily be adapted to deal with a single glued pair. Do it to two pairs of cubies, and trouble will begin to pile up, though).

I don't think "consisting of a cylinder" pinpoints the particular puzzle you're thinking about well enough to say something about it in particular. There are many different more-or-less cylindrical derivatives and variants of the cube.
